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Exploring emerging manycore architectures for uncertainty quantification through embedded stochastic Galerkin methods. (English) Zbl 1297.65011
Int. J. Comput. Math. 91, No. 4, 707-729 (2014); erratum ibid. 91, No. 8, 1861 (2014).
Summary: We explore approaches for improving the performance of intrusive or embedded stochastic Galerkin uncertainty quantification methods on emerging computational architectures. Our work is motivated by the trend of increasing disparity between floating-point throughput and memory access speed on emerging architectures, thus requiring the design of new algorithms with memory access patterns more commensurate with computer architecture capabilities. We first compare the traditional approach for implementing stochastic Galerkin methods to non-intrusive spectral projection methods employing high-dimensional sparse quadratures on relevant problems from computational mechanics, and demonstrate the performance of stochastic Galerkin is reasonable. Several reorganizations of the algorithm with improved memory access patterns are described and their performance measured on contemporary manycore architectures. We demonstrate these reorganizations can lead to improved performance for matrix-vector products needed by iterative linear system solvers, and highlight further algorithm research that might lead to even greater performance.

MSC:
65C30 Numerical solutions to stochastic differential and integral equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74B05 Classical linear elasticity
74S05 Finite element methods applied to problems in solid mechanics
Software:
DAKOTA; Stokhos
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