Dual-primal domain decomposition method for uncertainty quantification

Subber, Waad; Sarkar, Abhijit

doi:10.1016/j.cma.2013.07.007

Cited by 16 publications

(47 citation statements)

References 36 publications

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“…This chapter is closely based on the references [79,80,86]. In this chapter, we report are identical when using the same primal constraints [83].…”

Section: Introductionmentioning

confidence: 89%

“…For deterministic PDEs, BDDC and dual-primal finite element tearing and interconnect FETI-DP [64,81] are perhaps the most popular non-overlapping dom ain decom position techniques for the iterative solution o f large-scale determ inistic linear system s [82], It has already been demonstrated that the condition num ber and thus the parallel perform ance of BDDC and FETI-DP are quite sim ilar [65,67,78,[83][84][85]. It is therefore natural to ask whether the similarity o f BDDC and FETI-DP extends to stochastic systems To address this question, we form ulate a probabilistic version of the FETI-D P iterative substructuring technique for the intrusive SSFEM [79,80,86]. In the probabilistic setting, the operator of the dual interface system in the dual-prim al approach contains a coarse problem.…”

Section: Research Objective and Scopementioning

confidence: 99%

“…The numerical results presented in this chapter are from the references [75,77,86]. O ne of the most attractive features o f domain decom position m ethod is its ability to offer parallel scalable algorithms [7,72], The iterative algorithm is said to be scalable when its con vergence rate does not deteriorate as the num ber of subdomains increases (or the size o f subdomain H decreases) [4,5,7,64], On the other hand, the iterative algorithm is said to be optimal when its convergence rate is independent of the m esh size h [4,5].…”

Section: Introductionmentioning

confidence: 99%

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Domain decomposition methods for uncertainty quantification

Subber¹

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“…This chapter is closely based on the references [79,80,86]. In this chapter, we report are identical when using the same primal constraints [83].…”

Section: Introductionmentioning

confidence: 89%

Section: Research Objective and Scopementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Domain decomposition methods for uncertainty quantification

Subber¹

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“…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%

“…Considerations such as these motivated Sarkar et al [23] to develop the theoretical formulation for iterative substructuring methods for stochastic PDEs. Subsequently, Subber and Sarkar [17,18] formulated and employed one-level and two-level DDM solvers for stochastic PDEs with a few random variables. Taking motivation from their work, in this thesis, the primary attention is given to the development and efficient parallel implementation of scalable DD solvers for two and three dimensional elliptic stochastic PDEs with a large number of random variables.…”

Section: Introductionmentioning

confidence: 99%

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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A local hybrid surrogate‐based finite element tearing interconnecting dual‐primal method for nonsmooth random partial differential equations

Eigel

Gruhlke

2020

Numerical Meth Engineering

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A domain decomposition approach for high-dimensional random partial differential equations exploiting the localization of random parameters is presented. To obtain high efficiency, surrogate models in multielement representations in the parameter space are constructed locally when possible. The method makes use of a stochastic Galerkin finite element tearing interconnecting dual-primal formulation of the underlying problem with localized representations of involved input random fields. Each local parameter space associated to a subdomain is explored by a subdivision into regions where either the parametric surrogate accuracy can be trusted or where instead one has to resort to Monte Carlo. A heuristic adaptive algorithm carries out a problem-dependent hp-refinement in a stochastic multielement sense, anisotropically enlarging the trusted surrogate region as far as possible. This results in an efficient global parameter to solution sampling scheme making use of local parametric smoothness exploration for the surrogate construction. Adequately structured problems for this scheme occur naturally when uncertainties are defined on subdomains, for example, in a multiphysics setting, or when the Karhunen-Loève expansion of a random field can be localized. The efficiency of the proposed hybrid technique is assessed with numerical benchmark problems illustrating the identification of trusted (possibly higher order) surrogate regions and nontrusted sampling regions. K E Y W O R D S domain decomposition, FETI, nonsmooth elliptic partial differential equations, partial differential equations with random coefficients, stochastic finite element method, uncertainty quantification 1 INTRODUCTION In uncertainty quantification (UQ), numerical methods typically are either based on pointwise sampling, which is applicable to quite general problems but rather inefficient, or they rely on (an often analytic) smoothness of the parameter This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Dual-primal domain decomposition method for uncertainty quantification

Cited by 16 publications

References 36 publications

Domain decomposition methods for uncertainty quantification

Domain decomposition methods for uncertainty quantification

Scalable Domain Decomposition Algorithms for Uncertainty Quantification in High Performance Computing

A local hybrid surrogate‐based finite element tearing interconnecting dual‐primal method for nonsmooth random partial differential equations

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