Benchmarking preconditioned boundary integral formulations for acoustics

Wout, Elwin van ’t; Haqshenas, Seyyed R.; Gélat, Pierre; Betcke, Timo; Saffari, Nader

doi:10.1002/nme.6777

Cited by 14 publications

(23 citation statements)

References 88 publications

(180 reference statements)

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“…56 The BEM employs the Green's function of the Helmholtz equation to reformulate the volumetric wave problem into a boundary integral equation at the interfaces of piecewise homogeneous domains embedded in free space. 57 Benchmarks 3, 5 and 7 were modeled with the PMCHWT formulation, 58 benchmarks 4 and 6 were solved with a multi-trace formulation, 59 and a nested version of the PMCHWT formulation solves benchmarks 8 and 9. The numerical discretization leads to a dense system of linear equations, whose computational footprint is reduced through hierarchical matrix compression.…”

Section: Optimusmentioning

confidence: 99%

Benchmark problems for transcranial ultrasound simulation: Intercomparison of compressional wave models

Aubry¹,

Bates²,

Boehm³

et al. 2022

Preprint

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Section: Optimusmentioning

confidence: 99%

Benchmark problems for transcranial ultrasound simulation: Intercomparison of compressional wave models

Aubry¹,

Bates²,

Boehm³

et al. 2022

Preprint

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“…Second, the interior boundary integral formulation () can be written as

\begin{matrix} {right left}true \\ [\begin{matrix} center \\ I & 0 \\ 0 & \frac{ϵ_{int}}{ϵ_{ext}} I \end{matrix}] [\begin{matrix} center \\ \frac{1}{2} I + K_{int} & - V_{int} \\ - D_{int} & \frac{1}{2} I - T_{int} \end{matrix}] [\begin{matrix} center \\ I & 0 \\ 0 & \frac{ϵ_{ext}}{ϵ_{int}} I \end{matrix}] [\begin{matrix} center \\ γ_{D}^{-} ϕ_{int} \\ \frac{ϵ_{int}}{ϵ_{ext}} γ_{N}^{-} ϕ_{int} \end{matrix}] = [\begin{matrix} center \\ I & 0 \\ 0 & \frac{ϵ_{int}}{ϵ_{ext}} I \end{matrix}] [\begin{matrix} center \\ φ_{D} \\ φ_{N} \end{matrix}], \\ [\begin{matrix} center \\ \frac{1}{2} I + K_{int} & - \frac{ϵ_{ext}}{ϵ_{int}} V_{int} \\ - \frac{ϵ_{int}}{ϵ_{ext}} D_{int} & \frac{1}{2} I - T_{int} \end{matrix}] [\begin{matrix} center \\ γ_{D}^{+} ϕ_{ext} \\ γ_{N}^{+} ϕ_{ext} \end{matrix}] = [\begin{matrix} center \\ φ_{D} \\ \frac{ϵ_{int}}{ϵ_{ext}} φ_{N} \end{matrix}] . \end{matrix}

Note that four independent boundary integral equations are present while the four unknowns are reduced to two unknowns with the interface conditions. We can take linear combinations of the four boundary integral equations 20 in Equation (12), to produce different boundary integral formulations for the same Poisson–Boltzmann system …”

Section: Methodsmentioning

confidence: 99%

“…Examples of operator preconditioners include opposite‐order preconditioning, 53 Calderón preconditioning, 27 and OSRC preconditioning 54 . Their effectiveness to coupled systems has been shown for electromagnetics 55 and acoustics 20 . However, no operator preconditioning for the PBE is known so far.…”

Section: Methodsmentioning

confidence: 99%

“…Note that four independent boundary integral equations are present while the four unknowns are reduced to two unknowns with the interface conditions. We can take linear combinations of the four boundary integral equations 20 in Equation ( 12), to produce different boundary integral formulations for the same Poisson-Boltzmann system. 33…”

Section: Boundary Integral Formulations Of the Poisson-boltzmann Equa...mentioning

confidence: 99%

See 1 more Smart Citation

Towards optimal boundary integral formulations of the Poisson–Boltzmann equation for molecular electrostatics

Cooper

Wout

2022

J Comput Chem

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The Poisson‐Boltzmann equation offers an efficient way to study electrostatics in molecular settings. Its numerical solution with the boundary element method is widely used, as the complicated molecular surface is accurately represented by the mesh, and the point charges are accounted for explicitly. In fact, there are several well‐known boundary integral formulations available in the literature. This work presents a generalized expression of the boundary integral representation of the implicit solvent model, giving rise to new forms to compute the electrostatic potential. Moreover, it proposes a strategy to build efficient preconditioners for any of the resulting systems, improving the convergence of the linear solver. We perform systematic benchmarking of a set of formulations and preconditioners, focusing on the time to solution, matrix conditioning, and eigenvalue spectrum. We see that the eigenvalue clustering is a good indicator of the matrix conditioning, and show that they can be easily manipulated by scaling the preconditioner. Our results suggest that the optimal choice is problem‐size dependent, where a simpler direct formulation is the fastest for small molecules, but more involved second‐kind equations are better for larger problems. We also present a fast Calderón preconditioner for first‐kind formulations, which shows promising behavior for future analysis. This work sets the basis towards choosing the most convenient boundary integral formulation of the Poisson–Boltzmann equation for a given problem.

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“…In this work, we took inspiration from the computational acoustics community to develop a general approach to the BEM-BEM coupling algorithm 20 . Similar to the PBE, the Helmholtz equation for acoustic wave propagation can be solved efficiently with the BEM.…”

Section: Introductionmentioning

confidence: 99%

Towards optimal boundary integral formulations of the Poisson-Boltzmann equation for molecular electrostatics

Search¹,

Cooper²,

Wout³

2021

Preprint

View full text Add to dashboard Cite

The Poisson-Boltzmann equation offers an efficient way to study electrostatics in molecular settings. Its numerical solution with the boundary element method is widely used, as the complicated molecular surface is accurately represented by the mesh, and the point charges are accounted for explicitly. In fact, there are several well-known boundary integral formulations available in the literature. This work presents a generalized expression of the boundary integral representation of the implicit solvent model, giving rise to new forms to compute the electrostatic potential. Moreover, it proposes a strategy to build efficient preconditioners for any of the resulting systems, improving the convergence of the linear solver. We perform systematic benchmarking of a set of formulations and preconditioners, focusing on the time to solution, matrix conditioning, and eigenvalue spectrum. We see that the eigenvalue clustering is a good indicator of the matrix conditioning, and show that they can be easily manipulated by scaling the preconditioner. Our results suggest that the optimal choice is problem-size dependent, where a simpler direct formulation is the fastest for small molecules, but more involved second-kind equations are better for larger problems. We also present a fast Calderón preconditioner for first-kind formulations, which shows promising behavior for future analysis. This work sets the basis towards choosing the most convenient boundary integral formulation of the Poisson-Boltzmann equation for a given problem.

show abstract

Benchmarking preconditioned boundary integral formulations for acoustics

Cited by 14 publications

References 88 publications

Benchmark problems for transcranial ultrasound simulation: Intercomparison of compressional wave models

Benchmark problems for transcranial ultrasound simulation: Intercomparison of compressional wave models

Towards optimal boundary integral formulations of the Poisson–Boltzmann equation for molecular electrostatics

Towards optimal boundary integral formulations of the Poisson-Boltzmann equation for molecular electrostatics

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