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Keywords: stochastic finite elements, distributed memory, machine, hierarchical parallel Krylov subspace solver.

Summary

The example application is an elliptic partial differential equation for steady groundwater flow. Uncertainties in the conductivity may be quantified with a stochastic model. A discretisation by a Galerkin Ansatz with tensor products of finite element functions in space and stochastic ansatz functions leads to a certain type of stochastic finite element system (SFEM). This yields a large system of equations that can be efficiently solved by Krylov subspace methods. Due to its sheer size, parallel techniques are required, and we have implemented a "hierarchical parallel solver" on a distributed memory architecture 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 deterministic solver in parallel. Different possibilities for the distribution of data are investigated, and the efficiencies determined. On upto 128 processors systems with more than unknowns are solved.

Uncertainties remain in all models of the physical reality, and the quality of numerical prognoses is limited by the information available about the system. If prognoses are computed with an accuracy higher than merited by the available information, then the computing power at hand might be put to use more effectively by computing quantitative estimates for the uncertainties in the response.

Various parallelisation techniques are presented and tested. They allow to configure the parallelisation such that it is appropriate for the problem at hand: To work with problems that have a high number of spatial degrees of freedom the deterministic solvers may be a parallel program of which different instances are run in parallel on each processor group. The coarser levels of parallelism use these processor groups as smallest building blocks and thus allow to use large stochastic ansatz spaces and to speed up the solution.

The efficiency measurements shown here are computed for a fixed problem size. For a problem size scaled with the number of processors only the timings are shown as it is not clear how the effort of the block matrix-vector product increases with the number of unknowns.

It is shown that the parallel solver allows to tackle very large problems. Depending on the type of parallelisation, very good parallel efficiencies are obtained, and the hierarchical parallelism permits to adapt the solver to the properties of the discretisation at hand.

The example presented here looks simple, but the stochastic uncertainties increase its complexity considerably. The solver may integrate existing codes for the spatial discretisation and hence may be applied to more complex problems. Our goal is to implement a general-purpose version of the stochastic finite element method that may be applied to real-life problems, and as the resulting systems of equations are large, the parallel solver presented here is an important step towards this goal [4].

References

1: R. Ghanem and P. D. Spanos: "Stochastic Finite Elements--A Spectral Approach", Springer-Verlag, Berlin, 1991.
2: A. Keese: "A review of recent developments in the numerical solution of stochastic PDEs (stochastic finite elements)" Informatikbericht 2003-6, Technische Universität Braunschweig, Braunschweig, 2003. http://opus.tu-bs.de/opus/volltexte/2003/504/.
3: H. G. Matthies and A. Keese: "Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations", accepted for publication, Comp. Meth. Appl. Mech. Engrg., 2004. doi:10.1016/j.cma.2004.05.027
4: A. Keese: "Numerical Solution of Systems with Stochastic Uncertainties--A General Purpose Framework for Stochastic Finite Elements", Doctoral thesis, Technische Universität Braunschweig, 2003.
5: A. Keese and H. G. Matthies: "The hierarchical parallel solution of stochastic finite element equations." submitted to Computers & Structures, 2004.

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 79 PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and C.A. Mota Soares Paper 105 A Hierarchical Parallel Solver for Stochastic Finite Element Equations H.G. Matthies and A. Keese Institute of Scientific Computing, Technische Universität Braunschweig, Germany doi:10.4203/ccp.79.105 purchase the full-text of this paper Full Bibliographic Reference for this paper H.G. Matthies, A. Keese, "A Hierarchical Parallel Solver for Stochastic Finite Element Equations", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Seventh International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 105, 2004. doi:10.4203/ccp.79.105 Keywords: stochastic finite elements, distributed memory, machine, hierarchical parallel Krylov subspace solver. Summary The example application is an elliptic partial differential equation for steady groundwater flow. Uncertainties in the conductivity may be quantified with a stochastic model. A discretisation by a Galerkin Ansatz with tensor products of finite element functions in space and stochastic ansatz functions leads to a certain type of stochastic finite element system (SFEM). This yields a large system of equations that can be efficiently solved by Krylov subspace methods. Due to its sheer size, parallel techniques are required, and we have implemented a "hierarchical parallel solver" on a distributed memory architecture 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 deterministic solver in parallel. Different possibilities for the distribution of data are investigated, and the efficiencies determined. On upto 128 processors systems with more than unknowns are solved. Uncertainties remain in all models of the physical reality, and the quality of numerical prognoses is limited by the information available about the system. If prognoses are computed with an accuracy higher than merited by the available information, then the computing power at hand might be put to use more effectively by computing quantitative estimates for the uncertainties in the response. Various parallelisation techniques are presented and tested. They allow to configure the parallelisation such that it is appropriate for the problem at hand: To work with problems that have a high number of spatial degrees of freedom the deterministic solvers may be a parallel program of which different instances are run in parallel on each processor group. The coarser levels of parallelism use these processor groups as smallest building blocks and thus allow to use large stochastic ansatz spaces and to speed up the solution. The efficiency measurements shown here are computed for a fixed problem size. For a problem size scaled with the number of processors only the timings are shown as it is not clear how the effort of the block matrix-vector product increases with the number of unknowns. It is shown that the parallel solver allows to tackle very large problems. Depending on the type of parallelisation, very good parallel efficiencies are obtained, and the hierarchical parallelism permits to adapt the solver to the properties of the discretisation at hand. The example presented here looks simple, but the stochastic uncertainties increase its complexity considerably. The solver may integrate existing codes for the spatial discretisation and hence may be applied to more complex problems. Our goal is to implement a general-purpose version of the stochastic finite element method that may be applied to real-life problems, and as the resulting systems of equations are large, the parallel solver presented here is an important step towards this goal [4]. References 1 R. Ghanem and P. D. Spanos: "Stochastic Finite Elements--A Spectral Approach", Springer-Verlag, Berlin, 1991. 2 A. Keese: "A review of recent developments in the numerical solution of stochastic PDEs (stochastic finite elements)" Informatikbericht 2003-6, Technische Universität Braunschweig, Braunschweig, 2003. http://opus.tu-bs.de/opus/volltexte/2003/504/. 3 H. G. Matthies and A. Keese: "Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations", accepted for publication, Comp. Meth. Appl. Mech. Engrg., 2004. doi:10.1016/j.cma.2004.05.027 4 A. Keese: "Numerical Solution of Systems with Stochastic Uncertainties--A General Purpose Framework for Stochastic Finite Elements", Doctoral thesis, Technische Universität Braunschweig, 2003. 5 A. Keese and H. G. Matthies: "The hierarchical parallel solution of stochastic finite element equations." submitted to Computers & Structures, 2004. purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £135 +P&P)
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