Article2011

A Sparse Composite Collocation Finite Element Method for Elliptic SPDEs.

43 citations0/0 files storedSIAM Journal on Numerical Analysis

No readable copy stored yet — this record is metadata only.

No open-access location is known for this record yet.

Literature note Cite in a project Citation map

Probabilistic and Robust Engineering Design Advanced Numerical Methods in Computational Mathematics Mathematical Approximation and Integration

Abstract

This work presents a stochastic collocation method for solving elliptic PDEs with random coefficients and forcing term which are assumed to depend on a finite number of random variables. The method consists of a hierarchic wavelet discretization in space and a sequence of hierarchic collocation operators in the probability domain to approximate the solution's statistics. The selection of collocation points is based on a Smolyak construction of zeros of orthogonal polynomials with respect to the probability density function of each random input variable. A sparse composition of levels of spatial refinements and stochastic collocation points is then proposed and analyzed, resulting in a substantial reduction of overall degrees of freedom. Like in the Monte Carlo approach, the algorithm results in solving a number of uncoupled, purely deterministic elliptic problems, which allows the integration of existing fast solvers for elliptic PDEs. Numerical examples on two-dimensional domains will then demonstrate the superiority of this sparse composite collocation finite element method compared to the “full composite” collocation finite element method and the Monte Carlo method.

Paper brief

Loading…

Cite this work

Citation unavailable for this work.

BibTeX RIS CSL-JSON

No public copy is stored for this record, so only the metadata is cited.

Citations in this library

Citation map →

Stored only when both papers are in the corpus — 43 citations exist upstream.

References

1991Stochastic Finite Elements: A Spectral Approach 2005High-Order Collocation Methods for Differential Equations with Random Inputs 2007A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data 2008A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data 2008An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data 2001Solution of stochastic partial differential equations using Galerkin finite element techniques 2007Sparse grid collocation schemes for stochastic natural convection problems 2006Karhunen–Loève approximation of random fields by generalized fast multipole methods

Cited by

2016Multilevel Accelerated Quadrature for PDEs with Log-Normally Distributed Diffusion Coefficient 2015Approximate methods for stochastic eigenvalue problems 2016Multi-index Stochastic Collocation Convergence Rates for Random PDEs with Parametric Regularity 2017Stochastic Collocation Methods via $\ell_1$ Minimization Using Randomized Quadratures

Explore the map →

Papers that cite the same references — the relational fallback while embeddings are still backfilling.

Imported from OpenAlex (CC0) — nothing here is generated.