Error estimation and uncertainty quantification for first time to a threshold value

Chaudhry, Jehanzeb H.; Estep, Donald; Stevens, Zachary; Tavener, Simon

doi:10.1007/s10543-020-00825-0

Cited by 3 publications

(6 citation statements)

References 31 publications

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“…In addition to variational methods, many commonly used finite difference schemes, such as IMEX methods, Lax-Wendroff, Crank-Nicolson, etc. are also acceptable as they can be cast to nodally equivalent variational methods (see [9,17,20]).…”

Section: Numerical Scheme: Space-time Galerkin Finite Element Methodsmentioning

confidence: 99%

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Error estimation for the time to a threshold value in evolutionary partial differential equations

Chaudhry¹,

Estep²,

Giannini³

et al. 2021

Preprint

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We develop an a posteriori error analysis for a novel quantity of interest (QoI) evolutionary partial differential equations (PDEs). Specifically, the QoI is the first time at which a functional of the solution to the PDE achieves a threshold value signifying a particular event, and differs from classical QoIs which are modeled as bounded linear functionals. We use Taylor's theorem and adjoint based analysis to derive computable and accurate error estimates for linear parabolic and hyperbolic PDEs. Specifically, the heat equation and linearized shallow water equations (SWE) are used for the parabolic and hyperbolic cases, respectively. Numerical examples illustrate the accuracy of the error estimates.

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Section: Numerical Scheme: Space-time Galerkin Finite Element Methodsmentioning

confidence: 99%

“…Recent work in [17] applied a posteriori error analysis for such a QoI under the framework of ordinary differential equations (ODEs). In this work, we extend that analysis to evolutionary PDEs.…”

Section: Introductionmentioning

confidence: 99%

Error estimation for the time to a threshold value in evolutionary partial differential equations

Chaudhry¹,

Estep²,

Giannini³

et al. 2021

Preprint

Self Cite

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“…The second form of QoI for time-dependent equations is the nonstandard QoI which represents the time at which a certain event related to the solution of (3.4) occurs [12,19]. To express this quantity, let G(u; t) be a linear functional of u which is implicitly dependent on t through u, and expressed in terms of the inner product,…”

Section: Odes This Article Also Considers Ordinary Differential Equat...mentioning

confidence: 99%

“…The error estimation method is often used with a finite element or variational numerical method, however many finite difference and finite volume methods are acceptable, provided they be recast as an equivalent finite element method [13,17,19,22,23,25,26,30,34,44]. Nonlinear QoIs are typically handled by linearizing around a computed solution [6,9,20].…”

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confidence: 99%

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Adjoint-based Adaptive Multi-Level Monte Carlo for Differential Equations

Chaudhry¹,

Stevens²

2022

Preprint

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We present a multi-level Monte Carlo (MLMC) algorithm with adaptively refined meshes and accurately computed stopping-criteria utilizing adjoint-based a posteriori error analysis for differential equations. This is in contrast to classical MLMC algorithms that use either a hierarchy of uniform meshes or adaptively refined meshes based on Richardson extrapolation, and employ a stopping criteria that relies on assumptions on the convergence rate of the MLMC levels. This work develops two adaptive refinement strategies for the MLMC algorithm. These strategies are based on a decomposition of an error estimate of the MLMC bias and utilize variational analysis, adjoint problems and computable residuals.

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Error estimation for the time to a threshold value in evolutionary partial differential equations

Chaudhry

Estep²,

Giannini³

et al. 2023

Bit Numer Math

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Error estimation and uncertainty quantification for first time to a threshold value

Cited by 3 publications

References 31 publications

Error estimation for the time to a threshold value in evolutionary partial differential equations

Error estimation for the time to a threshold value in evolutionary partial differential equations

Adjoint-based Adaptive Multi-Level Monte Carlo for Differential Equations

Error estimation for the time to a threshold value in evolutionary partial differential equations

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