Iterative Solvers for a Spectral Galerkin Approach to Elliptic Partial Differential Equations with Fuzzy Coefficients

Corveleyn, Samuel; Rosseel, Eveline; Vandewalle, Stefan

doi:10.1137/120881592

Cited by 9 publications

(5 citation statements)

References 42 publications

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“…Indeed, it transforms the original problem into a parametric one, as done in the case of pure stochastic and pure fuzzy PDEs; see e.g. [5,38,16]. We can therefore extend the proofs for well-posedness and regularity of deterministic (see e.g.…”

Section: Solution Of the Fuzzy-stochastic Problemmentioning

confidence: 92%

“…Proof. Thanks to the uniform positivity assumption (13) Hence, by the Lax-Milgram theorem [20], there is a unique solution u(z) ∈ H 1 0 (D) ⊗ L 2 π (Γ) that satisfies (16). By setting v = u(z) in ( 16) and using Hölder and Poincaré inequalities on the right hand side, we obtain ∀z ∈ S z α ,…”

Section: Well-posednessmentioning

confidence: 99%

“…In a fuzzy framework the underlying mathematical models are often PDEs with fuzzy parameters; see e.g. [12,16,10]. Solving fuzzy PDEs leads to fuzzy computations that involve interval arithmetic [28,36] and optimization [31,40] at different membership levels.…”

Section: Introductionmentioning

confidence: 99%

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Fuzzy-Stochastic Partial Differential Equations

Motamed¹

2019

SIAM/ASA J. Uncertainty Quantification

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We introduce and study a new class of partial differential equations (PDEs) with hybrid fuzzy-stochastic parameters, coined fuzzy-stochastic PDEs. Compared to purely stochastic PDEs or purely fuzzy PDEs, which may treat either only random or only non-random uncertainty in physical systems, fuzzy-stochastic PDEs offer powerful models for accurate description and propagation of the hybrid random and non-random uncertainties inevitable in many real applications. We will use the level-set representation of fuzzy functions and define the solution to fuzzy-stochastic PDE problems through a corresponding parametric problem, and further present theoretical results on the well-posedness and regularity of such problems. We also propose a numerical strategy for computing output fuzzy-stochastic quantities, such as fuzzy failure probabilities and fuzzy probability distributions. We present two numerical examples to compute various fuzzy-stochastic quantities and to demonstrate the applicability of fuzzy-stochastic PDEs to complex engineering problems.

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Section: Solution Of the Fuzzy-stochastic Problemmentioning

confidence: 92%

Section: Well-posednessmentioning

confidence: 99%

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Fuzzy-Stochastic Partial Differential Equations

Motamed¹

2019

SIAM/ASA J. Uncertainty Quantification

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“…There are essentially two types of techniques that are used to quantify the uncertainty in the solution. Intrusive methods [9,10,11,12,13,14] are based on polynomial chaos expansions, where the solution is expressed as a spectral expansion of the random variables. The method is cheap in terms of function evaluations for smooth problems, but requires a code, dealing specifically with the uncertainty.…”

Section: Introductionmentioning

confidence: 99%

Robust boundary conditions for stochastic incompletely parabolic systems of equations

Wahlsten

Nordström

2018

Journal of Computational Physics

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We study an incompletely parabolic system in three space dimensions with stochastic boundary and initial data. We show how the variance of the solution can be manipulated by the boundary conditions, while keeping the mean value of the solution unaffected. Estimates of the variance of the solution is presented both analytically and numerically. We exemplify the technique by applying it to an incompletely parabolic model problem, as well as the one-dimensional compressible Navier-Stokes equations.

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“…where B = r i=0 f i g T i ∈ R n1×n2 and it is assumed that m, r n 1 , n 2 . The system matrices K i and G i obtained from discretization methods are typically sparse and, thus, for moderately large system matrices, Krylov subspace methods [29,30] and multigrid methods [3,9,21] have been natural choices to solve such systems.…”

mentioning

confidence: 99%

Alternating Energy Minimization Methods for Multi-term Matrix Equations

Lee,

Elman,

Powell

et al. 2020

Preprint

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We develop computational methods for approximating the solution of a linear multiterm matrix equation in low rank. We follow an alternating minimization framework, where the solution is represented as a product of two matrices, and approximations to each matrix are sought by solving certain minimization problems repeatedly. The solution methods we present are based on a rank-adaptive variant of alternating energy minimization methods that builds an approximation iteratively by successively computing a rank-one solution component at each step. We also develop efficient procedures to improve the accuracy of the low-rank approximate solutions computed using these successive rank-one update techniques. We explore the use of the methods with linear multi-term matrix equations that arise from stochastic Galerkin finite element discretizations of parameterized linear elliptic PDEs, and demonstrate their effectiveness with numerical studies.

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Iterative Solvers for a Spectral Galerkin Approach to Elliptic Partial Differential Equations with Fuzzy Coefficients

Cited by 9 publications

References 42 publications

Fuzzy-Stochastic Partial Differential Equations

Fuzzy-Stochastic Partial Differential Equations

Robust boundary conditions for stochastic incompletely parabolic systems of equations

Alternating Energy Minimization Methods for Multi-term Matrix Equations

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