Parallel Computation of Stochastic Groundwater Flow

If a stochastic model is used to describe uncertainties, the physical system may be described by a stochastic partial differential equation (SPDE). A discretisation by a Galerkin ansatz with tensor-products of finite element functions and stochastic ansatz functions yields a large system of equations that can be efficiently solved by iterative methods. Due to its sheer size, parallel techniques are required, and we have implemented a ``hierarchical parallel solver'' for this: our solver uses a (possibly parallel) deterministic solver for the spatial discretisation. Coarser grained levels of parallelism are implemented by distributing the unknowns over the processors and by running different instances of the (possibly parallel) deterministic solver in parallel.

[1]  R. Adler,et al.  The Geometry of Random Fields , 1982 .

[2]  Hermann G. Matthies,et al.  Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations , 2005 .

[3]  Rainer Helmig,et al.  Conditional statistical inverse modeling in groundwater flow by multigrid methods , 1999 .

[4]  Hierarchical parallel solution of stochastic systems , 2003 .

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

[6]  Hermann G. Matthies,et al.  Multilevel solvers for the analysis of stochastic systems , 2001 .

[7]  Gedeon Dagan,et al.  Subsurface flow and transport : a stochastic approach , 1997 .

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

[9]  H. Matthies,et al.  Uncertainties in probabilistic numerical analysis of structures and solids-Stochastic finite elements , 1997 .

[10]  Hermann G. Matthies,et al.  Numerical Methods and Smolyak Quadrature for Nonlinear Stochastic Partial Differential Equations , 2003 .

[11]  George Christakos,et al.  Random Field Models in Earth Sciences , 1992 .

[12]  Erik H. Vanmarcke,et al.  Random Fields: Analysis and Synthesis. , 1985 .

[13]  Andreas Keese,et al.  Review of Recent Developments in the Numerical Solution of Stochastic Partial Differential Equations (Stochastic Finite Elements)A , 2003 .

[14]  G. Schuëller.  A state-of-the-art report on computational stochastic mechanics , 1997 .

[15]  William Gropp,et al.  Efficient Management of Parallelism in Object-Oriented Numerical Software Libraries , 1997, SciTools.