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Rapid solution of boundary value problems on stochastic domains

English title Rapid solution of boundary value problems on stochastic domains
Applicant Harbrecht Helmut
Number 137669
Funding scheme Project funding (Div. I-III)
Research institution Fachbereich Mathematik Departement Mathematik und Informatik Universität Basel
Institution of higher education University of Basel - BS
Main discipline Mathematics
Start/End 01.11.2011 - 30.09.2014
Approved amount 154'699.00
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Keywords (4)

uncertainty quantification; boundary value problem; stochastic domain; Karhunen-Loeve expansion

Lay Summary (English)

Lead
Lay summary

Nowadays, many problems in science and engineering can be numerically solved highly accurate provided that the input data are known exactly. Often, however, the input parameters are not given exactly. The practical significance of highly accurate numerical solution of differential equation models must thus address how to account for uncertain input data. If a statistical description of the input data is available, one can mathematically describe data and solutions as random fields and aim at computation of corresponding deterministic statistics of the unknown random solution. In the present project, we model and compute the solution to problems where the computational domain is uncertain.

Direct link to Lay Summary Last update: 21.02.2013

Responsible applicant and co-applicants

Employees

Name Institute

Publications

Publication
On the quasi-Monte Carlo quadrature with Halton points for elliptic PDEs with log-normal diffusion
Harbrecht Helmut, Peters Michael, Siebenmorgen Markus (2017), On the quasi-Monte Carlo quadrature with Halton points for elliptic PDEs with log-normal diffusion, in Mathematics of Computation, 86, 771-797.
Analysis of the domain mapping method for elliptic diffusion problems on random domains
Harbrecht Helmut, Peters Michael, Siebenmorgen Markus (2016), Analysis of the domain mapping method for elliptic diffusion problems on random domains, in Numerische Mathematik, 134(4), 823-856.
Combination technique based second moment analysis for elliptic PDEs on random domains
Harbrecht Helmut, Peters Michael (2016), Combination technique based second moment analysis for elliptic PDEs on random domains, in Sparse grids and applications - Stuttgart 2014, StuttgartSpringer International Publishing, Cham.
Multilevel accelerated quadrature for PDEs with log-normally distributed random coefficient.
Harbrecht Helmut, Peters Michael, Siebenmorgen Markus (2016), Multilevel accelerated quadrature for PDEs with log-normally distributed random coefficient., in SIAM/ASA J. Uncertain. Quantif., 4(1), 520-551.
Efficient approximation of random fields for numerical applications
Harbrecht Helmut, Peters Michael, Siebenmorgen Markus (2015), Efficient approximation of random fields for numerical applications, in Numer. Linear Algebra Appl., 22(4), 596-617.
Approximation of bi-variate functions: singular value decomposition versus sparse grids
Griebel Michael, Harbrecht Helmut (2014), Approximation of bi-variate functions: singular value decomposition versus sparse grids, in IMA Journal of Numerical Analysis, 34(1), 28-54.
Combination technique based k-th moment analysis of elliptic problems with random diffusion
Harbrecht Helmut, Peters Michael, Siebenmorgen Markus (2013), Combination technique based k-th moment analysis of elliptic problems with random diffusion, in J. Comput. Phys., 252, 128-141.
Comparison of fast boundary element methods on parametric surfaces.
Harbrecht Helmut, Peters Michael (2013), Comparison of fast boundary element methods on parametric surfaces., in Comput. Methods Appl. Mech. Engrg., 261-262, 39-55.
First order second moment analysis for stochastic interface problems based on low-rank approximation
Harbrecht Helmut, Li Jingzhi (2013), First order second moment analysis for stochastic interface problems based on low-rank approximation, in ESAIM Math. Model. Numer. Anal., 47(5), 1533-1552.

Collaboration

Group / person Country
Types of collaboration
Prof. Dr. Reinhold Schneider (TU Berlin) Germany (Europe)
- in-depth/constructive exchanges on approaches, methods or results
Prof. Dr. Christopf Schwab (ETH Zuerich) Switzerland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Prof. Dr. Michael Griebel (Uni Bonn) Germany (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication

Scientific events

Active participation

Title Type of contribution Title of article or contribution Date Place Persons involved
27th Chemnitz FEM Symposium Talk given at a conference Numerical solution of elliptic diffusion problems on random domains 22.09.2014 Chemnitz, Germany Peters Michael;
NASPDE 2014 Workshop (Numerical Analysis of Stochastic Partial Differential Equations) Talk given at a conference Numerical solution of elliptic diffusion problems on random domains 09.09.2014 Lausanne , Switzerland Harbrecht Helmut;
Sparse Grids and Applications Talk given at a conference Combination technique based k-th moment analysis of elliptic problems with random diffusion 01.09.2014 Stuttgart, Germany Peters Michael;
SIAM UQ 2014 (SIAM Conference on Uncertainty Quantification) Talk given at a conference Multilevel quadrature for elliptic stochastic partial differential equations 31.01.2014 Savannah, United States of America Harbrecht Helmut;
11th Workshop on fast boundary element methods in industrial applications Talk given at a conference H–matrix accelerated second moment analysis for elliptic problems with rough correlation 26.09.2013 Hirschegg, Austria Peters Michael;
Discrepancy, Numerical Integration and Hyperbolic Cross Approximation Talk given at a conference Multilevel quadrature for elliptic stochastic partial differential equations 23.09.2013 Bonn, Germany Harbrecht Helmut;
ENUMATH 2013 Talk given at a conference Multilevel quadrature for elliptic stochastic partial differential equations 26.08.2013 Lausanne, Switzerland Harbrecht Helmut;
Multiscale and High-Dimensional Problems Talk given at a conference Multilevel quadrature for elliptic stochastic partial differential equations 08.07.2013 Oberwolfach, Germany Harbrecht Helmut;
Mathematisches Kolloquium am Institut fuer Mathematik Individual talk Modelling and simulation of elliptic PDEs on random domains 24.06.2013 Paderborn, Germany Harbrecht Helmut;
Numerical Methods for Uncertainty Quantification Talk given at a conference On multilevel quadrature for elliptic stochastic partial differential equations 13.05.2013 Bonn, Germany Harbrecht Helmut;
SAM Kolloquium an der ETH Zuerich Individual talk Modelling and simulation of elliptic PDEs on random domains 24.04.2013 Zuerich, Switzerland Harbrecht Helmut;
Numerical Methods for PDE Constrained Optimization with Uncertain Data Talk given at a conference Modelling and simulation of elliptic PDEs on random domains 28.01.2013 Oberwolfach, Germany Harbrecht Helmut;
WONAPDE 2013 (Fourth Chilean Workshop on Numerical Analysis of Partial Differential Equations) Talk given at a conference On multilevel quadrature for elliptic stochastic partial differential equations. 14.01.2013 Concepcion, Chile Harbrecht Helmut;
IABEM 2013 (Symposium of the International Association for Boundary Element Methods) Talk given at a conference Comparison of fast boundary element methods on parametric surfaces 09.01.2013 Santiago de Chile, Chile Harbrecht Helmut;
10th Workshop on Fast Boundary Element Methods in Industrial Applications Talk given at a conference Comparison of Fast Boundary Element Methods on Parametric Surfaces 27.09.2012 Hirschegg, Austria Peters Michael;
25th Chemnitz FEM-Symposium 2012 Talk given at a conference Combination technique based k-th moment analysis of elliptic problems with random diffusion 24.09.2012 Chemnitz, Germany Harbrecht Helmut;
ESCO 2012 (European Seminar on Computing) Talk given at a conference A fast deterministic method for stochastic interface problems 25.06.2012 Pilsen, Czech Republic Harbrecht Helmut;
Institutsvortrag am Fachbereich fuer Mathematik und Statistik Individual talk Gebietsvariationen in Optimierung und Ergebnisverifikation 08.12.2011 Konstanz, Germany Harbrecht Helmut;


Associated projects

Number Title Start Funding scheme
169599 Multilevel Methods and Uncertainty Quantification in Cardiac Electrophysiology 01.10.2016 Project funding (Div. I-III)
156101 H-matrix based first and second moment analysis 01.10.2014 Project funding (Div. I-III)

Abstract

We develop an efficient algorithm to compute the Karhunen-Loeve expansion of stochastic fields with nonsmooth covariance kernels. Since the eigenvalues of the covariance operator do not decay exponentially, a large number of eigenpairs need to be approximated. We will employ fast methods for nonlocal operators as for example known from boundary integral equation methods. Then, each eigenpair can be computed in essentially linear complexity.The Karhunen-Loeve expansion is applied to compute the approximate solutions of partial differential equations on stochastic domains. We considerboth, a domain transformation technique and a linearization based on the local shape derivative. Both approaches lead to a boundary value problemon a fixed nominal domain but with stochastic right hand side and/or stochastic coefficients. Stochastic Galerkin and collocation methods will be employed to solve the related boundary value problems.
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