Scalable domain decomposition solvers for stochastic PDEs in high performance computing

Desai, Ajit; Khalil, Mohammad; Pettit, Chris L.; Poirel, Dominique; Sarkar, Abhijit

doi:10.1016/j.cma.2017.09.006

Cited by 8 publications

(33 citation statements)

References 26 publications

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“…Further increase in stochastic dimensions can be accommodated by dividing the physical domain into more subdomains and accordingly using more computing cores to solve the problem. Therefore, the DDM can help us to accommodate increased problem size due to increase in stochastic dimensions [19,23,24,[125][126][127].…”

Section: Brief Overview Of Domain Decomposition Methods For Uncertainmentioning

confidence: 99%

“…On the other hand, in the intrusive SSFEM, the coupled sets of equations obtained by incorporating PCE of the random system parameters and the associated solution process into the governing equations are solved simultaneously [18,90,161]. From the implementational perspective, the intrusive approach using HPC requires the exchange of data among computing nodes in order to solve the given problem [18,19]. Therefore, the solution of the intrusive SSFEM system requires considerable efforts in terms of algorithmic development and HPC implementation [18,19,161].…”

Section: Brief Overview Of High-performance Computing In Uncertainty mentioning

confidence: 99%

“…From the implementational perspective, the intrusive approach using HPC requires the exchange of data among computing nodes in order to solve the given problem [18,19]. Therefore, the solution of the intrusive SSFEM system requires considerable efforts in terms of algorithmic development and HPC implementation [18,19,161].…”

Section: Brief Overview Of High-performance Computing In Uncertainty mentioning

confidence: 99%

“…The utilization of SSFEM for large-scale industrial applications are the ideal problems to exploit emerging exascale computing [161,163]. It is primarily because with increasing stochastic dimensions the size of the problems needed to be handled in the intrusive approach significantly increases and the number of sample evaluations grows rapidly in the cases of non-intrusive approaches rendering them impractical [19,161,163].…”

Section: Brief Overview Of High-performance Computing In Uncertainty mentioning

confidence: 99%

“…The FEM is used in the spatial domain and PCE is used in the stochastic domain to discretize the stochastic PDE using intrusive SSFEM, resulting in a large-scale system of linear equations [1,2,13]. Then, linear system solvers are employed to solve the system for PCE coefficients of the solution process [17][18][19].…”

Section: Introductionmentioning

confidence: 99%

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Scalable Domain Decomposition Algorithms for Uncertainty Quantification in High Performance Computing

Desai¹

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Uncertainty quantification of practical engineering applications using the intrusive spectral stochastic finite element methods (SSFEM) may involve solving a system of linear equations in the order of billions of unknowns. Therefore, in this thesis the intrusive polynomial chaos expansion (PCE) based two-level domain decomposition (DD) algorithms for stochastic partial differential equations (PDEs) are extended to handle high resolution numerical models using an in-house scalable parallel solvers toolkit. First, attention is given to facilitate the numerical simulation of the elliptic stochastic PDEs with a large number of random variables to address the so-called curse of dimensionality issue. Second, for three-dimensional coupled stochastic PDE systems such as equations of linear elasticity, the extended wirebasket-based coarse grid is developed to improve the performance and overcome the scalability issues of the DD based iterative solvers with a vertex-based coarse grid. Third, the developed DD solvers for the SSFEM are coupled with FEniCS deterministic finite element assembly routines in order to reduce the coding required for the implementation and generalize the application of these solvers to a variety of PDEs using FEniCS. Fourth, the intrusive SSFEM with scalable DD solver is shown to outperform the non-intrusive SSFEM with the sparse grid quadrature for a stochastic PDE with the non-Gaussian random variables. This highlights the advantages of the intrusive approach and demonstrates the necessity of scalable parallel solvers for uncertainty quantification. This thesis also elaborates on the HPC implementational aspects of the DD solvers for SSFEM. Three-level nested sparse iterative solvers, which employ an efficient DD based preconditioners are used to simulate two and three-dimensional scalar and vector-valued stochastic PDEs. The random system parameters and the solution process are modeled as a non-Gaussian II III stochastic process characterized by using up to 25 random variables with a third-order expansion resulting in 3276 PCE coefficients. Numerical and parallel scalabilities of these algorithms are investigated employing large-scale high-performance computing clusters with MPI, PETSc and FEniCS libraries. For all these scalability measurements, we report the results for different numbers of random variables and different orders of PCE to measure the sensitivity of the algorithms to the stochastic parameters.Dedicated to my wife, Pragati, and my son, Prajit. IV

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Section: Brief Overview Of Domain Decomposition Methods For Uncertainmentioning

confidence: 99%

Section: Brief Overview Of High-performance Computing In Uncertainty mentioning

confidence: 99%

Section: Brief Overview Of High-performance Computing In Uncertainty mentioning

confidence: 99%

Section: Brief Overview Of High-performance Computing In Uncertainty mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Scalable Domain Decomposition Algorithms for Uncertainty Quantification in High Performance Computing

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Uncertainty propagation in dynamic sub-structuring by model reduction integrated domain decomposition

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2020

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Seismic analysis of gravity dam-foundation systems using stochastic spectral finite element method

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Scalable domain decomposition solvers for stochastic PDEs in high performance computing

Cited by 8 publications

References 26 publications

Scalable Domain Decomposition Algorithms for Uncertainty Quantification in High Performance Computing

Scalable Domain Decomposition Algorithms for Uncertainty Quantification in High Performance Computing

Uncertainty propagation in dynamic sub-structuring by model reduction integrated domain decomposition

Seismic analysis of gravity dam-foundation systems using stochastic spectral finite element method

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