Schwarz preconditioners for stochastic elliptic PDEs

Abstract Increasingly the spectral stochastic finite element method (SSFEM) has become a popular computational tool for uncertainty quantification in numerous practical engineering problems. For large-scale problems however, the computational cost associated with solving the arising linear system in the SSFEM still poses a significant challenge. The development of efficient and robust preconditioned iterative solvers for the SSFEM linear system is thus of paramount importance for uncertainty quantification of large-scale industrially relevant problems. In the context of high performance computing, the preconditioner must scale to a large number of processors. Therefore in this paper, a two-level additive Schwarz preconditioner is described for the iterative solution of the SSFEM linear system. The proposed preconditioner can be viewed as a generalization of the mean based block-diagonal preconditioner commonly used in the literature. For the numerical illustrations, two-dimensional steady-state diffusion and elasticity problems with spatially varying random coefficients are considered. The performance of the algorithm is investigated with respect to the geometric parameters, strength of randomness, dimension and order of the stochastic expansion.

[1]  S. Adhikari,et al.  A Reduced Spectral Projection Method for Stochastic Finite Element Analysis , 2011 .

[2]  Howard C. Elman,et al.  ASSESSMENT OF COLLOCATION AND GALERKIN APPROACHES TO LINEAR DIFFUSION EQUATIONS WITH RANDOM DATA , 2011 .

[3]  I. Babuska,et al.  Solution of stochastic partial differential equations using Galerkin finite element techniques , 2001 .

[4]  T. Chan,et al.  Domain decomposition algorithms , 1994, Acta Numerica.

[5]  Andy J. Keane,et al.  Comparative study of projection schemes for stochastic finite element analysis , 2006 .

[6]  Roger Ghanem,et al.  Numerical solution of spectral stochastic finite element systems , 1996 .

[7]  S. Adhikari,et al.  A reduced polynomial chaos expansion method for the stochastic finite element analysis , 2012 .

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

[9]  M. Eldred,et al.  Comparison of Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Quantification , 2009 .

[10]  I. M. Soboĺ.  Quasi-Monte Carlo methods , 1990 .

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

[12]  Tarek P. Mathew,et al.  Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations , 2008, Lecture Notes in Computational Science and Engineering.

[13]  Stefan Vandewalle,et al.  Iterative Solvers for the Stochastic Finite Element Method , 2008, SIAM J. Sci. Comput..

[14]  O. L. Maître,et al.  Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics , 2010 .

[15]  Timothy Walsh,et al.  Experiences with FETI-DP in a Production Level Finite Element Application , 2002 .

[16]  Waad Subber,et al.  Domain Decomposition Methods of Stochastic PDEs , 2013, Domain Decomposition Methods in Science and Engineering XX.

[17]  Hermann G. Matthies,et al.  Parallel Computation of Stochastic Groundwater Flow , 2003 .

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

[19]  Habib N. Najm,et al.  A multigrid solver for two-dimensional stochastic diffusion equations , 2003 .

[20]  Ronald L. Iman.  Latin Hypercube Sampling , 2008 .

[21]  Barry Smith,et al.  Domain Decomposition Methods for Partial Differential Equations , 1997 .

[22]  Elisabeth Ullmann,et al.  A Kronecker Product Preconditioner for Stochastic Galerkin Finite Element Discretizations , 2010, SIAM J. Sci. Comput..

[23]  D. Xiu.  Fast numerical methods for stochastic computations: A review , 2009 .

[24]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[25]  A. Klawonn,et al.  Highly scalable parallel domain decomposition methods with an application to biomechanics , 2010 .

[26]  Charbel Farhat,et al.  A FETI‐preconditioned conjugate gradient method for large‐scale stochastic finite element problems , 2009 .

[27]  Andrea Toselli,et al.  Domain decomposition methods : algorithms and theory , 2005 .

[28]  Ionel Michael Navon,et al.  Domain decomposition and parallel processing of a finite element model of the shallow water equations , 1993 .

[29]  Andy J. Keane,et al.  Hybridization of stochastic reduced basis methods with polynomial chaos expansions , 2006 .

[30]  William Gropp,et al.  Parallel computing and domain decomposition , 1992 .

[31]  A. Sarkar,et al.  A Domain Decomposition Method of Stochastic PDEs: An Iterative Solution Technique Using A Two-level , 2013 .

[32]  Martin J. Gander,et al.  Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods , 2012 .

[33]  BabuskaIvo,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007 .

[34]  Waad Subber,et al.  Dual-primal domain decomposition method for uncertainty quantification , 2013 .

[35]  S. Adhikari.  A reduced spectral function approach for the stochastic finite element analysis , 2011 .

[36]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[37]  R. Ghanem,et al.  A stochastic projection method for fluid flow. I: basic formulation , 2001 .

[38]  Roger G. Ghanem,et al.  On the construction and analysis of stochastic models: Characterization and propagation of the errors associated with limited data , 2006, J. Comput. Phys..

[39]  A. Sarkar,et al.  Domain decomposition of stochastic PDEs: Theoretical formulations , 2009 .

[40]  I. Babuska,et al.  Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation , 2005 .

[41]  O. Widlund.  Domain Decomposition Algorithms , 1993 .

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

[43]  Waad Subber,et al.  Domain Decomposition of Stochastic PDEs: A Novel Preconditioner and Its Parallel Performance , 2009, HPCS.

[44]  D. Xiu.  Efficient collocational approach for parametric uncertainty analysis , 2007 .

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

[46]  K. A. Cliffe,et al.  Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients , 2011, Comput. Vis. Sci..

[47]  Elisabeth Ullmann,et al.  Computational aspects of the stochastic finite element method , 2007 .

[48]  I. Sobol,et al.  On quasi-Monte Carlo integrations , 1998 .

[49]  Andrea Barth,et al.  Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients , 2011, Numerische Mathematik.

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

[51]  D. Xiu.  Numerical Methods for Stochastic Computations: A Spectral Method Approach , 2010 .

[52]  R. Ghanem,et al.  Iterative solution of systems of linear equations arising in the context of stochastic finite elements , 2000 .

[53]  H. Matthies,et al.  Hierarchical parallelisation for the solution of stochastic finite element equations , 2005 .

[54]  H. Najm,et al.  Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection , 2003 .

[55]  D. Xiu,et al.  Modeling uncertainty in flow simulations via generalized polynomial chaos , 2003 .

[56]  Howard C. Elman,et al.  Solving the Stochastic Steady-State Diffusion Problem using , 2006 .

[57]  Habib N. Najm,et al.  Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics , 2009 .

[58]  A. Keane,et al.  Stochastic Reduced Basis Methods , 2002 .

[59]  Waad Subber,et al.  A domain decomposition method of stochastic PDEs: An iterative solution techniques using a two-level scalable preconditioner , 2014, J. Comput. Phys..

[60]  R. Ghanem,et al.  Polynomial Chaos in Stochastic Finite Elements , 1990 .

[61]  Barry F. Smith,et al.  Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations , 1996 .

[62]  Ina Ruck,et al.  USA , 1969, The Lancet.

[63]  Stefan Vandewalle,et al.  Algebraic multigrid for stationary and time‐dependent partial differential equations with stochastic coefficients , 2008, Numer. Linear Algebra Appl..

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

[65]  Howard C. Elman,et al.  Block-diagonal preconditioning for spectral stochastic finite-element systems , 2008 .