Stochastic Galerkin methods for the steady-state Navier–Stokes equations

Sousedík, Bedřich; Elman, Howard C.

doi:10.1016/j.jcp.2016.04.013

Cited by 26 publications

(40 citation statements)

References 28 publications

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“…The second example of a stochastic eigenvalue problem that we consider is used to estimate the inf-sup stability constant associated with a discrete stochastic Stokes problem. Consider the following stochastic incompressible Stokes equation in a two-dimensional domain Such problems and more general stochastic Navier-Stokes equations have been studied in [29,35]. As in the diffusion problem, we assume that the stochastic viscosity a(x, ω) is represented by a truncated KL expansion (4.2) with random variables {ξ l } m l=1 .…”

Section: Stochastic Stokes Equationmentioning

confidence: 99%

Low-Rank Solution Methods for Stochastic Eigenvalue Problems

Elman¹,

Su²

2019

SIAM J. Sci. Comput.

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We study efficient solution methods for stochastic eigenvalue problems arising from discretization of self-adjoint partial differential equations with random data. With the stochastic Galerkin approach, the solutions are represented as generalized polynomial chaos expansions. A low-rank variant of the inverse subspace iteration algorithm is presented for computing one or several minimal eigenvalues and corresponding eigenvectors of parameter-dependent matrices. In the algorithm, the iterates are approximated by low-rank matrices, which leads to significant cost savings. The algorithm is tested on two benchmark problems, a stochastic diffusion problem with some poorly separated eigenvalues, and an operator derived from a discrete stochastic Stokes problem whose minimal eigenvalue is related to the inf-sup stability constant. Numerical experiments show that the low-rank algorithm produces accurate solutions compared to the Monte Carlo method, and it uses much less computational time than the original algorithm without low-rank approximation.

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Section: Stochastic Stokes Equationmentioning

confidence: 99%

Low-Rank Solution Methods for Stochastic Eigenvalue Problems

Elman¹,

Su²

2019

SIAM J. Sci. Comput.

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“…One approach to reduce the curse of dimensionality is to sparsify explicitly the set of polynomials, by bounding, for example, the total degree. [4][5][6][12][13][14] Alternatively, one can consider the full Cartesian space of polynomials with bounded individual degree, but never store the expansion coefficients explicitly. Instead, one approximates the tensor of coefficients by a low-rank decomposition.…”

Section: Introductionmentioning

confidence: 99%

Low‐rank solution of an optimal control problem constrained by random Navier‐Stokes equations

Benner

Dolgov

Onwunta

et al. 2020

Numerical Methods in Fluids

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Many problems in computational science and engineering are simultaneously characterized by the following challenging issues: uncertainty, nonlinearity, nonstationarity and high dimensionality. Existing numerical techniques for such models would typically require considerable computational and storage resources. This is the case, for instance, for an optimization problem governed by time-dependent Navier-Stokes equations with uncertain inputs. In particular, the stochastic Galerkin finite element method often leads to a prohibitively high dimensional saddle-point system with tensor product structure. In this paper, we approximate the solution by the low-rank Tensor Train decomposition, and present a numerically efficient algorithm to solve the optimality equations directly in the low-rank representation. We show that the solution of the vorticity minimization problem with a distributed control admits a representation with ranks that depend modestly on model and discretization parameters even for high Reynolds numbers. For lower Reynolds numbers this is also the case for a boundary control. This opens the way for a reduced-order modeling of the stochastic optimal flow control with a moderate cost at all stages.

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“…During last decades there has been a rapid development in uncertainty quantification for solving partial differential equations (PDEs) with random inputs. These random inputs usually arise from lack of knowledge about the system or/and the measurements of realistic model parameters such as the permeability coefficients in diffusion problems [29,55], the viscosity parameters of incompressible flows [12,42,47,49], and shape parameters in acoustic scattering [58]. In particular, stochastic Helmholtz equations attracted a lot of interest recently and this paper aims at the development of efficient strategies for their solution.…”

Section: Introductionmentioning

confidence: 99%

“…This leads to linear systems in Kronecker formulation [10,41,43]. We note that for stochastic Galerkin linear systems, various efficient iterative solvers such as mean-based preconditioning methods [41,43], hierarchical preconditioners [47,48], preconditioned low-rank projection methods [36] are vigorously studied. Nevertheless, to the best of our knowledge, in the case of stochastic Helmholtz problems these methods have not been analysed.…”

Section: Introductionmentioning

confidence: 99%

Efficient Spectral Stochastic Finite Element Methods for Helmholtz Equations with Random Inputs

Liao¹

2019

EAJAM

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The implementation of spectral stochastic Galerkin finite element approximation methods for Helmholtz equations with random inputs is considered. The corresponding linear systems have matrices represented as Kronecker products. The sparsity of such matrices is analysed and a mean-based preconditioner is constructed. Numerical examples show the efficiency of the mean-based preconditioners for stochastic Helmholtz problems, which are not too close to a resonant frequency.

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Stochastic Galerkin methods for the steady-state Navier–Stokes equations

Cited by 26 publications

References 28 publications

Low-Rank Solution Methods for Stochastic Eigenvalue Problems

Low-Rank Solution Methods for Stochastic Eigenvalue Problems

Low‐rank solution of an optimal control problem constrained by random Navier‐Stokes equations

Efficient Spectral Stochastic Finite Element Methods for Helmholtz Equations with Random Inputs

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