uncertainty quantification; boundary value problem; stochastic domain; Karhunen-Loeve expansion
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.
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.
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.
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.
Harbrecht Helmut, Peters Michael, Siebenmorgen Markus (2015), Efficient approximation of random fields for numerical applications, in Numer. Linear Algebra Appl.
, 22(4), 596-617.
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.
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.
Harbrecht Helmut, Peters Michael (2013), Comparison of fast boundary element methods on parametric surfaces., in Comput. Methods Appl. Mech. Engrg.
, 261-262, 39-55.
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.
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.