A heterogeneous stochastic FEM framework for elliptic PDEs

We introduce a new concept of sparsity for the stochastic elliptic operator - div ( a ( x , ω ) ? ( ? ) ) , which reflects the compactness of its inverse operator in the stochastic direction and allows for spatially heterogeneous stochastic structure. This new concept of sparsity motivates a heterogeneous stochastic finite element method (HSFEM) framework for linear elliptic equations, which discretizes the equations using the heterogeneous coupling of spatial basis with local stochastic basis to exploit the local stochastic structure of the solution space. We also provide a sampling method to construct the local stochastic basis for this framework using the randomized range finding techniques. The resulting HSFEM involves two stages and suits the multi-query setting: in the offline stage, the local stochastic structure of the solution space is identified; in the online stage, the equation can be efficiently solved for multiple forcing functions. An online error estimation and correction procedure through Monte Carlo sampling is given. Numerical results for several problems with high dimensional stochastic input are presented to demonstrate the efficiency of the HSFEM in the online stage.

[1]  N. Cutland,et al.  On homogeneous chaos , 1991, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  Houman Owhadi,et al.  A non-adapted sparse approximation of PDEs with stochastic inputs , 2010, J. Comput. Phys..

[3]  Fabio Nobile,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007, SIAM Rev..

[4]  Marcel Bieri,et al.  A Sparse Composite Collocation Finite Element Method for Elliptic SPDEs , 2011, SIAM J. Numer. Anal..

[5]  Dongbin Xiu,et al.  High-Order Collocation Methods for Differential Equations with Random Inputs , 2005, SIAM J. Sci. Comput..

[6]  Wolfgang Hackbusch,et al.  A Sparse Matrix Arithmetic Based on H-Matrices. Part I: Introduction to H-Matrices , 1999, Computing.

[7]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[8]  Per-Gunnar Martinsson,et al.  Randomized algorithms for the low-rank approximation of matrices , 2007, Proceedings of the National Academy of Sciences.

[9]  Olaf Steinbach,et al.  Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements , 2007 .

[10]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[11]  Raúl Tempone,et al.  Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations , 2004, SIAM J. Numer. Anal..

[12]  Henry Stark,et al.  Probability, Random Processes, and Estimation Theory for Engineers , 1995 .

[13]  S. Ravindran,et al.  A Reduced-Order Method for Simulation and Control of Fluid Flows , 1998 .

[14]  Roger Ghanem,et al.  Stochastic model reduction for chaos representations , 2007 .

[15]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[16]  Michał Kleiber,et al.  The Stochastic Finite Element Method: Basic Perturbation Technique and Computer Implementation , 1993 .

[17]  C. Schwab,et al.  Sparse high order FEM for elliptic sPDEs , 2009 .

[18]  Peter E. Thornton,et al.  DIMENSIONALITY REDUCTION FOR COMPLEX MODELS VIA BAYESIAN COMPRESSIVE SENSING , 2014 .

[19]  Thomas Y. Hou,et al.  Wiener Chaos expansions and numerical solutions of randomly forced equations of fluid mechanics , 2006, J. Comput. Phys..

[20]  P. Frauenfelder,et al.  Finite elements for elliptic problems with stochastic coefficients , 2005 .

[21]  Christoph Schwab,et al.  Sparse Tensor Discretization of Elliptic sPDEs , 2009, SIAM J. Sci. Comput..

[22]  Thomas Y. Hou,et al.  A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations I: Derivation and algorithms , 2013, J. Comput. Phys..

[23]  Nathan Halko,et al.  Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions , 2009, SIAM Rev..

[24]  Fabio Nobile,et al.  A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data , 2008, SIAM J. Numer. Anal..

[25]  Thomas A. Zang,et al.  Stochastic approaches to uncertainty quantification in CFD simulations , 2005, Numerical Algorithms.

[26]  Fabio Nobile,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2010, SIAM Rev..

[27]  Michael Grüter,et al.  The Green function for uniformly elliptic equations , 1982 .

[28]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[29]  Kyle A. Gallivan,et al.  A compressed sensing approach for partial differential equations with random input data , 2012 .

[30]  P. L’Ecuyer,et al.  Random Number Generation and Quasi-Monte Carlo† , 2015 .

[31]  Zhiwen Zhang,et al.  A Data-Driven Stochastic Method for Elliptic PDEs with Random Coefficients , 2013, SIAM/ASA J. Uncertain. Quantification.

[32]  Boris N. Khoromskij,et al.  A Sparse H-Matrix Arithmetic. Part II: Application to Multi-Dimensional Problems , 2000, Computing.

[33]  Lexing Ying,et al.  Fast construction of hierarchical matrix representation from matrix-vector multiplication , 2009, J. Comput. Phys..

[34]  Thomas Y. Hou,et al.  A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations II: Adaptivity and generalizations , 2013, J. Comput. Phys..

[35]  Ivo Babuška,et al.  On solving elliptic stochastic partial differential equations , 2002 .

[36]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[37]  W. Hackbusch.  A Sparse Matrix Arithmetic Based on $\Cal H$-Matrices. Part I: Introduction to ${\Cal H}$-Matrices , 1999, Computing.

[38]  R. Caflisch.  Monte Carlo and quasi-Monte Carlo methods , 1998, Acta Numerica.

[39]  W. T. Martin,et al.  The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals , 1947 .

[40]  M. Lemaire,et al.  Stochastic Finite Elements , 2010 .

[41]  R. M. Brown,et al.  The Green Function for Elliptic Systems in Two Dimensions , 2012, 1205.1089.