Monte Carlo methods for linear and non-linear Poisson-Boltzmann equation

Bossy, Mireille; Champagnat, Nicolas; Leman, Hélène; Maire, Sylvain; Violeau, Laurent; Yvinec, Mariette

doi:10.1051/proc/201448020

Cited by 12 publications

(9 citation statements)

References 36 publications

(68 reference statements)

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“…For instance, WoS is not suited for PDEs with variable coefficients, and we do not here consider Neumann boundary conditions (needed for, e.g., linear elasticity). However, WoS can be generalized far beyond the basic PDEs considered in this paper, to include both linear and nonlinear elliptic, parabolic, and hyperbolic equations [Bossy et al 2015;Shakenov 2014;Pardoux and Tang 1999]. Moreover, Monte Carlo methods for PDEs are far broader than just WoS [Higham 2001;Kloeden and Platen 2013], and the basic framework presented here can be enhanced significantly by drawing on deep knowledge from fields like stochastic control, mathematical finance, and Monte Carlo rendering (Sec.…”

Section: Implicit Surfacesmentioning

confidence: 99%

“…In particular, PDEs with spatially inhomogeneous coefficients can be solved in the WoS framework via walks with smaller steps (akin to volumetric pathtracing). Some nonlinear PDEs can likewise be handled by, e.g., simulating branching diffusion [Bossy et al 2015], or by applying forward-backward stochastic differential equations [Pardoux and Tang 1999]. We also did not treat a variety of boundary conditions (Neumann, Robin, etc.)…”

Section: Limitations and Future Workmentioning

confidence: 99%

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Monte Carlo geometry processing

Sawhney

Crane

2020

ACM Trans. Graph.

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This paper explores how core problems in PDE-based geometry processing can be efficiently and reliably solved via grid-free Monte Carlo methods. Modern geometric algorithms often need to solve Poisson-like equations on geometrically intricate domains. Conventional methods most often mesh the domain, which is both challenging and expensive for geometry with fine details or imperfections (holes, self-intersections, etc. ). In contrast, grid-free Monte Carlo methods avoid mesh generation entirely, and instead just evaluate closest point queries. They hence do not discretize space, time, nor even function spaces, and provide the exact solution (in expectation) even on extremely challenging models. More broadly, they share many benefits with Monte Carlo methods from photorealistic rendering: excellent scaling, trivial parallel implementation, view-dependent evaluation, and the ability to work with any kind of geometry (including implicit or procedural descriptions). We develop a complete "black box" solver that encompasses integration, variance reduction, and visualization, and explore how it can be used for various geometry processing tasks. In particular, we consider several fundamental linear elliptic PDEs with constant coefficients on solid regions of R n. Overall we find that Monte Carlo methods significantly broaden the horizons of geometry processing, since they easily handle problems of size and complexity that are essentially hopeless for conventional methods.

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Section: Implicit Surfacesmentioning

confidence: 99%

Section: Limitations and Future Workmentioning

confidence: 99%

Monte Carlo geometry processing

Sawhney

Crane

2020

ACM Trans. Graph.

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“…A stochastic representation for these equations is well-understood and the methodology we have presented so far can be applied to them, see [41,Section 2.3]. Moreover, [17] use the polynomial representation of trigonometric functions to calculate the nonlinear Poisson-Boltzmann equation.…”

Section: More General Pdes Through Approximationmentioning

confidence: 99%

“…In [17] the authors deal with representations (and Monte Carlo simulation) for nonlinear divergence-form elliptic Poisson-Boltzmann PDE over the whole R 3 . Quasilinear parabolic PDEs (multidimensional and systems) admit a stochastic representation in terms of FBSDEs [55], [54] and many extensions exist ranging from representations for obstacle PDE problems [27] to representations of fully nonlinear elliptic and parabolic PDEs [20].…”

Section: A Brief Remark On Fbsdes and Stochastic Representationsmentioning

confidence: 99%

Hybrid PDE solver for data-driven problems and modern branching

Bernal¹,

Reis²,

Smith³

2017

Eur. J. Appl. Math

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The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often critical enough to warrant acknowledgement in the modelling phase. The purpose of this paper is to spread awareness of the Probabilistic Domain Decomposition (PDD) method, a fresh approach to the parallelization of PDEs with excellent scalability properties. The idea exploits the stochastic representation of the PDE and its approximation via Monte Carlo in combination with deterministic high-performance PDE solvers. We describe the ingredients of PDD and its applicability in the scope of data science. In particular, we highlight recent advances in stochastic representations for nonlinear PDEs using branching diffusions, which have significantly broadened the scope of PDD.We envision this work as a dictionary giving large-scale PDE practitioners references on the very latest algorithms and techniques of a non-standard, yet highly parallelizable, methodology at the interface of deterministic and probabilistic numerical methods. We close this work with an invitation to the fully nonlinear case and open research questions.

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“…On the other hand, the construction of numerical approximations to this problem for the case of κ (and λ) being piecewise constant has been the subject of several investigations, especially for the case of λ = 0. See [6,22,24,31] for recent results. Different schemes have been proposed, which include so-called kinetic schemes [21], mixing schemes [22], schemes relying on occupation times [20] and schemes using ideas of finite differences [22,25].…”

Section: Stochastic Analysis For the Two-dimensional Maxwell's Equationsmentioning

confidence: 99%

Probabilistic domain decomposition for the solution of the two-dimensional magnetotelluric problem

et al. 2016

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Stochastic domain decomposition is proposed as a novel method for solving the two-dimensional Maxwell's equations as used in the magnetotelluric method. The stochastic form of the exact solution of Maxwell's equations is evaluated using Monte-Carlo methods taking into consideration that the domain may be divided into neighboring sub-domains. These sub-domains can be naturally chosen by splitting the sub-surface domain into regions of constant (or at least continuous) conductivity. The solution over each sub-domain is obtained by solving Maxwell's equations in the strong form. The sub-domain solver used for this purpose is a meshless method resting on radial basis function based finite differences. The method is demonstrated by solving a number of classical magnetotelluric problems, including the quarter-space problem, the block-in-half-space problem and the triangle-in-half-space problem.

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Monte Carlo methods for linear and non-linear Poisson-Boltzmann equation

Cited by 12 publications

References 36 publications

Monte Carlo geometry processing

Monte Carlo geometry processing

Hybrid PDE solver for data-driven problems and modern branching

Probabilistic domain decomposition for the solution of the two-dimensional magnetotelluric problem

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