Student Projects and Theses

Our research group focuses on the study of stability and regularization of ill-posed problems which arise in various applications. For such problems conventional solvers yield unreliable outputs, so that regularization techniques are necessary to produce trustworthy solutions. Research topics include medical imaging, phase retrieval (with application in coherent diffraction imaging and audio processing) and the vulnerability of deep neural networks. Theses projects on these topics are available, with possible emphasis on either implementation or analysis.

Our goal is to develop new mathematical models and methods of broad utility to science and engineering and to make fundamental advances in the mathematical and physical sciences themselves. We have been developing mathematical and computational frameworks to achieve super-resolution in biomedical imaging. Our work on biomimetics focuses on elucidating mechanisms exploited by bats and weakly electric fish for imaging. Nanophotonics is the study of electromagnetic wave phenomena in media structured on the same lengthscale as the wavelength, and is also a very active area of study in our group. Thesis projects on these current topics of research are available on request.

List of projects see here

The research of the group of S. Mishra focuses on the design, analysis and efficient implementation on state of the art high performance computing platforms, of robust numerical methods for nonlinear partial differential equations, particularly of the hyperbolic type, and their application in fluid dynamics, plasma physics, astrophysics, climate science and geophysics.

We also address fundamental mathematical questions of well-posedness of nonlinear hyperbolic PDEs, particularly in the context of novel solution frameworks such as statistical solutions.

Numerical methods, of the finite volume, finite difference, spectral and discontinuous Galerkin finite element type are developed in the group. Another focus area is on computational uncertainty quantification i.e efficient calculation of uncertainties in the solutions of PDEs that are driven by stochastic initial/boundary conditions, parameters and chaotic dynamics.

Student projects for Bachelor, semester and Masters thesis are available on the whole range of topics from addressing fundamental mathematical questions to the design of efficient algorithms, HPC implementation and interesting applications.

Stu­dent pro­jects are avail­able in Math­em­at­ical Ana­lysis (er­ror es­tim­ates) and Com­puter Im­ple­ment­a­tion of ef­fi­cient dir­ect and in­verse prob­lem solu­tion for de­term­in­istic and stochastic Par­tial Dif­fer­en­tial Equa­tions with either many (sev­eral hun­dred) para­met­ers, or with solu­tions on very high di­men­sional state-​ and phase spaces.

Mathematical techniques comprise (but are not limited to):

Numerical Analysis of Partial Differential Equations
(Finite Element Galerkin discretizations)
see, e.g.
https://math.ethz.ch/sam/research/reports.html?id=1031
https://math.ethz.ch/sam/research/reports.html?id=2020-67
Sparse Grid and Polynomial Chaos techniques

Mathematics of Deep Learning
(neural network approximation theory, deep neural network training, etc.), for numerical PDE approximation
see, e.g.
https://math.ethz.ch/sam/research/reports.html?id=2021-40
https://math.ethz.ch/sam/research/reports.html?id=2022-11

Monte-Carlo and Quasi-Monte Carlo integration in high dimensions
see, e.g.
https://math.ethz.ch/sam/research/reports.html?id=2022-10
external page J. Dick and M. Longo and Ch. Schwab: Extrapolated Lattice Rule Integration in Computational Uncertainty Quantification, SIAM/ASA Journ. Uncertainty Quantification, 10/2 (2022), pp. 651-686

All Projects are closely related to current research topics, and can be extended to MSc and PhD thesis projects.

1. Yanchen He, Download Analytic regularity and hp-Discontinuous Galerkin approximation of viscous, incompressible flow in a polygon (PDF, 844 KB), 2021, supervisors Prof. Dr. Christoph Schwab and Dr. Carlo Marcati
   
2. Marcello Longo, Download Higher order QMC integration with scrambling for elliptic PDEs with random coeffcients (PDF, 1.3 MB), 2019, supervisor Prof. Ch. Schwab
   
3. Joost Opschoor, Download ReLU DNN expression of sparse gpc expansion in uncertainty quantification (PDF, 1 MB), 2018, supervisor Prof. Ch. Schwab
   
4. Simon Etter, Download Parallel Tensor-Formatted Numerics for the Chemical Master Equation (PDF, 2 MB), 2015, supervisors Robert Gantner and Prof. Dr. Christoph Schwab
5. Oded Stein, Download A Boundary Element Method for Eddy Currents (PDF, 2.6 MB), 2015, supervisor Prof. Ralf Hiptmair
6. Ingmar Getzner, Download Examination of Measure Valued Solutions for the Magnetohydrodynamic Equations (PDF, 2.6 MB), 2014, supervisors Prof. Siddhartha Mishra and Dr. Roger Käppeli
7. Samuel Lanthaler, Download Computation of Measure Valued Solutions of the Incompressible Euler Equations (PDF, 3.3 MB), supervisor Prof. Siddhatha Mishra
8. Simon Laumer, Download Finite Difference Approach for the Measure Valued Vanishing Dispersion Limit of Burgers’ Equation (PDF, 1.9 MB), 2014, supervisor Prof. Siddhartha Mishra
   
9. Jakob Zech, Download A Posteriori Error Estimation of hp-DG Finite Element Methods for Highly Indefinite Helmholtz Problems (PDF, 4.4 MB), 2014, supervisor Prof. Dr Stefan Sauter
10. Lukas Herrmann, Download Isotropic Random Fields on the Sphere - Stochastic Heat Equation and Regularity of Random Elliptic PDEs (PDF, 1.4 MB), 2013, supervisors Dr. Annika Lang, Prof. Dr. Christoph Schwab 
11. Yann Poltera, Download MLMC-FD method for 2D statistical solutions of the Navier-Stokes equations (PDF, 1.4 MB), 2013, supervisor Prof. Dr. Christoph Schwab
12. Helga Thum, Download Numerical Simulations of the Weak Approximation Error for Parabolic Stochastic Partial Differential Equations (PDF, 706 KB), 2013, supervisor Prof. Dr. Arnulf Jentzen
13. Yulia Smirnova, Download Calderon Preconditioning for Higher Order Boundary Element Method (PDF, 1.9 MB), 2013, supervisor Prof. Ralf Hiptmair
14. Raffael Casagrande, Download Sliding Interfaces for Eddy Current Simulations (PDF, 7.1 MB), 2013, supervisor Prof. Ralf Hiptmair
15. Simon Pintarelli, Download Local Multi-trace Boundary Element Formulation for Diffusion Problems (PDF, 2.9 MB), 2013, supervisor Prof. Ralf Hiptmair
16. Michel Verlinden, Download Adaptive Galerkin Methods for Infinite Dimensional Parabolic Equations (PDF, 580 KB), 2012, supervisor Prof. Christoph Schwab
17. Severin Thaler, Download The Criticality Spectral Problem in Neutron Transport (PDF, 1.4 MB), 2012, supervisor Prof. Christoph Schwab
18. Robbin Tops, Download Numerical Pricing of American Options for gerneral Bivariate Lévy Models (PDF, 3.8 MB), 2012, supervisor Prof. Christoph Schwab
19. Elke Spindler, Download Second Kind Single Trace Boundary Element Methods (PDF, 5.4 MB), 2012, Prof. Ralf Hiptmair
20. D. Bernhardsgrütter, Download Multi-Level Monte Carlo Methods for Stochastic Parabolic and 2nd Order Hyperbolic PDEs (PDF, 3.7 MB), 2011, supervisor Prof. Dr. Christoph Schwab
21. Daniel Kövi, Download hp Finite Element Method pricing algorithms for lookback options in Lévy Markets (PDF, 1.3 MB), 2011, supervisor Prof. Christoph Schwab
22. Barry Thornton, Download Electricity Spot Price Modelling and Derivatives Pricing (PDF, 5.3 MB), 2010, supervisor Prof. Christoph Schwab
23. Nathaniel Zollinger, Download Multi-Level Monte Carlo Finite Element Method for Elliptic Partial Differential Equations with Stochastic Data (PDF, 720 KB), 2010, supervisor Prof. Christoph Schwab
24. Alberto Paganini, Download Efficient Convolution Based Impedance Boundary Conditions (PDF, 731 KB), 2011, supervisor Prof. Ralf Hiptmair
25. Michael Spreng, Download Resolution of Skin Layer in EM Simulation (PDF, 865 KB), 2010, supervisor Prof. Ralf Hiptmair
26. Stefan Siebenmann, Download A Scalable Eddy-Current Solver for ANSYS (PDF, 1.3 MB), 2010, supervisor Prof. Ralf Hiptmair
27. Lisa Powers, Download Numerical Study of Small-Jump Regularization on Exotic Contracts in Lévy Markets (PDF, 1.4 MB), 2008, supervisor Prof. Christoph Schwab 
28. Franz Piehler, Download Boundary element discretisation of asymptotic layer models (PDF, 879 KB), 2008, supervisor Prof. Ralf Hiptmair
29. Claude Gittelson, Download Plane wave discontinuous Galerkin methods (PDF, 1.6 MB), 2008, supervisor Prof. Ralf Hiptmair
    
30. Chris Bardgett, Download Pricing Convertible Bonds using Finite Elements (PDF, 400 KB), 2007, supervisor Prof. Christoph Schwab
31. Simone Riva, Download Computation of electrostatic forces by the virtual work method (PDF, 1.4 MB), 2007, supervisor Prof. Ralf Hiptmair
32. Jan Kayatz, Download Cone Strategies for Formal Deformations of Non-Collapsible Simplicial Complexes (PDF, 412 KB), 2006,supervisor Prof. Ralf Hiptmair
33. Oliver Pfeifer, Download Asymptotische Methode zur Berechnung elektrostatischer Kräfte (PDF, 489 KB), 2006, supervisor Prof. Ralf Hiptmair
34. Julien Renggli, Download Quadtree techniques for the construction of structured auxiliary meshes (PDF, 429 KB), 2006, supervisor Prof. Ralf Hiptmair
35. Gisela Widmer, Download Auxiliary Space Method for Edge Elements (PDF, 706 KB), 2003/04, supervisor Prof. Ralf Hiptmair
36. Benjamin Guélat, Download Régularité des solutions des problèmes d'élasticité et d'écoulement de Stokes dans un polygone (PDF, 2.3 MB), 2001, supervisor Prof. Christoph Schwab

1. Thomas Haener, Download Numerical Solution of a Coefficient Identification Problem for the Poisson Equation (PDF, 3.9 MB), September 2014, supervisor Prof. Ralf Hiptmair
2. Elke Spindler, Download A Posteriori Error Estimates for hp-FEM: Comparison and Implementation (PDF, 545 KB), 2010, supervisor Prof. Ralf Hiptmair
3. Robert Gantner, Download Rayleigh Quotient Multigrid (PDF, 4.2 MB), 2011, supervisor Prof. Ralf Hiptmair
4. Simon Pintarelli, Download Duality based error estimation for electrostatic force computation (PDF, 13.6 MB), 2010, supervisor Prof. Ralf Hiptmair
5. Julia Slipantschuk, Download Numerical Validation of Emergent Behavior in Flocks (PDF, 4.5 MB), 2008, supervisors Prof. Christoph Schwab, Claude Jeffrey Gittelson
6. Sabine Elmiger, Download Convergence of Binomial Tree Methods for European/American Path-Dependent Options (PDF, 5.8 MB), 2008, supervisors Prof. Christoph Schwab
7. Claude Gittelson, Download Non-equidistant approximate DFT based on Z-Splines (PDF, 382 KB), 2006, supervisor Prof. Ralf Hiptmair

1. Nitya Afambo, Download Generalized Tensor Models for Recurrent Neural Networks (PDF, 1.2 MB), 2020, supervisors Dr. Maksim Rakhuba, Prof. Christoph Schwab
   
2. Simon Etter, Download MPI-Based Tensor Network Library (PDF, 528 KB), 2014, supervisors Robert Gantner, Vladimir Kazeev, Prof. Christoph Schwab (RW/CSE)
3. Jakob Zech, Download Nonlinear n-term approximation for the solution of the dirichlet problem (PDF, 288 KB), 2013, supervisors Dr. Markus Hansen, Prof. Christoph Schwab
4. Simon Härdi, Download Parallelization of a radiative transfer solver (PDF, 444 KB), 2012, supervisor Prof. Christoph Schwab
5. Yulia Smirnova, Download Dual Mesh Calderon Preconditioning for the Single Layer Boundary Integral Operator (PDF, 5 MB), 2012, supervisor Prof. Ralf Hiptmair
6. Raffael Casagrande, Download Implementation of impedance boundary conditions in HADAPT (PDF, 1.3 MB), 2012, supervisor Prof. Ralf Hiptmair
7. Anders Rosero, Download Spurious Solutions for transient Maxwell equations in 2D (PDF, 1.1 MB), 2011, supervisor Prof. Ralf Hiptmair
8. Yavor Stoev, Download Variance-optimal hedging error in geometric Lévy models (PDF, 530 KB), 2010, supervisor Prof. Christoph Schwab
9. Jonas Sukys, Download Discontinuous Galerkin discretization of magnetic convection (PDF, 2.5 MB), 2010, supervisor Prof. Ralf Hiptmair
10. Andreas Hiltebrand, Download Fast solvers for Eulerian convection schemes, (PDF, 328 KB) 2010, supervisor Prof. Ralf Hiptmair
11. Christoph Bossard, Download Hash Based Data Structures for Tetrahedral Meshes (PDF, 152 KB), 2005, supervisor Prof. Ralf Hiptmair
12. Mathias Hack, Download Angle conditions for face and edge elements (PDF, 427 KB), 2005, supervisor Prof. Ralf Hiptmair
13. Manuel Guidon, Download Transient Simulation of Eddy Currents in Ferromagnets (PDF, 3.7 MB), 2003/04, supervisor Prof. Ralf Hiptmair
14. Philippe Corboz, Download Micromagnetism (PDF, 3.7 MB), 2003, supervisor Prof. Ralf Hiptmair