A Posteriori Analysis and Efficient Refinement Strategies for the Poisson--Boltzmann Equation

Chaudhry, Jehanzeb H.

doi:10.1137/17m1119846

Cited by 10 publications

(4 citation statements)

References 65 publications

(117 reference statements)

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“…Adjoint-based analysis has been used for the error estimation of a variety of numerical methods and differential equations, [26][27][28][29] for example, finite element methods, [30][31][32][33] finite volume methods, 34 numerous time-integration schemes, [35][36][37][38][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: 98%

“…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. Residual‐based a posteriori analysis for BEM has been studied earlier, 44,45 however, they focused on error in some global norm of the solution, whereas here we focus on quantifying the error in a goal functional or quantity of interest.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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

2021

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“…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

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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“…Adjoint based analysis has been used for the error estimation of a variety of numerical methods and differential equations [24][25][26][27] , for example, finite element methods [28][29][30][31] , finite volume methods 32 , numerous time-integration schemes [33][34][35][36][37] , and parallel-in-time and domain decomposition methods 38,39 . A posteriori analysis of the finite element method for the PBE has been considered previously 40,41 , however, this work is the first such analysis for the PBE with BEM. Residual based a posteriori analysis for BEM has been studied earlier 42,43 , however, they focused on error in some global norm of the solution, whereas here we focus on quantifying the error in a goal functional or quantity of interest.…”

Section: Introductionmentioning

confidence: 99%

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

Ramm¹,

Chaudhry²,

Cooper³

2020

Preprint

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The Poisson-Boltzmann equation is a widely used model to study the 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 allows 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%. This result sets the basis towards efficient automatic mesh refinement schemes that produce optimal meshes for solvation energy calculations.

show abstract

A Posteriori Analysis and Efficient Refinement Strategies for the Poisson--Boltzmann Equation

Cited by 10 publications

References 65 publications

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

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

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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