Domain decomposition methods for uncertainty quantification

Subber, Waad

doi:10.22215/etd/2013-09627

Cited by 2 publications

(73 citation statements)

References 119 publications

(444 reference statements)

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“…a one-level preconditioner can be constructed using only the local interface coupling matrices A s rr in the following form [181]:…”

Section: L Parallel One-level Lumped Preconditionermentioning

confidence: 99%

“…Equiva The parallelization of the matrix-vector product, along with the parallel precondi-tioning algorithm detailed in the previous section, permits an efficient distributed imple mentation of Krylov subspace solvers to tackle the Schur complement system. Subber and Sarkar [84,87,181] developed a parallel preconditioned CGM solver for symmet ric systems based on the preconditioner and matrix-vector product algorithms presented above as detailed in the next section. Similarly, a parallel preconditioned BICGSTAB solver is also developed to tackle non-symmetric systems as presented later.…”

Section: L Parallel One-level Lumped Preconditionermentioning

confidence: 99%

“…To examine the performance of these methods for more realistic systems, the applications should consider time-dependent problems.• Parallel Solver for Non-Symmetric Systems Based on Preconditioned Bicon jugate Gradient Stabilized Method:-It is envisioned that the proposed parallel solver based on preconditioned bi conjugate gradient stabilized method will have difficulty in scaling to very large number of processors/cores due to the use o f a one-level precondi tioner. Two-level parallel preconditioners[181] should be used to enhance the scalability of the algorithm to very large number of processors/cores in order to tackle large-scale filtering problems. Inverse of a Partitioned Symmetric Matrix).Given a symmetric 2 x 2 block matrix A of dimension n x n inverse B = A 1 which can also be divided into four blocks asB = A 1 with the following dimensions • A n and Bn are r x r, • A22 and B22 are q x q, • A i 2 = A21 and Bi2 = B21 are r x q, • r + q = n, Bu -A |j or (An A i2A22 A12) B n -Ir or 1 A T Bn -(An -A i2A22Af2) Applying Theorem A.…”

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confidence: 99%

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Bayesian inference for complex and large-scale engineering systems

Khalil

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“…a one-level preconditioner can be constructed using only the local interface coupling matrices A s rr in the following form [181]:…”

Section: L Parallel One-level Lumped Preconditionermentioning

confidence: 99%

Section: L Parallel One-level Lumped Preconditionermentioning

confidence: 99%

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confidence: 99%

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Bayesian inference for complex and large-scale engineering systems

Khalil

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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%

“…Therefore, the iterative substructuring methods are preferred in the stochastic mechanics community [24,[140][141][142][143]. The iterative substructuring techniques are further categorized into the primal and dual methods [132,139].…”

Section: Brief Overview Of Domain Decomposition Methods For Uncertainmentioning

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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Domain decomposition methods for uncertainty quantification

Cited by 2 publications

References 119 publications

Bayesian inference for complex and large-scale engineering systems

Bayesian inference for complex and large-scale engineering systems

Scalable Domain Decomposition Algorithms for Uncertainty Quantification in High Performance Computing

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