A Posteriori Error Analysis of Two-Stage Computation Methods with Application to Efficient Discretization and the Parareal Algorithm

Chaudhry, Jehanzeb H.; Estep, Donald; Tavener, Simon; Carey, Varis; Sandelin, Jeff

doi:10.1137/16m1079014

Cited by 21 publications

(17 citation statements)

References 41 publications

(55 reference statements)

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“…To begin, we will employ simple implicit time integrators. We then aim to combine the current analysis with earlier work on parallel methods for initial value problems [10], to develop an a posteriori analysis for a method that is parallel in both space and time. Once again we will adopt a two stage solution approach, using the distribution of various sources of error estimated from an initial coarse solve to inform the discretization choices for a second "production" computation.…”

Section: Discussionmentioning

confidence: 99%

A posteriori error analysis for Schwarz overlapping domain decomposition methods

Chaudhry¹,

Estep²,

Tavener³

2019

Preprint

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Domain decomposition methods are widely used for the numerical solution of partial differential equations on parallel computers. We develop an adjoint-based a posteriori error analysis for overlapping multiplicative Schwarz and for overlapping additive Schwarz domain decomposition methods. In both cases the numerical error in a user-specified functional of the solution (quantity of interest), is decomposed into a component that arises as a result of the finite iteration between the subdomains, and a component that is due to the spatial discretization. The spatial discretization error can be further decomposed in to the errors arising on each subdomain. This decomposition of the total error can then be used as part of a two-stage approach to construct a solution strategy that efficiently reduces the error in the quantity of interest.

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Section: Discussionmentioning

confidence: 99%

A posteriori error analysis for Schwarz overlapping domain decomposition methods

Chaudhry¹,

Estep²,

Tavener³

2019

Preprint

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“…Theorem 1 provides a representation of the error (16) Theorem 1. For a PDE of the form (1) with weak form (3), if the function f (x, t) is continuous, the error in the computed QoI (2) is…”

Section: Error Formulation For Pdesmentioning

confidence: 99%

“…Classical a posteriori error analysis deals with QoIs that can be expressed as bounded functionals of the solution and has been widely explored [1][2][3][4][6][7][8][9][10][11][12][13]15,16,[19][20][21]24,26,28,30,31,35]. The error estimation utilizes generalized Green's functions solving an adjoint problem, computable residuals of the numerical solution, and variational analysis [1,4,21,27,30,31].…”

Section: Introductionmentioning

confidence: 99%

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

Chaudhry¹,

Estep²,

Giannini³

et al. 2021

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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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“…Adjoint‐based analysis has been used for the error estimation of a variety of numerical methods and differential equations, 26–29 for example, finite element methods, 30–33 finite volume methods, 34 numerous time‐integration schemes, 35–39 and parallel‐in‐time and domain decomposition methods 40,41 . A posteriori analysis of the finite element method for the PBE has been considered previously, 42,43 however, this work is the first such analysis for the PBE with BEM.…”

Section: Introductionmentioning

confidence: 99%

Efficient mesh refinement for the Poisson‐Boltzmann equation with boundary elements

2021

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The Poisson-Boltzmann equation is a widely used model to study electrostatics in molecular solvation. Its numerical solution using a boundary integral formulation requires a mesh on the molecular surface only, yielding accurate representations of the solute, which is usually a complicated geometry. Here, we utilize adjoint-based analyses to form two goal-oriented error estimates that allow us to determine the contribution of each discretization element (panel) to the numerical error in the solvation free energy. This information is useful to identify high-error panels to then refine them adaptively to find optimal surface meshes. We present results for spheres and real molecular geometries, and see that elements with large error tend to be in regions where there is a high electrostatic potential. We also find that even though both estimates predict different total errors, they have similar performance as part of an adaptive mesh refinement scheme. Our test cases suggest that the adaptive mesh refinement scheme is very effective, as we are able to reduce the error one order of magnitude by increasing the mesh size less than 20% and come out to be more efficient than uniform refinement when computing error estimations. This result sets the basis toward efficient automatic mesh refinement schemes that produce optimal meshes for solvation energy calculations.

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A Posteriori Error Analysis of Two-Stage Computation Methods with Application to Efficient Discretization and the Parareal Algorithm

Cited by 21 publications

References 41 publications

A posteriori error analysis for Schwarz overlapping domain decomposition methods

A posteriori error analysis for Schwarz overlapping domain decomposition methods

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

Efficient mesh refinement for the Poisson‐Boltzmann equation with boundary elements

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