Fast methods for determining the evolution of uncertain parameters in reaction-diffusion equations

Estep, D.; Neckels, David

doi:10.1016/j.cma.2006.10.048

Cited by 17 publications

(15 citation statements)

References 4 publications

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“…where, as previously, fz i g are n independent samples of Z and fC k g are subsets of C centered on fz k g. Dominant estimates fÊ k g of local errors can be used to identify subsets that need to be refined, as proposed in [8]. If C k is a subset that needs to be refined, samples of a local SROM for the conditional vector ZjðZ 2 C k Þ can be used to refine the approximation e U L ðZÞ in C k by following the method proposed in this section.…”

Section: A New Methods For Solving Stochastic Equationsmentioning

confidence: 99%

“…; A m Þ and F denotes the distribution of Z. It has been suggested in [8] to use c P m k¼1 PðA k ÞE k as an approximation for E jhðUðZÞÞ À hðU L ðZÞÞj ½ in Eq. 6, where…”

Section: Geometrically-based Partitionmentioning

confidence: 99%

“…An adaptive scheme has been used in [8] to identify members of the partition fC k g of C that need to be refined. The scheme is based on the relative magnitude of the local errors fE k g in Eq.…”

Section: Geometrically-based Partitionmentioning

confidence: 99%

“…. ; m, in these partitions [7,8]. The initial partition fC k g may be refined based on the information provided by deterministic solutions of the stochastic equation for Z ¼z k ; k ¼ 1; .…”

Section: Introductionmentioning

confidence: 99%

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A method for solving stochastic equations by reduced order models and local approximations

Grigoriu

2012

Journal of Computational Physics

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Section: A New Methods For Solving Stochastic Equationsmentioning

confidence: 99%

“…; A m Þ and F denotes the distribution of Z. It has been suggested in [8] to use c P m k¼1 PðA k ÞE k as an approximation for E jhðUðZÞÞ À hðU L ðZÞÞj ½ in Eq. 6, where…”

Section: Geometrically-based Partitionmentioning

confidence: 99%

Section: Geometrically-based Partitionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A method for solving stochastic equations by reduced order models and local approximations

Grigoriu

2012

Journal of Computational Physics

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“…This method has been successfully extended to estimate numerical errors due to operator splittings [11] and operator decomposition for multiscale/multiphysics applications [6,15,16]. adaptive sampling algorithms [13,14], stochastic approximations [20,4], and inverse sensitivity analysis [3,5].…”

Section: Introductionmentioning

confidence: 99%

Adjoint based a posteriori error estimation in Drekar::CFD.

Wildey¹,

Cyr²,

Pawlowski³

et al. 2012

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This document summarizes the results from a level 3 milestone study within the CASL VUQ effort. It demonstrates the capability to perform adjoint-based a posteriori error estimation within Drekar::CFD. The a posteriori error estimates require the solution of a (linear) adjoint problem in a higher order approximation space with data chosen based on the quantity of interest. The a posteriori error estimates are verified using an analytical solution to a laminar flow problem. Finally, the framework is demonstrated on the TH-M Test Case #2 involving a 3-dimensional axisymmetric sudden expansion with a moderate Reynolds numbers ( 5000).3 Acknowledgment

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A numerical method for solving a stochastic inverse problem for parameters

Butler

Estep

2013

Annals of Nuclear Energy

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We review recent work (Briedt et al., 2011., 2012) on a new approach to the formulation and solution of the stochastic inverse parameter determination problem, i.e. determine the random variation of input parameters to a map that matches specified random variation in the output of the map, and then apply the various aspects of this method to the interesting Brusselator model. In this approach, the problem is formulated as an inverse problem for an integral equation using the Law of Total Probability. The solution method employs two steps: (1) we construct a systematic method for approximating set-valued inverse solutions and (2) we construct a computational approach to compute a measure-theoretic approximation of the probability measure on the input space imparted by the approximate set-valued inverse that solves the inverse problem. In addition to convergence analysis, we carry out an a posteriori error analysis on the computed probability distribution that takes into account all sources of stochastic and deterministic error.

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Fast methods for determining the evolution of uncertain parameters in reaction-diffusion equations

Cited by 17 publications

References 4 publications

A method for solving stochastic equations by reduced order models and local approximations

A method for solving stochastic equations by reduced order models and local approximations

Adjoint based a posteriori error estimation in Drekar::CFD.

A numerical method for solving a stochastic inverse problem for parameters

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