numerical analysis phd topics

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Ph.D. in Mathematics, Specializing in Applied Math

Table of contents, overview of applied mathematics at the courant institute.

• PhD Study in Applied Mathematics
• Applied math courses

Applied mathematics has long had a central role at the Courant Institute, and roughly half of all our PhD's in Mathematics are in some applied field. There are a large number of applied fields that are the subject of research. These include:

• Atmosphere and Ocean Science
• Biology, including biophysics, biological fluid dynamics, theoretical neuroscience, physiology, cellular biomechanics
• Computational Science, including computational fluid dynamics, adaptive mesh algorithms, analysis-based fast methods, computational electromagnetics, optimization, methods for stochastic systems.
• Data Science
• Financial Mathematics
• Fluid Dynamics, including geophysical flows, biophysical flows, fluid-structure interactions, complex fluids.
• Materials Science, including micromagnetics, surface growth, variational methods,
• Stochastic Processes, including statistical mechanics, Monte-Carlo methods, rare events, molecular dynamics

PhD study in Applied Mathematics

PhD training in applied mathematics at Courant focuses on a broad and deep mathematical background, techniques of applied mathematics, computational methods, and specific application areas. Descriptions of several applied-math graduate courses are given below.

Numerical analysis is the foundation of applied mathematics, and all PhD students in the field should take the Numerical Methods I and II classes in their first year, unless they have taken an equivalent two-semester PhD-level graduate course in numerical computing/analysis at another institution. Afterwards, students can take a number of more advanced and specialized courses, some of which are detailed below. Important theoretical foundations for applied math are covered in the following courses: (1) Linear Algebra I and II, (2) Intro to PDEs, (3) Methods of Applied Math, and (4) Applied Stochastic Analysis. It is advised that students take these courses in their first year or two.

A list of the current research interests of individual faculty is available on the Math research page.

Courses in Applied Mathematics

The following list is for AY 2023/2024:

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(MATH-GA.2701) Methods Of Applied Math

Fall 2023, Oliver Buhler

Description:  This is a first-year course for all incoming PhD and Masters students interested in pursuing research in applied mathematics. It provides a concise and self-contained introduction to advanced mathematical methods, especially in the asymptotic analysis of differential equations. Topics include scaling, perturbation methods, multi-scale asymptotics, transform methods, geometric wave theory, and calculus of variations.

Prerequisites : Elementary linear algebra, ordinary differential equations; at least an undergraduate course on partial differential equations is strongly recommended.

(MATH-GA.2704) Applied Stochastic Analysis

Spring 2024, Jonathan Weare

This is a graduate class that will introduce the major topics in stochastic analysis from an applied mathematics perspective.  Topics to be covered include Markov chains, stochastic processes, stochastic differential equations, numerical algorithms, and asymptotics. It will pay particular attention to the connection between stochastic processes and PDEs, as well as to physical principles and applications. The class will attempt to strike a balance between rigour and heuristic arguments: it will assume that students have some familiarity with measure theory and analysis and will make occasional reference to these, but many results will be derived through other arguments. The target audience is PhD students in applied mathematics, who need to become familiar with the tools or use them in their research.

Prerequisites: Basic Probability (or equivalent masters-level probability course), Linear Algebra (graduate course), and (beginning graduate-level) knowledge of ODEs, PDEs, and analysis.

(MATH-GA.2010/ CSCI-GA.2420) Numerical Methods I

• Fall 2023, Benjamin Peherstorfer

Description:   This course is part of a two-course series meant to introduce graduate students in mathematics to the fundamentals of numerical mathematics (but any Ph.D. student seriously interested in applied mathematics should take it). It will be a demanding course covering a broad range of topics. There will be extensive homework assignments involving a mix of theory and computational experiments, and an in-class final. Topics covered in the class include floating-point arithmetic, solving large linear systems, eigenvalue problems, interpolation and quadrature (approximation theory), nonlinear systems of equations, linear and nonlinear least squares, and nonlinear optimization, and iterative methods. This course will not cover differential equations, which form the core of the second part of this series, Numerical Methods II.

Prerequisites:   A good background in linear algebra, and some experience with writing computer programs (in MATLAB, Python or another language).

(MATH-GA.2020 / CSCI-GA.2421) Numerical Methods II

Spring 2024, Aleksandar Donev

This course (3pts) will cover fundamental methods that are essential for the numerical solution of differential equations. It is intended for students familiar with ODE and PDE and interested in numerical computing; computer programming assignments in MATLAB/Python will form an essential part of the course. The course will introduce students to numerical methods for (approximately in this order):

• The Fast Fourier Transform and pseudo-spectral methods for PDEs in periodic domains
• Ordinary differential equations, explicit and implicit Runge-Kutta and multistep methods, IMEX methods, exponential integrators, convergence and stability
• Finite difference/element, spectral, and integral equation methods for elliptic BVPs (Poisson)
• Finite difference/element methods for parabolic (diffusion/heat eq.) PDEs (diffusion/heat)
• Finite difference/volume methods for hyperbolic (advection and wave eqs.) PDEs (advection, wave if time permits).

Prerequisites

This course requires Numerical Methods I or equivalent graduate course in numerical analysis (as approved by instructor), preferably with a grade of B+ or higher.

( MATH-GA.2011 / CSCI-GA 2945) Computational Methods For PDE

Fall 2023, Aleksandar Donev & Georg Stadler

This course follows on Numerical Methods II and covers theoretical and practical aspects of advanced computational methods for the numerical solution of partial differential equations. The first part will focus on finite element methods (FEMs), and the second part on finite volume methods (FVMs) including discontinuous Galerkin (FE+FV) methods. In addition to setting up the numerical and functional analysis theory behind these methods, the course will also illustrate how these methods can be implemented and used in practice for solving partial differential equations in two and three dimensions. Example PDEs will include the Poisson equation, linear elasticity, advection-diffusion(-reaction) equations, the shallow-water equations, the incompressible Navier-Stokes equation, and others if time permits. Students will complete a final project that includes using, developing, and/or implementing state-of-the-art solvers.

In the Fall of 2023, Georg Stadler will teach the first half of this course and cover FEMs, and Aleks Donev will teach in the second half of the course and cover FVMs.

A graduate-level PDE course, Numerical Methods II (or equivalent, with approval of syllabus by instructor(s)), and programming experience.

• Elman, Silvester, and Wathen: Finite Elements and Fast Iterative Solvers , Oxford University Press, 2014.
• Farrell: Finite Element Methods for PDEs , lecture notes, 2021.
• Hundsdorfer & Verwer: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations , Springer-Verlag, 2003.
• Leveque: Finite Volume Methods for Hyperbolic Problems , Cambridge Press, 2002.

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( MATH-GA.2012 ) Immersed Boundary Method For Fluid-Structure Interaction

Not offered AY 23/24.

The immersed boundary (IB) method is a general framework for the computer simulation of flows with immersed elastic boundaries and/or complicated geometry.  It was originally developed to study the fluid dynamics of heart valves, and it has since been applied to a wide variety of problems in biofluid dynamics, such as wave propagation in the inner ear, blood clotting, swimming of creatures large and small, and the flight of insects.  Non-biological applications include sails, parachutes, flows of suspensions, and two-fluid or multifluid problems. Topics to be covered include: mathematical formulation of fluid-structure interaction in Eulerian and Lagrangian variables, with interaction equations involving the Dirac delta function; discretization of the structure, fluid, and interaction equations, including energy-based discretization of the structure equations, finite-difference discretization of the fluid equations, and IB delta functions with specified mathematical properties; a simple but effective method for adding mass to an immersed boundary; numerical simulation of rigid immersed structures or immersed structures with rigid parts; IB methods for immersed filaments with bend and twist; and a stochastic IB method for thermally fluctuating hydrodynamics within biological cells.  Some recent developments to be discussed include stability analysis of the IB method and a Fourier-Spectral IB method with improved boundary resolution.

Course requirements include homework assignments and a computing project, but no exam.  Students may collaborate on the homework and on the computing project, and are encouraged to present the results of their computing projects to the class.

Prerequisite:   Familiarity with numerical methods and fluid dynamics.

(MATH-GA.2012 / CSCI-GA.2945) :  High Performance Computing

Not offered AY 23/24

This class will be an introduction to the fundamentals of parallel scientific computing. We will establish a basic understanding of modern computer architectures (CPUs and accelerators, memory hierarchies, interconnects) and of parallel approaches to programming these machines (distributed vs. shared memory parallelism: MPI, OpenMP, OpenCL/CUDA). Issues such as load balancing, communication, and synchronization will be covered and illustrated in the context of parallel numerical algorithms. Since a prerequisite for good parallel performance is good serial performance, this aspect will also be addressed. Along the way you will be exposed to important tools for high performance computing such as debuggers, schedulers, visualization, and version control systems. This will be a hands-on class, with several parallel (and serial) computing assignments, in which you will explore material by yourself and try things out. There will be a larger final project at the end. You will learn some Unix in this course, if you don't know it already.

Prerequisites for the course are (serial) programming experience with C/C++ (I will use C in class) or Fortran, and some familiarity with numerical methods.

(MATH-GA.2011) Monte Carlo Methods

Fall 2023, Jonathan Weare and Jonathan Goodman

Topics : The theory and practice of Monte Carlo methods. Random number generators and direct sampling methods, visualization and error bars. Variance reduction methods, including multi-level methods and importance sampling. Markov chain Monte Carlo (MCMC), detailed balance, non-degeneracy and convergence theorems. Advanced MCMC, including Langevin and MALA, Hamiltonian, and affine invariant ensemble samplers. Theory and estimation of auto-correlation functions for MCMC error bars. Rare event methods including nested sampling, milestoning, and transition path sampling. Multi-step methods for integration including Wang Landau and related thermodynamic integration methods. Application to sampling problems in physical chemistry and statistical physics and to Bayesian statistics.

Required prerequisites:

• A good probability course at the level of Theory of Probability (undergrad) or Fundamentals of Probability (masters)
• Linear algebra: Factorizations (especially Cholesky), subspaces, solvability conditions, symmetric and non-symmetric eigenvalue problem and applications
• Working knowledge of a programming language such as Python, Matlab, C++, Fortran, etc.
• Familiarity with numerical computing at the level of Scientific Computing (masters)

Desirable/suggested prerequisites:

• Numerical methods for ODE
• Applied Stochastic Analysis
• Familiarity with an application area, either basic statistical mechanics (Gibbs Boltzmann distribution), or Bayesian statistics

(MATH-GA.2012 / CSCI-GA.2945) Convex & Non Smooth Optimization

Spring 2024, Michael Overton

Convex optimization problems have many important properties, including a powerful duality theory and the property that any local minimum is also a global minimum. Nonsmooth optimization refers to minimization of functions that are not necessarily convex, usually locally Lipschitz, and typically not differentiable at their minimizers. Topics in convex optimization that will be covered include duality, CVX ("disciplined convex programming"), gradient and Newton methods, Nesterov's optimal gradient method, the alternating direction method of multipliers, the primal barrier method, primal-dual interior-point methods for linear and semidefinite programs. Topics in nonsmooth optimization that will be covered include subgradients and subdifferentials, Clarke regularity, and algorithms, including gradient sampling and BFGS, for nonsmooth, nonconvex optimization. Homework will be assigned, both mathematical and computational. Students may submit a final project on a pre-approved topic or take a written final exam.

Prerequisites: Undergraduate linear algebra and multivariable calculus

Q1: What is the difference between the Scientific Computing class and the Numerical Methods two-semester sequence?

The Scientific Computing class (MATH-GA.2043, fall) is a one-semester masters-level graduate class meant for graduate or advanced undergraduate students that wish to learn the basics of computational mathematics. This class requires a working knowledge of (abstract) linear algebra (at least at the masters level), some prior programming experience in Matlab, python+numpy, Julia, or a compiled programming language such as C++ or Fortran, and working knowledge of ODEs (e.g., an undergrad class in ODEs). It only briefly mentions numerical methods for PDEs at the very end, if time allows.

The Numerical Methods I (fall) and Numerical Methods II (spring) two-semester sequence is a Ph.D.-level advanced class on numerical methods, meant for PhD students in the field of applied math, masters students in the SciComp program , or other masters or advanced undergraduate students that have already taken at least one class in numerical analysis/methods. It is intended that these two courses be taken one after the other, not in isolation . While it is possible to take just Numerical Methods I, it is instead strongly recommended to take the Scientific Computing class (fall) instead. Numerical Methods II requires part I, and at least an undergraduate class in ODEs, and also in PDEs. Students without a background in PDEs should not take Numerical Methods II; for exceptions contact Aleks Donev with a detailed justification.

The advanced topics class on Computational Methods for PDEs follows on and requires having taken NumMeth II or an equivalent graduate-level course at another institution (contact Aleks Donev with a syllabus from that course for an evaluation), and can be thought of as Numerical Methods III.

Q2: How should I choose a first graduate course in numerical analysis/methods?

• If you are an undergraduate student interested in applied math graduate classes, you should take the undergraduate Numerical Analysis course (MATH-UA.0252) first, or email the syllabus for the equivalent of a full-semester equivalent class taken elsewhere to Aleks Donev for an evaluation.
• Take the Scientific Computing class (fall), or
• Take both Numerical Methods I (fall) and II (spring), see Q1 for details. This is required of masters students in the SciComp program .

Numerical Analysis Research Topics Ideas [MS PhD]

List of Research Topics and Ideas of Numerical Analysis for MS and Ph.D. Thesis.

• A comprehensive numerical analysis of heat and mass transfer phenomenons during cavitation sono-process
• Numerical analysis of mechanical behaviour of lattice and porous structures
• ATHENA: Advanced Techniques for High dimensional parameter spaces to Enhance Numerical Analysis
• Experimental and numerical analysis on coupled hygro-thermo-chemo-mechanical effect in early-age concrete
• Numerical analysis of natural rubber bearing equipped with steel and shape memory alloys dampers
• A numerical analysis of the effects of nanofluid and porous media utilization on the performance of parabolic trough solar collectors
• Numerical analysis of cross-flow plate type indirect evaporative cooler: Modeling and parametric analysis
• Numerical analysis of the effect of fire source configuration on fire-wind enhancement
• Numerical analysis
• An influence mechanism of shale barrier on heavy oil recovery using SAGD based on theoretical and numerical analysis
• Numerical analysis of the stability of arched sill mats made of cemented backfill
• Theoretical and numerical analysis of regular one-side oscillations in a single pendulum system driven by a magnetic field
• Numerical analysis of performance and emission behavior of CI engine fueled with microalgae biodiesel blend
• Numerical analysis of ultrasonic multiple scattering for fine dust number density estimation
• Numerical analysis of fractional Volterra integral equations via Bernstein approximation method
• Numerical analysis of a highly sensitive surface plasmon resonance sensor for sars-cov-2 detection
• Numerical analysis of enhanced conductive deep borehole heat exchangers
• Numerical analysis of the robustness of clinching process considering the pre-forming of the parts
• Experimental study and numerical analysis on seismic performance of FRP confined high-strength rectangular concrete-filled steel tube columns
• Numerical analysis of twin-precipitate interactions in magnesium alloys
• A two-step procedure for the numerical analysis of curved masonry structures
• Numerical analysis of subsurface deformation induced by groundwater level changes in the Bangkok aquifer system
• Numerical analysis and applications of explicit high order maximum principle preserving integrating factor Runge-Kutta schemes for Allen-Cahn equation
• Numerical analysis of the cyclic mechanical damage of Li-ion battery electrode and experimental validation
• Experimental and numerical analysis of circular concrete-filled double steel tubular stub columns with inner square hollow section
• Numerical analysis of sand erosion for a pelton turbine injector at high concentration
• Deviations in yield and ultimate tensile strength estimation with the Small Punch Test: Numerical analysis of pre-straining and Bauschinger effect influence
• Numerical analysis applied to the study of soil stress and compaction due to mechanised sugarcane harvest
• Numerical analysis of combined electroosmotic-pressure driven flow of a viscoelastic fluid over high zeta potential modulated surfaces
• Numerical analysis of the flow-induced vibrations in the laminar wake behind a blunt body with rear flexible cavities
• Regular perturbation solution of Couette flow (non-Newtonian) between two parallel porous plates: a numerical analysis with irreversibility
• Numerical Analysis of the Coupling between Heat Transfer and Deformation in Rotary Air Preheater
• Numerical analysis of tiny-focal-spot generation by focusing linearly, circularly, and radially polarized beams through a micro/nanoparticle
• An investigation on viscoelastic characteristics of 3D-printed FDM components using RVE numerical analysis
• Experimental and static numerical analysis on bumper beam to be proposed for Indian passenger car
• Numerical analysis of special concentric braced frames using experimentally-validated fatigue and fracture model under short and long duration earthquakes
• Theoretical and numerical analysis of the evaporation of mono-and multicomponent single fuel droplets
• Investigation of numerical analysis and seismic performance of underground loess cave with traditional dwellings
• Unsteady-state numerical analysis of advanced Savonius wind turbine
• Numerical analysis of dynamic compaction using FEM-SPH coupling method
• Numerical Analysis of Macro-fiber Composite Structures
• Numerical analysis of a monotube mixed mode magnetorheological damper by using a new rheological approach in CFD
• Laboratory and numerical analysis of geogrid encased stone columns
• Numerical analysis of thermal-hydraulic characteristics of steam-air condensation in vertical sinusoidal corrugated tubes
• Numerical analysis of dynamic characteristics of in-arm type hydropneumatic suspension unit
• Numerical Analysis of the Hydrogen Dispersion Behavior in Different Directions in a Naturally Ventilated Space
• Material non-linearity in the numerical analysis of SLJ bonded with ductile adhesives: A meshless approach
• Numerical analysis of the shear behavior of FRP-strengthened continuous RC beams having web openings
• Numerical Analysis of Phosphorus Concentration Distribution in a Silicon Crystal during Directional Solidification Process
• An improved numerical analysis of the transient oil de-congealing process in a heat exchanger under low temperature conditions
• Numerical Analysis and Prediction of FLD For Al Alloy-6063
• Improvement of ammonia mixing in an industrial scale selective catalytic reduction De-NOx system of a coal-fired power plant: A numerical analysis
• Experimental and Numerical Analysis of Water Hammer Phenomenon in Pipeline with Fiber Optic Cable
• Numerical analysis of the effect of air pressure and oil flow rate on droplet size and tool temperature in MQL machining
• Numerical Analysis of Viscoelastic Rotating Beam with Variable Fractional Order Model Using Shifted Bernstein–Legendre Polynomial Collocation Algorithm
• Numerical linear algebra and optimization
• Numerical Analysis of Serrated Chip Formation Mechanism with Johnson-Cook Parameters in Micro-Cutting of Ti6Al4V
• Numerical Analysis of the Mechanical Response of Anchored Wire Meshes
• Numerical Analysis for the Assessment of Factors Influencing the Breakdown of Swirl Flow in a Cylinder Driven by a Rotating End Wall
• Numerical analysis of the ultimate compressibility of concrete with indirect reinforcement for plotting a stress-strain diagram
• Numerical analysis and prediction of Aero-optical effects
• Numerical analysis for stochastic time-space fractional diffusion equation driven by fractional Gaussion noise
• Numerical analysis and performance improvement of nanostructured Cu2O/TiO2pn heterojunction solar cells using SCAPS
• Numerical Analysis of Mechanical Characteristics of Joint Structure of Steel Pipe Sheet Pile Foundation
• Numerical Analysis of Rotor Dynamics of Dredge Pump Shafting
• A numerical analysis of convection heat transfer and friction factor for oscillating corrugated channel flows
• Numerical Analysis on Aerodynamic Characteristics of Slender Body with Asymmetric Double Protuberance
• Mixed Finite Element-Second Order Upwind Fractional Step Difference Scheme of Darcy–Forchheimer Miscible Displacement and Its Numerical Analysis
• Numerical Analysis of JAXA Standard Model in High Lift Configuration
• Numerical Analysis of Shock Speed Attenuation in Expansion Tube
• Experimental and Numerical Analysis of the Mechanical Properties of a Pretreated Shape Memory Alloy Wire in a Self-Centering Steel Brace
• Numerical Analysis of Shallow Foundations Considering Hydraulic Hysteresis and Deformation Dependent Soil-Water Retention
• Experimental and Numerical Analysis of Laterally Loaded Pile Subjected to Earthquake Loading
• Numerical analysis of static behavior of caisson-type quay wall deepened by grouting rubble-mound
• Experimental and Numerical Analysis of Steel Beam-Column Connections
• Numerical Analysis of Shear and Particle Crushing Characteristics in Ring Shear System Using the PFC2D. Materials 2021, 14, 229
• Numerical Analysis on a Selection of Horn Material for the Design of Cylindrical Horn in Ultrasonic Machining
• A Simplified Francis Turbine for Micro Hydro Application: Design and Numerical Analysis
• Numerical Analysis of Nonlinear Dynamics Based on Spin-VCSELs with Optical Feedback. Photonics 2021, 8, 10
• Numerical analysis of the flow behavior in the throat section of an experimental conical nozzle
• Numerical Analysis OF PCM Within a Square Enclosure Having Different Wall Heating Conditions
• Numerical Analysis of Dynamic Hysteresis in Tape Springs for Space Applications
• Numerical Analysis of Autofrettaged High-Pressure Aluminium Cylinder
• Numerical analysis of steel columns subject to eccentric loadings
• Numerical Analysis of Flow and Radiative Transfer with Excitation Nonequilibrium in an Arc-jet Facility
• Numerical Analysis of Scramjet Intake with Spoon-shaped Isolator
• Numerical Analysis on Gas Turbine Blade of a Nickel-Based Alloy with Titanium Alloy
• Experimental and Numerical Analysis on the Mechanical Behaviour of Box Beam Cold-Formed Steel Built-up C-Sections
• Numerical Analysis of Segmental Tunnel Linings employing a Hybrid Modeling Approach
• Numerical Analysis of Long-span Floor Vibration Due to Crowd Synchronized Walking
• Numerical analysis of the carbon-containing pellet direct reduction process with central heat transfer enhancement
• Numerical analysis of coupled systems of ODEs and applications to enzymatic competitive inhibition by product
• Mathematical model of SIR epidemic system (COVID-19) with fractional derivative: stability and numerical analysis
• Numerical analysis of Vertical Axis Wind Turbine performance at varied Angle-of-Attack
• Numerical analysis of fretting wear in lateral contact of sphere/sphere
• Hybrid Algorithms for Numerical Solution of Optoelectronics Applications
• Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games I: The Ergodic Case
• Numerical–Experimental Analysis of Polyethylene Pipe Deformation at Different Load Values
• Bending behavior of octet-truss lattice structures: Modelling options, numerical characterization and experimental validation
• Numerical Simulation of Photovoltaic Cell
• Numerical Solving for Nonlinear Problems Using Iterative Techniques
• An introduction to numerical methods and analysis
• On Solving Nonlinear Equation Via Numerical Analysis for Photovoltaic Cell (HM-DM)
• Application of Numerical Analysis for Solving Nonlinear Equation (AHM-DM)
• On the Globalization of ASPIN Employing Trust-Region Control Strategies–Convergence Analysis and Numerical Examples
• Numerical modeling of two microwave sensors for biomedical applications
• Numerical modeling of solid-liquid phase change under the influence an external electric field
• The damage-failure criteria for numerical stability analysis of underground excavations: A review
• Numerical and chemical kinetic analysis to evaluate the effect of steam dilution and pressure on combustion of n-dodecane in a swirling flow environment
• Numerical simulation and stability analysis for the fractional-order dynamics of COVID-19
• Scenarios in the experimental response of a vibro-impact single-degree-of-freedom system and numerical simulations
• Experimental and Numerical Investigation of One-Dimensional Infiltration into Unsaturated Soil
• Numerical Modeling of Heat and Mass Transfer during Cryopreservation Using Interval Analysis
• Numerical techniques for infrared spectra analysis of organic and inorganic volatile compounds for biomedical applications
• Two-Dimensional Numerical Model for Stability Analysis of Tunnel Face Based on Particle Flow Code
• Numerical simulation and CFD-based correlations for artificially roughened solar air heater
• Numerical study on the thermo-hydraulic performance analysis of fly ash nanofluid
• Numerical Modeling and Stability Analysis of Surrounding Rock of Yuanjue Cave
• The theoretical performance analysis and numerical simulation of the cylindrical vane pump
• Investigation of novel multi-layer sandwich panels under quasi-static indentation loading using experimental and numerical analyses
• Numerical approximations for a fully fractional Allen–Cahn equation
• Quasi-static numerical modeling of miniature RF circuits based on lumped equivalent circuits
• A comparative Analysis of PV Cell Mathematical Model
• Experimental, analytical and numerical performance of RC beams with V-shaped reinforcement
• Shape and topology optimization. to appear in Handbook of Numerical Analysis, 22
• Geotechnical Properties of Stabilized Sedimentary Formation for Numerical Analysis , 14 Chapters of Research Work
• Numerical analysis of a distributed control problem with a stochastic parabolic equation
• Numerical Analysis of Reinforced Exterior Concrete Beam-Column Joints Retrofitted Using FRP under Cyclic Loads
• Experimental and Numerical Analysis of Energy Absorption of Hollow and Foam filled thick-wall Aluminum Tubes Considering Different Damage Models
• The dynamical model for COVID-19 with asymptotic analysis and numerical implementations
• Hyperbolic models for the spread of epidemics on networks: kinetic description and numerical methods
• A novel approach for the numerical approximation to the solution of singularly perturbed differential-difference equations with small shifts
• Hybrid bistable composite laminates for structural assemblies: A numerical and experimental study
• The relative salience of numerical and non-numerical dimensions shifts over development: A re-analysis of
• Numerical analyses on geogrid-reinforced cushion in pile-supported composite foundation
• On the numerical solution of stochastic oscillators driven by time-varying and random forces
• A New Blast Absorbing Sandwich Panel with Unconnected Corrugated Layers—Numerical Study
• Combined numerical and experimental study on the use of Gurney flaps for the performance enhancement of NACA0021 airfoil in static and dynamic conditions
• Supporting Data: Numerical Simulation of Sleeve Fracturing for In-Situ Stress Measurement using Cohesive Elements
• Numerical modeling of post-flood water flow in pavement structures
• Fatigue Analysis of Actuators with Teflon Impregnated Coating—Challenges in Numerical Simulation
• Seismic assessment of wind turbines: How crucial is rotor-nacelle-assembly numerical modeling?
• A Numerical Study on Interference Effects of Closely Spaced Strip Footings on Cohesionless Soils
• Numerical Investigation of a Phase Change Material Including Natural Convection Effects
• CSCI, MATH 6820 Numerical Solution of Ordinary Differential Equations Lecture Notes: Spring 1999
• A New Technique for Solar Cell Parameters Estimation of The Single-Diode Model
• Analysis and numerical simulation of fractional Biswas–Milovic model
• Thermal and Hydrodynamic Phenomena in the Stagnation Zone—Impact of the Inlet Turbulence Characteristics on the Numerical Analyses
• Elasto-Plastic Numerical Analyses for Predicting Cave-Ins of Tunnels and Caverns
• … pass location on thermo-fluidic characteristic on the novel air-cooled branched wavy minichannel heat sink: A comprehensive numerical and experimental analysis
• Numerical and experimental analysis of the structural performance of AM components built by fused filament fabrication
• Experimental and numerical simulation of the piston engine fueled with alternative fuel blends: CFD approach
• Comparison between experimental digital image processing and numerical methods for stress analysis in dental implants with different restorative materials
• Numerical investigation for heat transfer enhancement of nanofluid in the solar flat plate collector with insertion of multi-channel twisted tape
• A numerical study on energy absorption capability of lateral corrugated composite tube under axial crushing
• Parameters Estimation of Photovoltaic Model Using Nonlinear Algorithms
• The Influence of the Mesh Element Size on Critical Bending Speeds of a Rotor in the Finite Element Analysis
• Numerical simulations of Pelton turbine flow to predict large head variation influence
• Use of Vertical and Inclined Walls to Mitigate the Interaction of Reverse Faulting and Shallow Foundations: Centrifuge Tests and Numerical Simulation
• Numerical simulation and experimental study on the drilling process of 7075-t6 aerospace aluminum alloy
• A numerical study of field strength and clay morphology impact on NMR transverse relaxation in sandstones
• NUMERICAL SIMULATIONS FOR SHALLOW WATER FLOWS OVER ERODIBLE BEDS BY CENTRAL DG METHODS.
• Numerical modeling and parametric study of piled rafts foundations
• Direct numerical simulation of electroconvection with thin Debye layer matching canonical experiments
• A novel numerical approach to time-fractional parabolic equations with nonsmooth solutions
• Experimental and numerical investigation of the thermo-mechanical behaviour of an energy sheet pile wall
• Experimental and numerical investigation on dynamic responses of the umbrella membrane structure excited by heavy rainfall
• A novel Covid-19 model with fractional differential operators with singular and non-singular kernels: Analysis and numerical scheme based on Newton …
• Numerical investigation of sloshing in tank with horivert baffles under resonant excitation using CFD code
• Parametric investigations of magnetic nanoparticles hyperthermia in ferrofluid using finite element analysis
• Experimental and numerical investigations on seismic performance of RC bridge piers considering buckling and low-cycle fatigue of high-strength steel bars
• Numerical assessment of effective width in steel-concrete composite box girder bridges
• Structure-preserving, energy stable numerical schemes for a liquid thin film coarsening model
• Stability analysis of thin-walled composite plate in unsymmetrical configuration subjected to axial load
• Numerical Simulation of Swelling in Tunnels
• Numerical study on the behaviour of AL2O3/water nanofluid at multiple flows and concentration
• Weighted finite element method for elasticity problem with a crack
• The impact of a multilevel protection column on the propagation of a water wave and pressure distribution during a dam break: Numerical simulation
• Analysis and numerical simulation of novel coronavirus (COVID-19) model with Mittag-Leffler Kernel
• Numerical study of mist film cooling on a flat plate with various numbers of deposition
• Effect of Reinforcing Steel on the Impact Resistance of Reinforced Concrete Panel Subjected to Hard-Projectile Impact
• Comparative numerical and experimental analysis of thermal and hydraulic performance of improved plate fin heat sinks
• Study of the Effect of Welding Current on Heat Transfer and Melt Pool Geometry on Mild Steel Specimen Through Finite Element Analysis
• Sensitivity analysis of design parameters for erythritol melting in a horizontal shell and multi-finned tube system: Numerical investigation
• Numerical investigation on a dual loop EGR optimization of a light duty diesel engine based on water condensation analysis
• Ducted Fuel Injection: Experimental and numerical investigation on fuel spray characteristics, air/fuel mixing and soot mitigation potential
• Numerical simulation and experimental investigation on laser beam welding of Inconel 625
• Numerical limit analysis-based modelling of masonry structures subjected to large displacements
• Numerical performance of blind-bolted demountable square CFST K-joints
• Analysis and development of novel data-driven drag models based on direct numerical simulations of fluidized beds
• Experimental and numerical fretting fatigue using a new test fixture
• Numerical treatment for dynamics of second law analysis and magnetic induction effects on ciliary induced peristaltic transport of hybrid nanomaterial
• Experimental and numerical investigation of Low-Velocity impact on steel wire reinforced foam Core/Composite skin sandwich panels
• Laboratory and numerical experimentation for masonry in compression
• A radiatively induced neutrino mass model with hidden local U (1) and LFV processes l i? l j ?, µ? eZ’ and µe? ee
• Influence of lowering groundwater level on the behavior of pile in soft soil
• Crushing of Single-Walled Corrugated Board during Converting: Experimental and Numerical Study
• Numerical optimization of key design parameters of a thermoelectric microfluidic sensor for ultrasensitive detection of biochemical analytes
• Numerical modeling and analysis of the effect of pressure on the performance of an alkaline water electrolysis system
• Numerical Simulation of the Daikai Station Subway Structure Collapse due to Sudden Uplift during Earthquake
• Advanced Numerical Modelling of Geogrids and Steel Wire Meshes
• Explicit Numerical Model of Solar Cells to Determine Current and Voltage
• Numerical Dynamic Programming for Continuous States
• A Numerical-Analytical Method for Dynamic Analysis of Piles in Non-homogeneous Transversely Isotropic Media
• Failure mode analysis of post-seismic rockfall in shattered mountains exemplified by detailed investigation and numerical modelling
• Estimating water balance components in irrigated agriculture using a combined approach of soil moisture and energy balance monitoring, and numerical modelling
• Discovery of dynamics using linear multistep methods
• A numerical study of arsenic contamination at the Bagnoli bay seabed by a semi-anthropogenic source. Analysis of current regime
• Model uncertainty in non-linear numerical analyses of slender reinforced concrete members
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• Numerical modeling of axially loaded circular concrete-filled double-skin steel tubular short columns incorporating a new concrete confinement model
• Structural Safety and Stability of the Bridge on the Paraopeba River in Moeda, Minas Gerais, Brazil: Case Study
• Modeling and analysis of the coupling in discrete fracture matrix models
• An inclusive analysis of inductive dielectric and/or metallic discontinuities in a rectangular waveguide
• Numerical study on blast responses of rubberized concrete slabs using the Karagozian and Case concrete model
• A decision tree lifted domain for analyzing program families with numerical features
• Numerical simulation of nonlinear thermal radiation on the 3D flow of a couple stress Casson nanofluid due to a stretching sheet
• Experimental and numerical investigation on compressive and flexural behavior of structural steel tubular beams strengthened with AFRP composites
• Horizontal Displacement of Urban Deep Excavated Walls Supported by Multistrands Anchors, Steel Piles, and In Situ Concrete Piles: Case Study
• A study on transmission dynamics of the emerging Candida Auris infections in Intensive Care Units: Optimal control analysis and numerical computations
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• A combined convection carreau–yasuda nanofluid model over a convective heated surface near a stagnation point: a numerical study
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• Analysis of the triggering mechanism of landslide in the village Podi, Montenegro
• Numerical prediction of temperature distribution and residual stresses on plasma arc welded thin titanium sheets
• Significance of Interface Modeling in the Analysis of Laterally Loaded Deep Foundations
• Asymptotically compatible reproducing kernel collocation and meshfree integration for nonlocal diffusion
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• A study on rock mass crack propagation and coalescence simulation based on improved numerical manifold method (NMM)
• Numerical investigation of the recent Chenhecun landslide (Gansu, China) using the discrete element method
• Closure to “Experimental Evaluation and Numerical Modeling of Wide-Flange Steel Columns Subjected to Constant and Variable Axial Load Coupled with Lateral Drift …
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• Bearing Capacity of Soft Clays Improved by Stone Columns: A Parametric Analysis
• On strong stability of explicit Runge–Kutta methods for nonlinear semibounded operators
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• Numerical and Experimental Analysis of Hardening Distortions of Drawpieces Produced in Hot Stamping Process. Metals 2021, 11, 457
• Three-Dimensional Modeling of Laterally Loaded Pile Embedded in Unsaturated Sandy Soil
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• Low-velocity drop weight impact behavior of Twaron® fabric investigated using experimental and numerical simulations
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• The effect of a hot-wire in the tandem GMAW process ascertained by developing a multiphysics simulation model
• Numerical Study on CO2 Injection in Indian Geothermal Reservoirs Using COMSOL Multiphysics 5.2a
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• MAP123-EPF: A mechanistic-based data-driven approach for numerical elastoplastic modeling at finite strain
• On the Construction and Analysis of Finite Volume Element Schemes with Optimal L 2 Convergence Rate
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• Theoretical, Numerical, and Experimental Study on an Unsteady Venturi Flowmeter for Incompressible Flows
• Numerical Modelling and Thermohydraulic Analysis of Circular Pipe Having Internal Vortex Generators
• Variational analysis of the discontinuous Galerkin time-stepping method for parabolic equations
• New Exact Operational Shifted Pell Matrices and Their Application in Astrophysics
• Laboratory and Numerical Studies on the Performance of Geocell Reinforced Base Layer Overlying Soft Subgrade
• EXPERIMENTAL AND NUMERICAL STUDY OF INSERTING AN INTERNAL HOLLOW CORE TO FINNED HELICAL COIL TUBE-SHELL HEAT EXCHANGER
• A smoothed iFEM approach for efficient shape-sensing applications: Numerical and experimental validation on composite structures
• New analysis and application of fractional order Schrödinger equation using with Atangana–Batogna numerical scheme
• Experimental and Numerical Study of Low-Velocity Impact and Tensile after Impact for CFRP Laminates Single-Lap Joints Adhesively Bonded Structure
• Basalt Fibre Reinforcement of Bent Heterogeneous Glued Laminated Beams
• Mid-infrared supercontinuum generation in a low-loss germanium-on-silicon waveguide
• Mapping sequence to feature vector using numerical representation of codons targeted to amino acids for alignment-free sequence analysis
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• Finite Element Analysis of Ballistic Impact on Monolithic and Multi-layered Target Plate with and Without Air Gap
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• Numerical investigation of free convection through a horizontal open-ended axisymmetric cavity
• Spectral monic chebyshev approximation for higher order differential equations
• Dynamic Characteristics of Woven Flax/Epoxy Laminated Composite Plate. Polymers 2021, 13, 209
• Numerical Solution of Interval Volterra-Fredholm-Hammerstein Integral Equations via Interval Legendre Wavelets? Method?
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• Numerical investigation on a multi-channel micro combustor fueled with hydrogen for a micro-thermophotovoltaic system
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• Blown Wing Aerodynamic Coefficient Predictions Using Traditional Machine Learning and Data Science Approaches
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• Numerical Investigation of Heat Transfer in Aircraft Engine Blade Using k-? and SST k- ? Model
• Advanced complex analysis of the thermal softening of nitrided layers in tools during hot die forging
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Computational Science & Numerical Analysis

Computational science is a key area related to physical mathematics. The problems of interest in physical mathematics often require computations for their resolution. Conversely, the development of efficient computational algorithms often requires an understanding of the basic properties of the solutions to the equations to be solved numerically. For example, the development of methods for the solution of hyperbolic equations (e.g. shock capturing methods in, say, gas-dynamics) has been characterized by a very close interaction between theoretical, computational, experimental scientists, and engineers.

Department Members in This Field

• Laurent Demanet Applied analysis, Scientific Computing
• Alan Edelman Parallel Computing, Numerical Linear Algebra, Random Matrices
• Steven Johnson Waves, PDEs, Scientific Computing
• Pablo Parrilo Optimization, Control Theory, Computational Algebraic Geometry, Applied Mathematics
• Gilbert Strang Numerical Analysis, Partial Differential Equations
• John Urschel Matrix Analysis, Numerical Linear Algebra, Spectral Graph Theory

Instructors & Postdocs

• Pengning Chao Scientific computing, Nanophotonics, Inverse problems, Fundamental limits
• Ziang Chen applied analysis, applied probability, statistics, optimization, machine learning
• Andrew Horning Numerical Analysis, Scientific Computing, Large-Scale And Infinite-Dimensional Spectral Problems
• Adam Kay Hydrodynamic Quantum Analogues

Researchers & Visitors

• Keaton Burns PDEs, Spectral Methods, Fluid Dynamics
• Raphaël Pestourie Surrogate Models, AI, Electromagnetic Design, End-to-end Optimization, Inverse Design

Graduate Students*

• Rodrigo Arrieta Candia Numerical methods for PDEs, Numerical Analysis, Scientific Computing, Computational Electromagnetism
• Mo Chen Optimization, Scientific Computing
• Max Daniels High-dimensional statistics, optimization, sampling algorithms, machine learning
• Sarah Greer Imaging, inverse problems, signal processing
• George Stepaniants Statistical Learning of Differential Equations, Optimal Transport in Biology
• Songchen Tan computational science, numerical analysis, differentiable programming

*Only a partial list of graduate students

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​Numerical Analysis and Scientific Computation is one of the largest and most active groups within the Department, with most of the major research areas represented. Our group is interested in a wide range of topics, from the study of fundamental, theoretical issues in numerical methods; to algorithm development; to the numerical solution of large-scale problems in fluid mechanics, solid mechanics, electromagnetism and materials science.

​Departmental Colloquia Calendar

Applied Mathematics and Mathematical Medicine and Biology Seminar

Numerical Analysis and PDE Seminar​

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Department of Mathematics

• Numerical analysis and scientific computing

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Areas of expertise

Research seminars, phd research opportunities, research publications.

Our researchers develop and analyse algorithms that compute numerical approximations and apply them to real-world problems.

Numerical analysis is the branch of rigorous mathematics that concerns the development and analysis of methods to compute numerical approximations to the solutions of mathematical problems. It is a broadly based discipline that sits at the interface between mathematical analysis and scientific computing. Scientific computing describes the use of numerical simulation to study natural phenomena, complementing the more traditional experimental and theoretical approaches. Another broad discipline, it spans all the sciences with strong links to numerical analysis, computer science and software engineering. Our work covers the breadth of these disciplines from fundamental theory and algorithm development through to implementation in open source software. Our researchers have expertise in the following areas.

Approximation theory

Approximation theory is a key component of contemporary algorithms used in computational science and engineering.

Mathematical foundations of data science and AI

Data science refers to the study of theory, methods, algorithms, and applications focused around data, and is a highly interdisciplinary subject which relies on solid foundations of mathematical and statistical fundamentals.

Numerical linear algebra

Numerical linear algebra is at the heart of computational algorithms used in science and engineering, and in industry.

Scientific computing

Scientific computing is the study of the techniques that underpin discipline-specific fields of computational science.

Uncertainty quantification

Uncertainty quantification is a modern inter-disciplinary science that cuts across traditional research groups and combines statistics, numerical analysis and computational applied mathematics. 

Research seminars on topics associated with numerical analysis and scientific computing take place regularly in the following series:

• Applied maths (informal)

We welcome applications for PhD study in all areas of mathematics in the life sciences. PhD enquiries related to this theme can be directed to  Sean Holman .

To discover the PhD opportunities available in the Department of Mathematics, explore our  Postgraduate research in mathematics .

Our staff, students and postgraduate researchers have access to a fantastic range of facilities across the University.

The Department's recent publications in the University's database.

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Discover more about postgraduate research

PhD Numerical Analysis / Overview

Year of entry: 2024

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The standard academic entry requirement for this PhD is an upper second-class (2:1) honours degree in a discipline directly relevant to the PhD (or international equivalent) OR any upper-second class (2:1) honours degree and a Master’s degree at merit in a discipline directly relevant to the PhD (or international equivalent).

Other combinations of qualifications and research or work experience may also be considered. Please contact the admissions team to check.

Full entry requirements

Apply online

In your application you’ll need to include:

• The name of this programme
• Your research project title (i.e. the advertised project name or proposed project name) or area of research
• Your proposed supervisor’s name
• If you already have funding or you wish to be considered for any of the available funding
• A supporting statement (see 'Advice to Applicants for what to include)
• Details of your previous university level study
• Names and contact details of your two referees.

Programme options

Programme description.

The Department of Mathematics has an outstanding research reputation. The research facilities include one of the finest libraries in the country, the John Rylands University Library. This library has recently made a very large commitment of resources to providing comprehensive online facilities for the free use of the University's research community. Postgraduate students in the Department benefit from direct access to all the Library electronic resources from their offices.

Many research seminars are held in the Department on a weekly basis and allow staff and research students to stay in touch with the latest developments in their fields. The Department is one of the lead partners in the MAGIC project and research students can attend any of the postgraduate courses offered by the MAGIC consortium.

For entry in the academic year beginning September 2024, the tuition fees are as follows:

• PhD (full-time) UK students (per annum): Band A £4,786; Band B £7,000; Band C £10,000; Band D £14,500; Band E £24,500 International, including EU, students (per annum): Band A £28,000; Band B £30,000; Band C £35,500; Band D £43,000; Band E £57,000
• PhD (part-time) UK students (per annum): Band A £2393; Band B £3,500; Band C £5,000; Band D £7,250; Band E 12,250 International, including EU, students (per annum): Band A £14,000; Band B £15,000; Band C £17,750; Band D £21,500; Band E £28,500

Further information for EU students can be found on our dedicated EU page.

The programme fee will vary depending on the cost of running the project. Fees quoted are fully inclusive and, therefore, you will not be required to pay any additional bench fees or administration costs.

All fees for entry will be subject to yearly review and incremental rises per annum are also likely over the duration of the course for Home students (fees are typically fixed for International students, for the course duration at the year of entry). For general fees information please visit the postgraduate fees page .

Always contact the Admissions team if you are unsure which fees apply to your project.

Scholarships/sponsorships

There are a range of scholarships, studentships and awards at university, faculty and department level to support both UK and overseas postgraduate researchers.

To be considered for many of our scholarships, you’ll need to be nominated by your proposed supervisor. Therefore, we’d highly recommend you discuss potential sources of funding with your supervisor first, so they can advise on your suitability and make sure you meet nomination deadlines.

For more information about our scholarships, visit our funding page or use our funding database to search for scholarships, studentships and awards you may be eligible for.

Contact details

Our internationally-renowned expertise across the School of Natural Sciences informs research led teaching with strong collaboration across disciplines, unlocking new and exciting fields and translating science into reality.  Our multidisciplinary learning and research activities advance the boundaries of science for the wider benefit of society, inspiring students to promote positive change through educating future leaders in the true fundamentals of science. Find out more about Science and Engineering at Manchester .

Programmes in related subject areas

Use the links below to view lists of programmes in related subject areas.

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You can find regulations and policies relating to student life at The University of Manchester, including our Degree Regulations and Complaints Procedure, on our regulations website .

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University of Wisconsin-Milwaukee

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College of Letters & Science Mathematical Sciences

Numerical analysis, numerical analysis research group, numerical analysis at uwm.

The Numerical Analysis Group of the University of Wisconsin-Milwaukee Department of Mathematical Sciences offers many opportunities for students to enter the important and exciting field of research in numerical analysis and scientific computing. The cornerstone of the program is the PhD in Mathematics with specialty in Numerical Analysis (Applied Mathematics). It is also possible to receive a PhD in Industrial Mathematics while emphasizing scientific computing. There are five graduate research faculty and one senior lecturer, with a variety of fields of expertise and scholarly activities among the faculty. The research involves numerical analysis for integral equations, partial differential equations (finite difference, finite element, domain decomposition), and optimization.

Research Relationships

The Numerical Analysis Group is closely related to the Center Industrial Mathematics (CIM) and Atmospheric Science Groups, each of which is part of the Department of Mathematical Sciences. The CIM acts as a liaison between academic and corporate units, assists researchers with non-disclosure and patent issues, works to gather funding support, and sponsors conferences and workshops on industrial mathematics. The Atmospheric Science Group is distinguished with its mathematical approach within its field of discipline. Hence they create solid opportunities for interdisciplinary work for students of numerical analysis and scientific computation.

Student Opportunities

There are opportunities for teaching assistantships, research assistantships, and fellowships in numerical analysis, scientific computing, and industrial mathematics; if interested, please contact the Associate Chair for the Graduate Program .

The Department of Mathematical Sciences also offers an option within the PhD with industrial emphasis; students work on dissertations solving advanced mathematical problems with industrial sources. More information may be obtained at the Web site of the Center for Industrial Mathematics (CIM) .

Milwaukee is a very good location to carry out scientific computational research and industrial mathematics activities because of its excellent universities, easy access to many other universities (e.g., Chicago and Madison), as well as large variety of industry. UWM Numerical Analysis and Center for Industrial Mathematics faculty have published in areas of finite difference and finit element methods for ordinary and partial differential equations, computational aspects of bio-mathematics, computational aspects of control, numerical analysis and computational analysis for integral equations, domain decomposition for PDE, optimization, Computerized Tomography and Magnetic Resonance Imaging, Nonlinear PDE modeling and simulation (various types), statistics as well as computational statistics, and applications of artificial neural networks. UWM faculty and students have worked with industries in activities as diverse as airline scheduling, electrical power systems, engine performance modeling and simulation, finance, industrial controls, industrial printing, medical imaging (CT and MRI), paint production, refrigeration, telephone queuing systems, and travel industry data analysis.

Mathematics

Numerical Analysis and Scientific Computing

The numerical analysis group at UCL is a relatively recent addition. The main focus is on the design and implementation of efficient and accurate computational methods for the approximation of solutions to partial differential equations. A wide range of applications are considered, including continuum mechanics, electro-magnetics, stochastic optimisation and inverse problems. Research interest and staff can be broadly classed into the following categories:

• Computational methods in continuum mechanics - finite element methods for fluid and solid mechanics including multiphysics coupling, such as fluid-structure interaction or contact, and ficititious domain methods.
• Analysis and computational methods for wave scattering - boundary integral equations, wavenumber dependent stability, fractal domains, boundary element methods, fast solvers.
• Inverse problems - robust and accurate finite element methods with Carleman estimate based analysis, medical imaging. An example of a multidisciplinary research project in this context is the computational optimisation of treatment plans for high intensity focused ultrasound (HIFU) cancer treatment.
• Numerical methods in stochastic optimal control, a posteriori error estimation and adaptivity, parallel solvers.

The group has also participated in several research software developments, such as BEM++ and the Fenics based Cutfemlib.

Researchers

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Numerical Analysis

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• Weekly seminar series in data science and applied and computational mathematics

Douglas Arnold

McKnight Presidential Professor

[email protected] Numerical analysis, differential equations, mechanics, computational relativity

Jeffrey Calder 

Associate Professor 

[email protected] partial differential equations, numerical analysis, applied probability, machine learning, image processing and computer vision

Bernardo Cockburn

Distinguished McKnight University Professor

[email protected] numerical analysis

Jasmine Foo

Northrop Professor

[email protected] mathematical biology and applied mathematics

William Leeb

Assistant Professor

[email protected] applied mathematics, computational harmonic analysis, signal and image processing, data analysis

Gilad Lerman

[email protected] computational harmonic analysis, analysis of large data sets and statistical learning, bio-informatics

[email protected] Mathematical foundations of machine learning and data sciences, applied probability and stochastic dynamics, applied analysis and PDEs, Bayesian and computational statistics, inverse problems and uncertainty quantification

Mitchell Luskin

[email protected] numerical analysis, scientific computing, applied mathematics, computational physics

Peter Olver

[email protected] Lie groups, differential equations, computer vision, applied mathematics, differential geometry, mathematical physics

Associate Professor

[email protected] numerical analysis, scientific computing, applied math

Alex Watson

[email protected] Partial differential equations, mathematical physics, numerical analysis, computational physics, data science

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Mathematical Modeling Doctor of Philosophy (Ph.D.) Degree

Request Info about graduate study Visit Apply

The mathematical modeling Ph.D. enables you to develop mathematical models to investigate, analyze, predict, and solve the behaviors of a range of fields from medicine, engineering, and business to physics and science.

STEM-OPT Visa Eligible

Overview for Mathematical Modeling Ph.D.

Mathematical modeling is the process of developing mathematical descriptions, or models, of real-world systems. These models can be linear or nonlinear, discrete or continuous, deterministic or stochastic, and static or dynamic, and they enable investigating, analyzing, and predicting the behavior of systems in a wide variety of fields. Through extensive study and research, graduates of the mathematical modeling Ph.D. will have the expertise not only to use the tools of mathematical modeling in various application settings, but also to contribute in creative and innovative ways to the solution of complex interdisciplinary problems and to communicate effectively with domain experts in various fields.

Plan of Study

The degree requires at least 60 credit hours of course work and research. The curriculum consists of three required core courses, three required concentration foundation courses, a course in scientific computing and high-performance computing (HPC), three elective courses focused on the student’s chosen research concentration, and a doctoral dissertation. Elective courses are available from within the School of Mathematics and Statistics as well as from other graduate programs at RIT, which can provide application-specific courses of interest for particular research projects. A minimum of 30 credits hours of course work is required. In addition to courses, at least 30 credit hours of research, including the Graduate Research Seminar, and an interdisciplinary internship outside of RIT are required.

Students develop a plan of study in consultation with an application domain advisory committee. This committee consists of the program director, one of the concentration leads, and an expert from an application domain related to the student’s research interest. The committee ensures that all students have a roadmap for completing their degree based on their background and research interests. The plan of study may be revised as needed. Learn more about our mathematical modeling doctoral students and view a selection of mathematical modeling seminars hosted by the department.

Qualifying Examinations

All students must pass two qualifying examinations to determine whether they have sufficient knowledge of modeling principles, mathematics, and computational methods to conduct doctoral research. Students must pass the examinations in order to continue in the Ph.D. program.

The first exam is based on the Numerical Analysis I (MATH-602) and Mathematical Modeling I, II (MATH-622, 722). The second exam is based on the student's concentration foundation courses and additional material deemed appropriate by the committee and consists of a short research project.

Dissertation Research Advisor and Committee

A dissertation research advisor is selected from the program faculty based on the student's research interests, faculty research interest, and discussions with the program director. Once a student has chosen a dissertation advisor, the student, in consultation with the advisor, forms a dissertation committee consisting of at least four members, including the dissertation advisor. The committee includes the dissertation advisor, one other member of the mathematical modeling program faculty, and an external chair appointed by the dean of graduate education. The external chair must be a tenured member of the RIT faculty who is not a current member of the mathematical modeling program faculty. The fourth committee member must not be a member of the RIT faculty and may be a professional affiliated with industry or with another institution; the program director must approve this committee member.

The main duties of the dissertation committee are administering both the candidacy exam and final dissertation defense. In addition, the dissertation committee assists students in planning and conducting their dissertation research and provides guidance during the writing of the dissertation.

Admission to Candidacy

When a student has developed an in-depth understanding of their dissertation research topic, the dissertation committee administers an examination to determine if the student will be admitted to candidacy for the doctoral degree. The purpose of the examination is to ensure that the student has the necessary background knowledge, command of the problem, and intellectual maturity to carry out the specific doctoral-level research project. The examination may include a review of the literature, preliminary research results, and proposed research directions for the completed dissertation. Requirements for the candidacy exam include both a written dissertation proposal and the presentation of an oral defense of the proposal. This examination must be completed at least one year before the student can graduate.

Dissertation Defense and Final Examination

The dissertation defense and final examination may be scheduled after the dissertation has been written and distributed to the dissertation committee and the committee has consented to administer the final examination. Copies of the dissertation must be distributed to all members of the dissertation committee at least four weeks prior to the final examination. The dissertation defense consists of an oral presentation of the dissertation research, which is open to the public. This public presentation must be scheduled and publicly advertised at least four weeks prior to the examination. After the presentation, questions will be fielded from the attending audience and the final examination, which consists of a private questioning of the candidate by the dissertation committee, will ensue. After the questioning, the dissertation committee immediately deliberates and thereafter notifies the candidate and the mathematical modeling graduate director of the result of the examination.

All students in the program must spend at least two consecutive semesters (summer excluded) as resident full-time students to be eligible to receive the doctoral degree.

Maximum Time Limitations

University policy requires that doctoral programs be completed within seven years of the date of the student passing the qualifying exam. All candidates must maintain continuous enrollment during the research phase of the program. Such enrollment is not limited by the maximum number of research credits that apply to the degree.

National Labs Career Fair

Hosted by RIT’s Office of Career Services and Cooperative Education, the National Labs Career Fair is an annual event that brings representatives to campus from the United States’ federally funded research and development labs. These national labs focus on scientific discovery, clean energy development, national security, technology advancements, and more. Students are invited to attend the career fair to network with lab professionals, learn about opportunities, and interview for co-ops, internships, research positions, and full-time employment.

Students are also interested in: Applied and Computational Mathematics MS

Join us for Fall 2024

Many programs accept applications on a rolling, space-available basis.

Learn what you need to apply

The College of Science consistently receives research grant awards from organizations that include the National Science Foundation , National Institutes of Health , and NASA , which provide you with unique opportunities to conduct cutting-edge research with our faculty members.

Faculty in the School of Mathematics and Statistics conducts research on a broad variety of topics including:

• applied inverse problems and optimization
• applied statistics and data analytics
• biomedical mathematics
• discrete mathematics
• dynamical systems and fluid dynamics
• geometry, relativity, and gravitation
• mathematics of earth and environment systems
• multi-messenger and multi-wavelength astrophysics

Learn more by exploring the school’s mathematics research areas .

Michael Cromer

Basca Jadamba

Moumita Das

Carlos Lousto

Matthew Hoffman

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Mathematical Modeling, Curtain Coating, and Glazed Donuts

Bridget Torsey (Mathematical Modeling)

In her research, Bridget Torsey, a Math Modeling Ph.D. student, developed a mathematical model that can optimize curtain coating processes used to cover donuts with glaze so they taste great.

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Mathematics is a powerful tool for answering questions. From mitigating climate risks to splitting the dinner bill, Professor Wong shows students that math is more than just a prerequisite.

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May 8, 2023

Squishing the barriers of physics

Four RIT faculty members are opening up soft matter physics, sometimes known as “squishy physics,” to a new generation of diverse scholars. Moumita Das, Poornima Padmanabhan, Shima Parsa, and Lishibanya Mohapatra are helping RIT make its mark in the field.

RIT to award record number of Ph.D. degrees

RIT will confer a record 69 Ph.D. degrees during commencement May 12, marking a 53 percent increase from last year.

Curriculum for 2023-2024 for Mathematical Modeling Ph.D.

Current Students: See Curriculum Requirements

Mathematical Modeling, Ph.D. degree, typical course sequence

Concentrations, applied inverse problems, biomedical mathematics, discrete mathematics, dynamical systems and fluid dynamics, geometry, relativity and gravitation, admissions and financial aid.

This program is available on-campus only.

Full-time study is 9+ semester credit hours. International students requiring a visa to study at the RIT Rochester campus must study full‑time.

Application Details

To be considered for admission to the Mathematical Modeling Ph.D. program, candidates must fulfill the following requirements:

• Complete an online graduate application .
• Submit copies of official transcript(s) (in English) of all previously completed undergraduate and graduate course work, including any transfer credit earned.
• Hold a baccalaureate degree (or US equivalent) from an accredited university or college.
• A recommended minimum cumulative GPA of 3.0 (or equivalent).
• Submit a current resume or curriculum vitae.
• Submit a statement of purpose for research which will allow the Admissions Committee to learn the most about you as a prospective researcher.
• Submit two letters of recommendation .
• Entrance exam requirements: None
• Writing samples are optional.
• Submit English language test scores (TOEFL, IELTS, PTE Academic), if required. Details are below.

English Language Test Scores

International applicants whose native language is not English must submit one of the following official English language test scores. Some international applicants may be considered for an English test requirement waiver .

International students below the minimum requirement may be considered for conditional admission. Each program requires balanced sub-scores when determining an applicant’s need for additional English language courses.

How to Apply   Start or Manage Your Application

Cost and Financial Aid

An RIT graduate degree is an investment with lifelong returns. Ph.D. students typically receive full tuition and an RIT Graduate Assistantship that will consist of a research assistantship (stipend) or a teaching assistantship (salary).

Additional Information

Foundation courses.

Mathematical modeling encompasses a wide variety of scientific disciplines, and candidates from diverse backgrounds are encouraged to apply. If applicants have not taken the expected foundational course work, the program director may require the student to successfully complete foundational courses prior to matriculating into the Ph.D. program. Typical foundation course work includes calculus through multivariable and vector calculus, differential equations, linear algebra, probability and statistics, one course in computer programming, and at least one course in real analysis, numerical analysis, or upper-level discrete mathematics.

DisCoMath Seminar: Well, well, well..."well"ness in graph theory, especially well-forcedness

• Applied Mathematics

The graduate program in Applied Mathematics comprises the study and application of mathematics to problems motivated by a wide range of application domains. Areas of concentration include the analysis of data in very high-dimensional spaces, the geometry of information, computational biology, mathematical physics (optical and condensed matter physics), and randomized algorithms. Topics covered by the program include classical and modern applied harmonic analysis, linear and nonlinear partial differential equations, inverse problems, quantum optics, imaging, numerical analysis, scientific computing and applications, discrete algorithms, combinatorics and combinatorial optimization, graph algorithms, geometric algorithms, discrete mathematics and applications, cryptography, statistical theory and applications, probability theory and applications, information theory, econometrics, financial mathematics, statistical computing, and applications of mathematical and computational techniques to fluid mechanics, combustion, and other scientific and engineering problems.

• Programs of Study
• PhD - Doctor of Philosophy
• Applied Mathematics Program

Anna Gilbert

Director of Graduate Studies

• [email protected]

Departmental Registrar

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• 203-432-1278

Admission Requirements

Standardized testing requirements.

GRE is optional.

English Language Requirement

TOEFL iBT or IELTS Academic is required of most applicants whose native language is not English.

You may be exempt from this requirement if you have received (or will receive) an undergraduate degree from a college or university where English is the primary language of instruction, and if you have studied in residence at that institution for at least three years.

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The Graduate School's academic calendar lists important dates and deadlines related to coursework, registration, financial processes, and milestone events such as graduation.

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Registration Information and Dates

https://registration.yale.edu/

Students must register every term in which they are enrolled in the Graduate School. Registration for a given term takes place the semester prior, and so it's important to stay on top of your academic plan. The University Registrar's Office oversees the systems that students use to register. Instructions about how to use those systems and the dates during which registration occurs can be found on their registration website.

Financial Information

Phd stipend & funding.

PhD students at Yale are normally full-funded for a minimum of five years. During that time, our students receive a twelve-month stipend to cover living expenses and a fellowship that covers the full cost of tuition and student healthcare.

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• Mathematics

Podlesny, Joscha: Multiscale modelling and simulation of deformation accumulation in fault networks

Kahnt, max: numerical approximation of non-isothermal multi-component, multi-phase field systems.

The subject of this thesis is the derivation and analysis of numerical approximations of multi-component, multi-phase field systems. Recent approximations of solutions to such models are mostly based on explicit time stepping schemes and require the computation of many time steps. Implicit methods exhibit inherent numerical challenges, in particular due to the non-smoothness of the underlying energy functionals.

Our focus lies on the derivation of numerical approximations within the thermodynamically consistent context with high efficiency and robustness. We aim to exploit the special mathematical structure of the model and the underlying thermodynamics without introducing additional regularizations.

We introduce the thermodynamic and multi-phase setting in chapter 2 and continue by motivating and presenting a thermodynamically consistent multi-component, multi-phase field model in chapter 3. Based on Rothe’s method, we obtain a semi-discretization allowing for adaptive meshes in chapter 4 and the implicit problems are analyzed. In chapter 5, a full discretization with adaptive finite elements based on hierarchical a posteriori error estimation is set up. We transition to a purely algebraic formulation and present the iterative approximation of solutions with a nonsmooth Schur–Newton multigrid approach in chapter 6. Finally, in chapter 7, we perform numerical experiments to underline the thermodynamical consistency and numerical efficiency of our method.

Kies, Tobias: Gradient methods for membrane-mediated particle interactions

Discrete-continuous hybrid models are a popular means for describing elastic membrane-mediated particle interactions in and on lipid bilayers. Here, the continuous part is usually given by an approximation of the lipid membrane by an infinitely thin and sufficiently smooth hypersurface, whose elastic energy is determined by a Canham-Helfrich type functional. The discrete component results from modeling non-membrane particles as rigid discrete entities, which, depending on their configuration, induce local constraints on the membrane along the membrane-particle interfaces. In this context, the interaction potential describes the optimal elastic energy of such hybrid systems with a fixed particle configuration. Correspondingly, the energy minimization principle yields that stationary particle configurations are given by the local minima of the interaction potential. The main goal of this work is the proof of differentiability of the interaction potential for a selected class of models. This is accomplished using a variational approach that is already established in the literature in order to develop and apply robust numerical optimization methods for computing stationary particle configurations. Correspondingly, an additional focus is the derivation of a numerically accessible representation of the gradient, including its discretization and relevant numerical analysis. The proof of differentiability is brought forward by an application of the implicit function theorem. The basis for this is so-called boundary preserving domain transformations, which are induced by suitable families of vector fields and which locally admit the reformulation of the minimization problem that is implicitly defined by the interaction potential with respect to a fixed particle configuration. This subsequently enables the representation of the gradient as a volume integral using matrix analysis methods. The discretization of the partial differential equations for describing optimal membrane shapes is done via finite element methods. For particle methods with so-called curve restrictions a fictitious domain stabilized Nitsche method is developed, and for models with point value restrictions a conforming Galerking discretization is made possible by local QR transformations of the nodal finite element basis. For both cases suitable a priori error estimates are proven, and in addition also error estimates for the volume representation of the gradient are shown within that context. These developed methods open up the domain of efficient simulation of macro structures by isotropic and anisotropic particles, which is illustrated with the aid of various example applications and by means of perturbed gradient methods.

Djurdjevac, Ana: Random partial differential equations on evolving hypersurfaces

Partial differential equations with random coefficients (random PDEs) is a very developed and popular field. The variety of applications, especially in biology, motivate us to consider the random PDEs on curved moving domains. We introduce and analyse the advection-diffusion equations with random coefficients on moving hypersurfaces. We consider both cases, uniform and log-normal distributions of coefficients. Furthermore, we will introduce and analyse a surface finite element discretisation of the equation. We show unique solvability of the resulting semi-discrete problem and prove optimal error bounds for the semi-discrete solution and Monte Carlo samplings of its expectation. Our theoretical findings are illustrated by numerical experiments. In the end we present an outlook for the case when the velocity of a hypersurface is an uniformly bounded random field and the domain is flat.

Youett, Evgenia: Adaptive multilevel Monte Carlo methods for random elliptic problems

In this thesis we introduce a novel framework for uncertainty quantification in problems with random coefficients. The developed framework utilizes the ideas of multilevel Monte Carlo (MLMC) methods and allows for exploiting the advantages of adaptive finite element techniques. In contrast to the standard MLMC method, where levels are characterized by a hierarchy of uniform meshes, we associate the MLMC levels with a chosen sequence of tolerances. Each deterministic problem corresponding to a MC sample on a given level is then approximated up to the corresponding accuracy. This can be done, for example, using pathwise a posteriori error estimation and adaptive mesh refinement techniques. We further introduce an adaptive MLMC finite element method for random linear elliptic problems based on a residual-based a posteriori error estimation technique. We provide a careful analysis of the novel method based on a generalization of existing results, for deterministic residual-based error estimation, to the random setting. We complement our theoretical results by numerical simulations illustrating the advantages of our approach compared to the standard MLMC finite element method when applied to problems with random singularities.

Youett, Jonathan William: Dynamic large deformation contact problems and applications in virtual medicine

Dynamic large deformation contact problems arise in many industrial applications like auto mobile engineering or biomechanics but only few methods exists for their numerical solution, all having their advantages and disadvantages. In this thesis the numerical solution of large deformation contact problems is tackled from an optimisation point of view and an application of this approach within a femoroacetabular impingement analysis is described. In this thesis we use a non-smooth Hamilton principle and Fréchet subdifferential calculus to derive a weak formulation of the problem. The resulting subdifferential inclusion is discretised in time by constructing a contact-stabilised midpoint rule. For the spatial discretisation the state-of- the-art dual mortar method is applied which results in non-convex constrained minimisation problems that have to be solved solved during each time step. For the solution of these problems an inexact filter trust-region method is derived which allows to use inexact linearisations of the non-penetration constraints. This method in combination with fast monotone multigrid method is then shown to be globally convergent.

Hardering, Hanne: Intrinsic Discretization Error Bounds for Geodesic Finite Elements

This work is concerned with the proof of optimal error bounds for the discretization of $H^1$-elliptic minimization problem with solutions taking values in a Riemannian manifold. The discretization is done using Geodesic Finite Elements, a method of arbitrary order that is invariant under isometries. The discretization error is considered both intrinsically in a specially introduced Sobolev-distance as well as extrinsically. Optimal estimates of $H^1$- and $L^2$-type are shown, that have been observed experimentally in previous works of other authors. Using the Rothe method consisting of an implicit Euler method for the time discretization and Geodesic Finite Elements for the spatial discretization, error estimates for $L^2$-gradient flows of $H^1$-elliptic energies are derived as well. The core of the work is formed by the discretization error estimates for minimization problems in instrinsic $H^1$- and $L^2$-distances. To derive these, inverse estimates and interpolation errors for Geodesic Finite Elements and their discrete variations are shown. Using a nonlinear Cea's Lemma, this leads to the $H^1$-error estimate for minimizers of $H^1$-elliptic energies. A generalization of the Aubin-Nitsche-Lemma shows optimal $L^2$-error estimates for (essentially) semilinear energies, as long as the dimension of the domain of the minimizer is limited to $d<4$ for technical reasons. All results are illustrated using harmonic maps into a smooth Riemannian manifold satisfying certain curvature bounds as an example.

Pipping, Elias: Dynamic problems of rate-and-state friction in viscoelasticity

In this work, the model of rate-and-state friction, which can be viewed as central to the numerical simulation of earthquakes, is considered from a mathematical point of view. First, a framework is presented through which a general class of such friction laws can be understood and analysed. A prototypical viscoelastic problem of earthquake rupture is then formulated, both in strong and in variational form. Analysis of this problem is difficult, since the incorporation of rate-and-state friction leads to a coupling of variables. In a time-discrete setting, nonetheless, results on existence, uniqueness, and continuous parameter dependence of solutions can be obtained. The principal idea is to reformulate the variable interdependence as a fixed point problem and to prove convergence for a corresponding iteration. With that in mind, next, a numerical algorithm is proposed that resolves the coupling through a fixed point iteration. Since it puts a state-of-the-art solver and adaptive time stepping to use, it is not only stable but also fast. Its applicability to problems of interest is demonstrated in the penultimate chapter, which focuses on simulations of megathrust earthquakes that form at the base of a subduction zone. The main assumptions made throughout this work are summarised and discussed in the last chapter.

Sack, Uli: Numerical Simulation of Phase Separation in Binary and Multicomponent Systems

The ban of lead in electronics solder by EU directives results in the technological challenge to develop lead-free alternatives with comparable life span and processing properties. Numerical simulations of the microstructure evolution may contribute to identify promising candidates and thus focus the immense experimental effort. Aim of this work is on the one hand to develop a numerical framework for the efficient and robust simulation of the microstructure evolution in binary alloys combining adaptive finite element methods with fast solvers for the Cahn-Hilliard model. On the other hand we will extend the existing fast solvers for the discrete scalar Cahn-Hilliard equation to the vector-valued case. After some preliminary remarks on phase diagrams, phase separation, and phasefield models in Chapter 1 we will firstly discuss anisotropic Allen-Cahn equations in Chapter 2. Alle-Cahn-like problems arise as subproblems in the Nonsmooth Schur-Newton (NSNMG) method for Cahn- Hilliard equations in Chapters 3 and 4. Here we prove existence and uniqueness of solutions to the anisotropic Allen-Cahn equation with logarithmic potential using the theory of maximal monotone operators. For the numerical solution we introduce an adaptive spatial mesh refinement cycle for evolution problems and several variants of implicit Euler time discretization. We prove stability for the latter and numerical experiments conclude the chapter. Chapter 3 combines existing and newly developed numerical tools to a simulation software for microstructure evolution in binary alloys. Key ingredients are the adaptive mesh refinement cycle of Chapter 2, the NSNMG solver, a quantification algorithm for measuring "coarseness" of microstructures and a quotient space multigrid method for indefinite problems. An application of this software to simulate the microstructure evolution in a eutectic AgCu alloy shows only marginal impact of elastic stresses on coarsening in the setting considered; while the use of a smooth interpolant of the logarithmic potential affects the coarsening dynamics considerably. In the final chapter we consider the multicomponent Cahn-Hilliard equation and derive a unified formulation for the discrete problems which allows a direct application of the NSNMG method. Existence and uniqueness of discrete solution are proved and numerical examples illustrate the robustness of the scheme with respect to temperature, mesh size, and number of components.

Gräser, Carsten: Convex minimization and phase field models

Phase field models are a widely used approach to describe physical processes that are characterized by thin interfacial regions between large almost homogeneous domains. Important application areas of phase field models are transition processes of the state of matter and the separation of alloys. A fundamental property of these models is, that the transition and separation of phases is driven by a double-well potential with distinct minima for the different phases. Already the pioneering work of Cahn and Hilliard used a temperature dependent logarithmic potential that is differentiable with singular derivatives. If the temperature tends to zero it degenerates to the non-differentiable obstacle potential. The goal of this thesis is to develop methods for the efficient numerical solution of such equations that are also robust for nonsmooth potentials and anisotropic surface energies. These methods are derived for the Cahn-Hilliard equation that are prototypic for a multitude of such models. The main result of the thesis is the development of a fast iterative solver for nonlinear saddle point problems like the ones that arise from finite element discretization of Cahn-Hilliard equations. The solver relies on a reformulation of the problem as dual minimization problem whose energy functional is differentiable. The gradient of this functional turns out to be the nonlinear Schur complement of the saddle point problem. Generalized linearizations for the Schur complement are derived and used for a nonsmooth Newton method. Global convergence for this 'Schur Nonsmooth Newton' method and inexact versions is proved using the fact the equivalence to a descent method for the dual minimization problem. Each step of this method requires the solution of a nonlinear convex minimization problem. To tackle this problem the 'Truncated Nonsmooth Newton Multigrid' (TNNMG) method is developed. In contrast to other nonlinear multigrid methods the TNNMG method is significantly easier to implement and can also be applied to anisotropic problems while its convergence speed is in general comparable or sometimes even faster. Numerical examples show that the derived methods exhibit mesh independent convergence. Furthermore they turn out to be robust with respect to the temperature including the limiting case zero. The reason for this robustness is, that all methods do not rely on smoothness but on the inherent convex structure of the problems.

Forster, Ralf: On the stochastic Richards equation

A frequent problem during numerical computations consists in the uncertainty of certain model parameters due to measuring errors or their high variability. In the last years, one could observe an increasing interest in the quantification of these uncertainties and their effects to the solution of numerical simulations; a powerful tool which has been proven to be an efficient approach in this context is the so-called polynomial chaos method which is based on a spectral decomposition of the covariance function of the uncertain parameters and a representation of the solution in a polynomial basis. The aim of this thesis is the application of this method to the Richards equation modeling groundwater flow in saturated and unsaturated porous media. The main difficulty consists in the saturation and the hydraulic conductivity appearing in the time derivative and in the spatial derivatives, since both depend nonlinearly on the pressure. Considering uncertain parameters like random initial and boundary conditions and, in particular, a random permeability leads to a stochastic variational inequality of second kind with obstacle conditions and a nonlinear convex functional as superposition operator. Considering variational inequalities in the context of uncertain parameters and the polynomial chaos method is new, and we start by deriving a weak formulation of the problem and approximating the parameters by a Karhunen-Loève expansion. The existence of a unique solution u in a tensor space can be proven for the time-discrete problem by reformulation as a convex minimization problem. We proceed by discretizing with finite elements and polynomial ansatz functions and by approximating the convex functional with Gaussian quadrature. The convergence of the solution of the discretized problem to the solution u is proved in a special case for a stochastic obstacle problem. Moreover, we perform numerical experiments to determine the discretization error. In the second part of this thesis, we develop an efficient numerical method to solve the discretized minimization problems. It is based on a global converging Block Gauß Seidel method and exploits a transformation which decouples the stochastic coefficients and connects the stochastic Galerkin with the stochastic collocation approach. This also allows us to establish a multigrid solver to accelerate the convergence. We conclude this thesis by demonstrating the power of our approach on a realistic example with lognormal permeability and exponential covariance.

Sander, Oliver: Multidimensional coupling in a human knee model

The thesis presents a new model for the numerical simulation of the mechanics of the human knee. In this model bones are described using linear elasticity. Ligaments instead are modelled as one-dimensional Cosserat rods. The simulations give insight into the mechanical behavior of human joints. This can be helpful for a number of applications. For example, it is possible to estimate the long-term effect of certain surgical interventions. Also, the design of prosthetic devices can be improved. The main mathematical focus is on the correct formulation of the coupling conditions between one- and three- dimensional objects. Starting from the case of two three-dimensional objects, for which coupling conditions can be derived rigorously, conditions for the multidimensional case are formulated. A solution algorithm for this coupled problem is presented, and the existence of solutions is shown under certain symmetry assumptions. For the subproblems, large contact problems and minimization problems on Riemannian manifolds have to be solved. For both problems, robust and efficient numerical methods are introduced. Numerical experiments show the applicability for real-world problems.

Berninger, Heiko: Domain decomposition methods for elliptic problems with jumping nonlinearities and application to the Richards equation

The thesis presents a new method for the solution of saturated-unsaturated groundwater flow problems in heterogeneous porous media. Concretely, highly nonlinear degenerate elliptic problems arising from a certain time discretization of the Richards equation are the basis of this work. The problems are considered as homogeneous in subdomains where a single soil prevails and, therefore, the parameter functions do not depend on space. These nonlinearities, however, may jump across the interfaces between the subdomains and, thus, account for the heterogeneous setting of different soil types in different subdomains. As a consequence, non-overlapping domain decomposition problems in which subproblems are coupled via nonlinear transmission conditions are obtained. In this work these problems are solved without any linearization. By Kirchhoff transformation the homogeneous subproblems are transformed into convex minimization problems. Here, additional constraints like Signorini-type boundary conditions, which occur on seapage faces around lakes, can be taken into account. Finite elements are chosen for the space discretization, and convex analysis is applied as the solution theory. Finally, monotone multigrid methods provide efficient solvers which are robust with respect to degenerating soil parameters. In order to deal with the coupling of the homogeneous subproblems, nonlinear Dirichlet-Neumann and Robin methods are used. Here, the thesis provides new convergence results for these iterations applied to nonlinear elliptic problems in 1D as well as well- posedness results, which generalize existing linear theory. On the other hand, detailed numerical experiments demonstrate that the methods can also be applied successfully to problems in 2D. Finally, based on the artificial viscosity method, an upwind discretization with finite elements is developed in order to account for gravity. Hence, stability of the numerical solutions is obtained. In a closing numerical example the Richards equation is solved in 2D with four different soils and coupled to a surface water reservoir. The result demonstrates the applicability of the developed solution technique to a heterogeneous problem with realistic hydrological data.

Gebauer, Susanna: Hierarchical Domain Decomposition Methods for saturated Groundwaterflow in fractured porous Media

The focus of this thesis ist the numerical computation of flow in special geometries dominated by jumps in the flow coefficients and large differences in the scales of the main flow pathes and the surrounding materials. These characteristics result in difficulties in the numerical computation of the modelling equations. By means of groundwaterflow in fractured porous media we present a hierarchical domain decomposition method for the numerical computation of flow. Under certain assumptions this new method converges independently of the fracture width, the refinement depth and the jump in the flow coefficient. The theoretical results are confirmed by practical computations of a model problem and a fracture network. Thus for a new class of completely overlapping domain decomposition methods multigrid efficiency is shown for a class of problems, for which so far no comparably theoretically validated method existed.

Krause, Rolf H.: Monotone Multigrid Methods for Signorini's Problem with Friction

In this work, we consider the numerical simulation of contact problems. Since the numerical realization of contact problems is of high importance in many application areas, there is a strong demand for fast and reliable simulation method. We introduce and analyze a new nonlinear multigrid method for solving contact problems with and without friction. As it turns out, by means of our new method nonlinear contact problems can be solved with a computational amount comparable that of linear problems. In particular, in our numerical experiments we observe our method to be of optimal complexity. Moreover, since we do not use any regularization techniques, the computed discrete boundary stresses as well as the computed displacements turn out to be highly accurate. The new method is based on the succesive minimization of the associated energy functional in direction of properly choosen functions. We show the global convergence of our method and give several numerical examples in two and three space dimensions, illustrating the robustness and the performance of the method. In addition to the theoretical analysis, the method has been implemented in an object oriented way. We explain the concepts of our implementation and show the flexibility of our approach by deriving a nonlinear algebraic multigrid method. To include frictional effects, we use a discrete fixed point iteration. As a faster alternative, also a Gauss-Seidel like iteration scheme is proposed. Both methods are compared in numerical examples. The resulting nonlinear algorithm turns out to be fast and reliable. Finally, we consider the case of contact between elastic bodies. Here, the information transfer at the interface is realized by means of non conforming domain decomposition methods (mortar methods). This gives rise to a non-linear Dirichlet Neumann Algorithm.

• Dissertationsserver der Freien Universität Berlin

Welcome to the Homepage of the research group Numerical Analysis and Uncertainty Quantification at University of Heidelberg

Postal address

Institute of Applied Mathematics and Interdisciplinary Center for Scientific Computing (IWR) Universität Heidelberg Im Neuenheimer Feld 205 69120 Heidelberg, Germany

Herta Fitzer NumOpt [at] uni-heidelberg.de +49 6221 5414111 Room: 1 / 318

Many physical models from the natural sciences and from engineering involve sources of uncertainty that affect their outputs. An example are variations in the orientation of layers of carbon fibres occurring naturally in the manufacturing process of aircraft wings. As a consequence, the locations and types of possible defects and cracks are difficult to predict precisely. The goal of uncertainty quantification is to use mathematical and computational methods to account for such uncertainties, and to understand how they propagate through to model outputs.

The research of our group focuses on developing innovative numerical methods to efficiently quantify uncertainty. We apply these techniques to tackle data-driven, large-scale problems that are typically modelled in the form of differential equations. Our methods strive for a balance between efficiency, a rigorous mathematical foundation and realistic model problems. As a brief overview, some of the techniques we develop are based on Monte Carlo/quasi-Monte Carlo sampling, stochastic collocation, sophisticated hierarchical/multilevel strategies and dimension reduction through low-rank tensor approximations. The applications we study range from groundwater flow, to nuclear physics and carbon fibre composites in manufacturing.

When studying real world processes mathematically, essentially all problems can be roughly put into two classes: Forward and inverse problems. A forward problem concerns the calculation of the state of a physical system, given all the necessary parameters, as well as boundary and initial conditions. Inverse problems, on the other hand, are concerned with computing parameters given observations of the state of the system. Good estimates of these parameters give us an insight into hidden quantities that typically cannot be observed directly and are otherwise very difficult to grasp. Consequently, inverse problems are among the most important in mathematical applications and uncertainty quantification plays a crucial role here.

A very popular approach to address inverse problems is Bayesian inference, a subbranch of statistics and data science. It facilitates the quantification of uncertainties regarding the model and its parameters. This resolves the inherent ill-posedness of inverse problems and has proven to be exceedingly fruitful in many applications. A particular focus in our group is the design and analysis of efficient numerical techniques for high- and infinite-dimensional Bayesian inverse problems, especially those constrained by differential equations. We look at the effect of choosing high-level priors, develop efficient multilevel algorithms and surrogates, incorporate novel ideas from numerical analysis into this setting and explore links to machine learning.

Natural or engineered materials often contain two or more key constituents, arranged in a heterogeneous structure varying at different scales. Such materials are desirable because their macroscopic properties can be superior to the properties of the individual constituents. It is even possible to explicitly design them for a particular purpose by changing the composition of the constituents. An example are carbon fibre composites for lightweight structures and vehicles. The mathematical modelling of such heterogeneous or composite materials naturally leads to partial differential equations (PDEs) with highly oscillating coefficients. Direct numerical solution of such problems with traditional methods, such as finite elements is computationally expensive. Just to compute the correct qualitative behaviour, the mesh resolution would need to be sufficiently high to capture all the fine scale variation.

In our group, we study and develop multiscale numerical methods that do not suffer from this drawback. We are particularly interested in systems without periodic structure or scale separation and in problems with a high contrast in the constituent material properties. Such problems require customized approximation spaces, computable via localised boundary value or eigen-problems. There are also strong links to model order reduction and domain decomposition methods. Examples include multiscale finite elements, generalised multiscale finite elements, or the localizable orthogonal decomposition method. Target applications are again subsurface flow and carbon fibre composites, but also biological materials such as bone or cells.

Scientific computing and numerical simulation are playing an ever more important role in science and technology. Hardly any new developments, e.g., in engineering or the geosciences, take place without careful mathematical modelling, analysis and optimisation. More and more complicated systems are being tackled, in particular in the life sciences or in the context of climate change. This requires continued research into efficient and robust numerical methods, especially in the context of heterogeneous or random media, and in their careful and rigorous numerical analysis. Increased efficiency requires a redesign of traditional algorithms to harness the power of modern many-core computing architectures (hardware-aware scientific computing), while data-driven (predictive) scientific computing poses new challenges for the robustness of existing methods.

Here, the focus of our group encompasses

• traditional topics, such as efficient preconditioning and discretisation methods for heterogeneous and anisotropic PDEs, or efficient algorithms for large-scale eigenproblems, as well as
• more modern topics that arise naturally in the context of uncertainty quantification or Bayesian inference, such as high-dimensional approximation and quadrature, including Monte Carlo, quasi-Monte Carlo, sparse grid, low-rank tensor approximation or deep learning.

In terms of novel software, we contribute in particular to DUNE (the Distributed and Unified Numerical Environment) and to MUQ (the MIT Uncertainty Quantification Library).

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