Multi-element probabilistic collocation method in high dimensions

Foo, Jasmine; Karniadakis, George Em

doi:10.1016/j.jcp.2009.10.043

Cited by 203 publications

(180 citation statements)

References 49 publications

(55 reference statements)

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“…The scheme is based on a tensor-product construction and hence suffers from the "curse of dimensionality". Future work is to couple the AMR and WENO methodologies in random space with sparse grid [23,26] and adaptive Analysis Of Variance (ANOVA) [11,18] techniques which are efficient tools known for approximating high-dimensional problems.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…In such cases, Gibbs-type phenomenon will be observed in gPC approximation, which causes the slow convergence. To alleviate such a challenging issue, multi-element gPC methods [16,17] and multi-element stochastic collocation methods [18,19] were developed to decompose the random space into sub-domains, then employ a gPC expansion in each element. The challenge of the standard multi-element gPC or multi-element stochastic collocation method is to detect the discontinuities in high-dimensional random space and the computational cost.…”

Section: Introductionmentioning

confidence: 99%

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An Adaptive WENO Collocation Method for Differential Equations with Random Coefficients

et al. 2016

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Abstract:The stochastic collocation method for solving differential equations with random inputs has gained lots of popularity in many applications, since such a scheme exhibits exponential convergence with smooth solutions in the random space. However, in some circumstance the solutions do not fulfill the smoothness requirement; thus a direct application of the method will cause poor performance and slow convergence rate due to the well known Gibbs phenomenon. To address the issue, we propose an adaptive high-order multi-element stochastic collocation scheme by incorporating a WENO (Weighted Essentially non-oscillatory) interpolation procedure and an adaptive mesh refinement (AMR) strategy. The proposed multi-element stochastic collocation scheme requires only repetitive runs of an existing deterministic solver at each interpolation point, similar to the Monte Carlo method. Furthermore, the scheme takes advantage of robustness and the high-order nature of the WENO interpolation procedure, and efficacy and efficiency of the AMR strategy. When the proposed scheme is applied to stochastic problems with non-smooth solutions, the Gibbs phenomenon is mitigated by the WENO methodology in the random space, and the errors around discontinuities in the stochastic space are significantly reduced by the AMR strategy. The numerical experiments for some benchmark stochastic problems, such as the Kraichnan-Orszag problem and Burgers' equation with random initial conditions, demonstrate the reliability, efficiency and efficacy of the proposed scheme.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An Adaptive WENO Collocation Method for Differential Equations with Random Coefficients

et al. 2016

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“…For a moderate number of dimensions, effective approaches are multi-element and sparse adaptive probabilistic collocation [22,23,34].…”

Section: (B) High-dimensional Random Forcingmentioning

confidence: 99%

“…To compute the statistical properties of the solution to stochastic Burgers problem, we need a representation of the functional relation between the state variables of the system and the random input processes. Well-known approaches are polynomial chaos [19][20][21], multi-element and sparse adaptive probabilistic collocation [22,23], high-dimensional model representations [24], stochastic biorthogonal expansions [25,26] and separated representations [27,28]. These techniques can provide considerable speed-up in computational time when compared to Monte Carlo (MC) or quasi-MC methods, and they are usually effective for the determination of the first few statistical moments of the solution.…”

Section: Introductionmentioning

confidence: 99%

Statistical analysis and simulation of random shocks in stochastic Burgers equation

2014

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We study the statistical properties of random shock waves in stochastic Burgers equation subject to random space-time perturbations and random initial conditions. By using the response-excitation probability density function (PDF) method and the Mori-Zwanzig (MZ) formulation of irreversible statistical mechanics, we derive exact reducedorder equations for the one-point and two-point PDFs of the solution field. In particular, we compute the statistical properties of random shock waves in the inviscid limit by using an adaptive (shock-capturing) discontinuous Galerkin method in both physical and probability spaces. We consider stochastic flows generated by high-dimensional random initial conditions and random additive forcing terms, yielding multiple interacting shock waves collapsing into clusters and settling down to a similarity state. We also address the question of how random shock waves in space and time manifest themselves in probability space. The proposed new mathematical framework can be applied to different conservation laws, potentially leading to new insights into highdimensional stochastic dynamical systems and more efficient computational algorithms.

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“…The PCE offers an efficient and accurate high-order way of including non-linear effects in stochastic analysis, see e.g. Fajraoui et al (2011);Foo & Karniadakis (2010); Zhang & Lu (2004). One of the attractive features of PCE is the high-order approximation of error propagation (Ghanem & Spanos, 1991; as well as its computational speed when compared to MC (Oladyshkin, Class, Helmig & Nowak, 2011b).…”

Section: Response Surface Via Polynomial Chaos Expansionmentioning

confidence: 99%

Polynomial Response Surfaces for Probabilistic Risk Assessment and Risk Control via Robust Design

Oladyshkin¹,

Nowak²

2012

Novel Approaches and Their Applications in Risk Assessment

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Multi-element probabilistic collocation method in high dimensions

Cited by 203 publications

References 49 publications

An Adaptive WENO Collocation Method for Differential Equations with Random Coefficients

An Adaptive WENO Collocation Method for Differential Equations with Random Coefficients

Statistical analysis and simulation of random shocks in stochastic Burgers equation

Polynomial Response Surfaces for Probabilistic Risk Assessment and Risk Control via Robust Design

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