 |
|
||||||||||
Sparse Tensor Approximations of High-Dimensional
and stochastic Partial Differential Equations
| Principal Investigator: | Prof. C. Schwab |
| Researchers: |
Andrea Barth (Seminar for Applied Mathematics, ETH Zurich) Phillip Grohs (Seminar for Applied Mathematics, ETH Zurich) Markus Hansen (Seminar for Applied Mathematics, ETH Zurich) Annika Lang (Seminar for Applied Mathematics, ETH Zurich) |
| Project period: | 01.01.2010 to 31.12.2014 (60 months) |
| Call reference: | ERC-2009-AdG_20090325 |
Summary
The present project addresses numerical analysis and algorithmic realization of sparse, adaptive
tensor product discretizations of partial di fferential equations (PDEs) in high dimensions
with stochastic data. The aim of the project is to develop mathematically founded adaptive
algorithms which are based on sparse tensorization of hierarchic Riesz bases or frames. These
will be hierarchic multilevel bases in the physical domain, either Finite Element wavelet type
bases or hierarchical, multilevel Finite Element frames. In the parameter domains corresponding
either to random inputs or to phase spaces in transport problems, spectral representations
of "polynomial chaos" type shall be employed. The mathematical aim is to obtain for broad
classes of elliptic and parabolic and certain hyperbolic PDEs on high or possibly infi nite dimensional
parameter spaces adaptive, deterministic and dimension independent numerical solution
methods with convergence rates superior to those afforded by sampling Methods, in terms of
accuracy vs. complexity. Algorithmic work in the project will address design of datastructures
with minimal overhead for the efficient realization of the sparse tensor approximations.
Applications include space-time adaptive solvers for elliptic, parabolic and certain parametric
hyperbolic PDEs. Applications include numerical solution of multiscale models in homogenization,
macro-micro polymer models in material science, stochastic di fferential equations arising
in bioengineering, Uncertainty Quanti cation and economics.
Mathematically, new results are expected on nonlinear approximate spectral representations
of nonstationary random fi elds, scale-resolving discretizations of multiscale elliptic and parabolic
problems featuring complexity independent of the number of scales. The project will be in
collaboration with coworkers in France, Germany, Singapore, UK, and The Netherlands. The
project involves mentoring postdocs and predocs who will be actively involved in all aspects of
the research, as well as a sustained graduate teaching component.
Research Reports
2011
- F.Y. Kuo, Ch. Schwab and I.H. Sloan, Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients, SAM Report 2011/52
http://www.sam.math.ethz.ch/reports/2011/52 - A. Chernov and Ch. Schwab, First order k-th moment finite element analysis of nonlinear operator equations with stochastic data, SAM Report 2011/51
http://www.sam.math.ethz.ch/reports/2011/51 - K. Grella and Ch. Schwab, Sparse discrete ordinates method in radiative transfer, SAM Report 2011/46
http://www.sam.math.ethz.ch/reports/2011/46 - A. Chkifa, A. Cohen, R. DeVore and Ch. Schwab, Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs, SAM Report 2011/44
http://www.sam.math.ethz.ch/reports/2011/44 - Ph. Grohs and Ch. Schwab, Sparse twisted tensor frame discretization of parametric transport operators, SAM Report 2011/41
http://www.sam.math.ethz.ch/reports/2011/41 - C.J. Gittelson, Adaptive wavelet methods for elliptic partial differential equations with random operators, SAM Report 2011/37
http://www.sam.math.ethz.ch/reports/2011/37 - A. Barth and A. Lang, Milstein approximation for advection-diffusion equations driven by multiplicative noncontinuous martingale noises, SAM Report 2011/36
http://www.sam.math.ethz.ch/reports/2011/36 - A. Lang, Almost sure convergence of a Galerkin approximation for SPDEs of Zakai type driven by square integrable martingales, SAM Report 2011/35
http://www.sam.math.ethz.ch/reports/2011/35 - J. Sukys, S. Mishra and Ch. Schwab, Static load balancing for multi-level Monte Carlo finite volume solvers, SAM Report 2011/32
http://www.sam.math.ethz.ch/reports/2011/32 - C.J. Gittelson, J. Könnö, Ch. Schwab and R. Stenberg, The multi-level Monte Carlo Finite Element Method for a stochastic Brinkman problem, SAM Report 2011/31
http://www.sam.math.ethz.ch/reports/2011/31 - A. Barth, A. Lang and Ch. Schwab, Multi-level Monte Carlo Finite Element method for parabolic stochastic partial differential equations, SAM Report 2011/30
http://www.sam.math.ethz.ch/reports/2011/30 - M. Hansen and Ch. Schwab, Analytic regularity and nonlinear approximation of a class of parametric semilinear elliptic PDEs, SAM Report 2011/29
http://www.sam.math.ethz.ch/reports/2011/29 - Ph. Grohs, Shearlets and microlocal analysis, SAM Report 2011/27
http://www.sam.math.ethz.ch/reports/2011/27 - C.J. Gittelson, Uniformly convergent adaptive methods for parametric operator equations, SAM Report 2011/19
http://www.sam.math.ethz.ch/reports/2011/19 - D. Marazzina, O. Reichmann and Ch. Schwab
hp-DGFEM for Kolmogorov-Fokker-Planck equations of multivariate Lévy processes, SAM Report 2011/17
http://www.sam.math.ethz.ch/reports/2011/17 - Ch. Schwab and A.M. Stuart, Sparse deterministic approximation of Bayesian inverse problems, SAM Report 2011/16
http://www.sam.math.ethz.ch/reports/2011/16 - A. Barth and A. Lang, $L^p$ and almost sure convergence of a Milstein scheme for stochastic partial differential equations, SAM Report 2011/15
http://www.sam.math.ethz.ch/reports/2011/15 - C.J. Gittelson, Adaptive stochastic Galerkin methods: beyond the elliptic case, SAM Report 2011/12
http://www.sam.math.ethz.ch/reports/2011/12 - C.J. Gittelson, An adaptive stochastic Galerkin method, SAM Report 2011/11
http://www.sam.math.ethz.ch/reports/2011/11 - C.J. Gittelson, Stochastic Galerkin approximation of operator equations with infinite dimensional noise, SAM Report 2011/10
http://www.sam.math.ethz.ch/reports/2011/10 - W. Dahmen, C. Huang, Ch. Schwab and G. Welper, Adaptive Petrov-Galerkin methods for first order transport equations, SAM Report 2011/08
http://www.sam.math.ethz.ch/reports/2011/08 - V.H. Hoang and Ch. Schwab, Analytic regularity and polynomial approximation of stochastic, parametric elliptic multiscale PDEs, SAM Report 2011/07
http://www.sam.math.ethz.ch/reports/2011/07 - S. Mishra, Ch. Schwab and J. Šukys, Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions, SAM Report 2011/02
http://www.sam.math.ethz.ch/reports/2011/02
2010
- P. Grohs, Ridgelet-type frame decompositions for Sobolev spaces related to linear transport, SAM Report 2010/46
http://www.sam.math.ethz.ch/reports/2010/46 - P. Grohs, Tree approximation and optimal image coding with shearlets, SAM Report 2010/45
http://www.sam.math.ethz.ch/reports/2010/45 - P. Grohs, Tree approximation with anisotropic decompositions, SAM Report 2010/44
http://www.sam.math.ethz.ch/reports/2010/44 - R. Andreev and Ch. Schwab, Sparse tensor approximation of parametric eigenvalue problems, SAM Report 2010/40
http://www.sam.math.ethz.ch/reports/2010/40 - H. Harbrecht and Ch. Schwab, Sparse tensor finite elements for elliptic multiple scale problems, SAM Report 2010/34
http://www.sam.math.ethz.ch/reports/2010/34 - K. Grella and Ch. Schwab, Sparse tensor spherical harmonics approximation in radiative transfer, SAM Report 2010/33
http://www.sam.math.ethz.ch/reports/2010/33 - M. Hansen, On tensor products of quasi-Banach spaces, SAM Report 2010/31
http://www.sam.math.ethz.ch/reports/2010/31 - S. Mishra and Ch. Schwab, Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random intitial data, SAM Report 2010/24
http://www.sam.math.ethz.ch/reports/2010/24 - V.H. Hoang and Ch. Schwab, Regularity and generalized polynomial chaos approximation of parametric and random 2nd order hyperbolic partial differential equations, SAM Report 2010/19
http://www.sam.math.ethz.ch/reports/2010/19 - A. Barth, C. Schwab and N. Zollinger, Multi-Level Monte Carlo Finite Element method for elliptic PDE's with stochastic coefficients, SAM Report 2010/18
http://www.sam.math.ethz.ch/reports/2010/18 - C.J. Gittelson, Representation of Gaussian fields in series with independent coefficients, SAM Report 2010/15
http://www.sam.math.ethz.ch/reports/2010/15 - H. Brandsmeier, K. Schmidt and Ch. Schwab, A multiscale hp-FEM for 2D photonic crystal band, SAM Report 2010/12
http://www.sam.math.ethz.ch/reports/2010/12 - V.H. Hoang and Ch. Schwab, Sparse tensor Galerkin discretizations for parametric and random parabolic PDEs. I: Analytic regularity and gpc-approximation, SAM Report 2010/11
http://www.sam.math.ethz.ch/reports/2010/11 - B. Pentenrieder and Ch. Schwab, hp-FEM for second moments of elliptic PDEs with stochastic data. Part 2: Exponential convergence, SAM Report 2010/09
http://www.sam.math.ethz.ch/reports/2010/09 - B. Pentenrieder and Ch. Schwab, hp-FEM for second moments of elliptic PDEs with stochastic data.
Part 1: Analytic regularity, SAM Report 2010/08
http://www.sam.math.ethz.ch/reports/2010/08 - C. Schwab and O. Reichmann, Numerical analysis of additive, Lévy and Feller processes with applications to option pricing, SAM Report 2010/06
http://www.sam.math.ethz.ch/reports/2010/06 - C. Schwab and R. Stevenson, Fast evaluation of nonlinear functionals of tensor product wavelet expansions, SAM Report 2010/05
http://www.sam.math.ethz.ch/reports/2010/05 - B.N. Khoromskij and C. Schwab, Tensor-structured Galerkin approximation of parametric and stochastic elliptic PDEs, SAM Report 2010/04
http://www.sam.math.ethz.ch/reports/2010/04 - A. Cohen, R. DeVore and C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs, SAM Report 2010/03
http://www.sam.math.ethz.ch/reports/2010/03 - N. Hilber, S. Kehtari, C. Schwab and C. Winter, Wavelet finite element method for option pricing in highdimensional diffusion market models, SAM Report 2010/01
http://www.sam.math.ethz.ch/reports/2010/01
Contact
Wichtiger Hinweis:
Diese Website wird in älteren Versionen von Netscape ohne
graphische Elemente dargestellt. Die Funktionalität der
Website ist aber trotzdem gewährleistet. Wenn Sie diese
Website regelmässig benutzen, empfehlen wir Ihnen, auf
Ihrem Computer einen aktuellen Browser zu installieren. Weitere
Informationen finden Sie auf
folgender
Seite.
Important Note:
The content in this site is accessible to any browser or
Internet device, however, some graphics will display correctly
only in the newer versions of Netscape. To get the most out of
our site we suggest you upgrade to a newer browser.
More
information
