Efficient simulation of mixed boundary value problems and conformal mappings

We present a stochastic method for the simulation of Laplace's equation with a mixed boundary condition in planar domains that are polygonal or bounded by circular arcs. We call this method the Reflected Walk-on-Spheres algorithm. The method combines a traditional Walk-on-Spheres algorithm with...

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Vydáno v:Applied mathematics and computation Ročník 488; s. 129119
Hlavní autoři: Han, Qiansheng, Rasila, Antti, Sottinen, Tommi
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier Inc 01.03.2025
Témata:
Brownian motion
Conformal mappings
Conformal modulus
Harmonic measure
Heat flow
Monte Carlo methods
Numerical algorithm
Walk on spheres
60J65
30C20
31A15
Harmonic measure
Heat flow
Conformal mappings
Walk on spheres
Conformal modulus
Brownian motion
31-08
Monte Carlo methods
65E05
65C05
30-08
Numerical algorithm
ISSN:0096-3003
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Popis
Shrnutí:We present a stochastic method for the simulation of Laplace's equation with a mixed boundary condition in planar domains that are polygonal or bounded by circular arcs. We call this method the Reflected Walk-on-Spheres algorithm. The method combines a traditional Walk-on-Spheres algorithm with use of reflections at the Neumann boundaries. We apply our algorithm to simulate numerical conformal mappings from certain quadrilaterals to the corresponding canonical domains, and to compute their conformal moduli. Finally, we give examples of the method on three dimensional polyhedral domains, and use it to simulate the heat flow on an L-shaped insulated polyhedron. •We provide an efficient stochastic algorithm for numerical computation of conformal mappings in the plane.•Our algorithm is based on the Walk-on-Spheres approach that is extended to certain types of mixed boundary value problems.•Harmonic solutions to mixed boundary value problems can also be obtained in higher dimensions.•Random walks are simulated independently of each other, allowing efficient use of parallel computation.•Algorithm does not require pre-existing mesh making it well-suited for conformal mesh generation.
ISSN:0096-3003
DOI:10.1016/j.amc.2024.129119