Dr. Martin Eigel
| Short CV | Publications | Teaching |
Research interests
Association to the Mathematical Topic "Numerical Methods for PDEs with Stochastic Data".
- Statistical and Deep Learning methods
- Adaptive functional approximations for random PDEs
- Low-rank tensor methods for SDEs
- Statistical inverse problems
- Topology and shape optimisation under uncertainties
- FEM a posteriori error estimators
- Spatial models in computational biology
Contact details
| Martin.Eigel@wias-berlin.de | |
| Phone | +49 (0) 30 20372 413 |
| Fax | +49 (0) 30 20372 412 |
Current projects
- Scientific Machine Learning with deep network representations: Analysis and development of statistical learning methods with adapted tensor and neural networks.
- Optimal sampling for nonlinear reconstruction
- Reinforcement Learning: Tensor approaches for (stochastic) dynamical programming in the ReLkat project and dynamics recovery with GPs.
- Adaptive numerical methods for stochastic PDE: Adaptive spectral methods for problems with stochastic data.
- Low-rank methods for Stochastic FEM: Efficient adaptive low-rank tensor solvers for SGFEM discretisations of stochastic PDE.
- Functional Bayesian inversion: Inverse problems with stochastically perturbed measurements based on low-rank tensor approximations.
- Multi-scale failure analysis with polymorphic uncertainties for optimal design of rotor blades: DFG SPP1886 sub-project 4.
Head of project together with D. Hömberg and J. Petryna.
Short CV
| Since March 2013 | Researcher in group Hömberg, WIAS, Berlin |
| November 2010 - February 2013 | PostDoc in group Peterseim, Matheon project C33, Humboldt Universität, Berlin |
| July 2008 - November 2010 | PostDoc in group Carstensen, Humboldt Universität, Berlin |
| 2008 | Ph.D., University of Warwick, United Kingdom |
| June 2003 | Diploma, Universität Heidelberg |
Publications
Articles
- M. Eigel, N. Farchmin, S. Heidenreich, P. Trunschke
Efficient approximation of high-dimensional exponentials by tensor networks
WIAS preprint 2844, submitted - Ch. Bayer, M. Eigel, L. Sallandt, P. Trunschke
Pricing high-dimensional Bermudan options with hierarchical tensor formats
WIAS preprint 2821, submitted - M. Eigel, O. Ernst, B. Sprungk, L. Tamellini
On the convergence of adaptive stochastic collocation for elliptic partial differential equations with affine diffusion
WIAS preprint 2753, submitted - M. Eigel, R. Schneider, P. Trunschke
Convergence bounds for empirical nonlinear least-squares
WIAS preprint 2714, submitted - M. Eigel, R. Gruhlke, M. Marschall
Low-rank tensor reconstruction of concentrated densities with application to Bayesian inversion
WIAS preprint 2672, submitted - M. Kirstein, M. Eigel, D. Bariamis
Bayesian Tensor Regression Of Random Fields
Workshop on Quantum Tensor Networks in Machine Learning, 34th Conference on Neural Information Processing Systems (NeurIPS 2020) - M. Eigel, L. Grasedyck, R. Gruhlke, D. Moser
Low rank surrogates for polymorphic fields with application to fuzzy-stochastic partial differential equations
WIAS preprint 2580, submitted - M. Drieschner, M. Eigel, R. Gruhlke, D. Hömberg, Y. Petryna
Comparison of various uncertainty models with experimental investigations regarding the failure of plates with holes
WIAS preprint 2579, Reliability Engineering & System Safety, 2020 - M. Eigel, M. Marschall, M. Multerer
An adaptive stochastic Galerkin tensor train discretization for randomly perturbed domains
WIAS preprint 2566, SIAM JUQ 2020 - M. Eigel, R. Gruhlke
A local hybrid surrogate-based finite element tearinginterconnecting dual-primal method for nonsmoothrandom partial differential equations
WIAS preprint 2565, Int J Numer Methods Eng., 2021 - M. Eigel, R. Schneider, P. Trunschke, S. Wolf
Variational Monte Carlo -- Bridging concepts of machine learning and high dimensional partial differential equations
WIAS preprint 2544, Adv. Comp. Math. 45, 2019 - M. Eigel, M. Marschall, M. Pfeffer, R. Schneider
Adaptive stochastic Galerkin FEM for log-normal coefficients in hierarchical tensor representations
WIAS preprint 2515, Numerische Mathematik 2020 - M. Eigel, M. Marschall, R. Schneider
Bayesian inversion with a hierarchical tensor representation
WIAS preprint 2363, Inverse Problems 2018 - M. Eigel, J. Neumann, R. Schneider, S. Wolf
Stochastic topology optimisation with hierarchical tensor reconstruction
WIAS preprint 2362, CMAM 2018 - C. Carstensen, M. Eigel
Reliable Averaging for the Primal Variable in the Courant FEM and Hierarchical Error Estimators on Red-Refined Meshes
WIAS preprint 2251, Comput. Methods Appl. Math. 2016 - M. Eigel, K. Sturm
Reproducing kernel Hilbert spaces and variable metric algorithms in PDE constrained shape optimisation
WIAS preprint 2244, OMS 2018 - F. Anker, Ch. Bayer, M. Eigel, M. Ladkau, J. Neumann, J.G. Schoenmakers
A fully adaptive interpolated stochastic sampling method for linear random PDEs
WIAS preprint 2200, IJUQ 2017 - M. Eigel, R. Müller
A posteriori error control for stationary coupled bulk-surface equations
WIAS preprint 2196, IMA J. Numerical Analysis 2017 - F. Anker, Ch. Bayer, M. Eigel, M. Ladkau, J. Neumann, J.G. Schoenmakers
SDE based regression for random PDEs
WIAS preprint 2192, SISC 2017 - M. Eigel, M. Pfeffer, R. Schneider
Adaptive stochastic Galerkin FEM with hierarchical tensor representations
WIAS preprint 2153, Numerische Mathematik 2017 - M. Eigel, Ch. Merdon, J. Neumann
An adaptive multi level Monte-Carlo method with stochastic bounds for quantities of interest in groundwater flow with uncertain data
WIAS preprint 2060, SIAM JUQ 2016 - M. Eigel, Ch. Merdon
Local equilibration error estimators for guaranteed error control in adaptive stochastic higher-order Galerkin FEM
WIAS preprint 1997, SIAM JUQ 2016 - W. Giese, M. Eigel, S. Westerheide, Ch. Engwer, E. Klipp
Influence of cell shape, inhomogeneities and diffusion barriers in cell polarization models
WIAS preprint 1959, Physical Biology 12, 6, 2015 - M. Eigel, C. J. Gittelson, Ch. Schwab, E. Zander
A convergent adaptive stochastic Galerkin finite element method with quasi-optimal spatial meshes
SAM Report 2014-01, ESAIM M2AN, 49 5, 2015 - M. Eigel, Ch. Merdon
Robust equilibration a posteriori error estimation for convection-diffusion-reaction problems
WIAS preprint 1822, J Sci Comput, 2015 - M. Eigel, C. J. Gittelson, Ch. Schwab, E. Zander
Adaptive Stochastic Galerkin FEM
SAM Report 2013-01, CMAME 270, 2014 - M. Eigel, T. Samrowski
Functional A Posteriori Error Estimation for Stationary Reaction-Convection-Diffusion Problems
WIAS preprint 1936, CMAM 14 2, 2014 -
M. Eigel, D. Peterseim
Simulation of composite materials by a Network FEM with error control
WIAS preprint 1833, CMAM 15 1, 2015 -
C. Carstensen, M. Eigel, R.H.W. Hoppe and C. Löbhard
A review of unified a posteriori finite element error control
Numer. Math. Theor. Meth. Appl. 5, 509-558, 2012 -
C. Carstensen, M. Eigel, and J. Gedicke
Computational competition of symmetric mixed FEM in linear elasticity
Comput. Methods Appl. Mech. Engrg., 200(41-44):2903 - 2915, 2011 -
M. Eigel, E. George and M. Kirkilionis
A mesh-free partition of unity method for diffusion equations on complex domains
IMA J. Numer. Anal. 30/3, 629-653, 2010 -
M. Kirkilionis, M. Domijan, M. Eigel, E. George, M. Li and L. Sbano
A definition of cellular interface problems
LNCS 5391, 36-62, 2009 -
M. Eigel, E. George and M. Kirkilionis
The Partition of Unity Meshfree Method for solving transport-reaction equations on complex domains: implementation and applications in the life sciences
LNCS 65, 69-93, 2008 -
D. Volz, M. Eigel, Ch. Athale, P. Bastian, H. Hermann, C. Kappel and R. Eils
Spatial modeling and simulation of diffusion in nuclei of living cells
Lecture Notes in Bioinformatics, LNBI, 3082: 161-71, 2005
Proceedings
-
J. Gedicke, C. Carstensen, and M. Eigel
Computational competition of symmetric mixed FEM in linear elasticity
In Mathematische Forschungsinstitut Oberwolfach. Report No. 09/2012: Advanced Computational Engineering, 12.2.-18.2.2012.
Thesis
-
M. Eigel,
An Adaptive Meshfree Method for Reaction-Diffusion Processes on Complex Domains,
University of Warwick, Ph.D. Thesis, 2008 -
M. Eigel,
Numerische Simulation von Transportvorgängen in der Zelle,
Universität Heidelberg, Diploma Thesis, 2003
Teaching
Uncertainty Quantification and Statistical Learning WS 2018/19 (with R. Schneider, TUB)
Realisierungen von Zufallsfeldern
(30.10.) Die Vorlesung findet diese Woche entgegen der Information im Vorlesungsverzeichnis statt!
(19.10.) Info Raumänderung: dienstags ab jetzt in E-N 193!
(22.12) Hier noch der letzte kleine Teil zur Vorlesung über a posteriori Fehlerschätzer. Alles Gute für 2019!
Referenzen
Uncertainty Quantification and Parametric PDEs
- [SG11] Schwab, Gittelson: Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs
- [CDS10] Cohen, DeVore, Schwab: Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs<\li>
- [CD15] Cohen, DeVore: Approximation of high-dimensional PDEs
- [LPS14] Lord, Powell, Shardlow: An introduction to computational stochastic PDEs
- [A15] Alexanderian: A brief note on the Karhunen-Loeve expansion
- [ES16] Ernst, Sprungk: Numerische Methoden der Unsicherheitsquantifizierung (Vorlesungsunterlagen, U Chemnitz)
- [EMSU12] Ernst, Mugler, Starkloff, Ullmann: On the convergence of generalized polynomial chaos
- [EGSZ] Eigel, Gittelson, Schwab, Zander: Adaptive Stochastic Galerkin FEM
Bayesian Inversion
- [S10] Stuart: Inverse problems: A Bayesian perspective
- [DS15] Dashti, Stuart: The Bayesian Approach to Inverse Problems
- [DS11] Dashti, Stuart: Uncertainty Quantification and weak approximation of an elliptic inverse problem
Statistical Learning, Hierarchical Tensors and Neural Nets
- [BSU17] Bachmayr, Schneider, Uschmajev: Tensor Networks and Hierarchical Tensors for the Solution of High-dimensional Partial Differential Equations
- [N15] Nouy: Low-rank methods for high-dimensional approximation and model order reduction
- [CS01] Cucker, Smale: One the mathematical foundations of learning
- [SZ18] Schwab, Zech: Deep learning in high dimension: Neural network expression rates for generalized polynomial chaos expansions in UQ
- [ESTW] Eigel, Schneider, Trunschke, Wolf: Variational Monte Carlo -- Bridging concepts of machine learning and high dimensional partial differential equations
Uncertainty Quantification and Tensor Approximation WS 2016/17 (with R. Schneider, TUB)
- Lecture notes (not revised, Mitschrift by Manuel Marschall)
- additional references