Kelvin transformations for simulations on infinite domains

Abstract

Solving partial differential equations (PDEs) on infinite domains has been a challenging task in physical simulations and geometry processing. We introduce a general technique to transform a PDE problem on an unbounded domain to a PDE problem on a bounded domain. Our method uses the Kelvin Transform, which essentially inverts the distance from the origin. However, naive application of this coordinate mapping can still result in a singularity at the origin in the transformed domain. We show that by factoring the desired solution into the product of an analytically known (asymptotic) component and another function to solve for, the problem can be made continuous and compact, with solutions significantly more efficient and well-conditioned than traditional finite element and Monte Carlo numerical PDE methods on stretched coordinates. Specifically, we show that every Poisson or Laplace equation on an infinite domain is transformed to another Poisson (Laplace) equation on a compact region. In other words, any existing Poisson solver on a bounded domain is readily an infinite domain Poisson solver after being wrapped by our transformation. We demonstrate the integration of our method with finite difference and Monte Carlo PDE solvers, with applications in the fluid pressure solve and simulating electromagnetism, including visualizations of the solar magnetic field. Our transformation technique also applies to the Helmholtz equation whose solutions oscillate out to infinity. After the transformation, the Helmholtz equation becomes a tractable equation on a bounded domain without infinite oscillation. To our knowledge, this is the first time that the Helmholtz equation on an infinite domain is solved on a bounded grid without requiring an artificial absorbing boundary condition.

Walk on Stars: A Grid-Free Monte Carlo Method for PDEs with Neumann Boundary Conditions

“…This has resulted in rapid advances along two main thrusts: First, increasing efficiency through optimized implementations [Mossberg 2021], bidirectional formulations [Qi et al 2022], and sample caching techniques [Miller et al 2023]. Second, increasing generality through new estimators that can solve PDEs in infinite domains [Nabizadeh et al 2021] or with variable coefficients [Sawhney et al 2022], simulate fluids [Rioux-Lavoie et al 2022], or enable differentiability for inverse problems [Yılmazer et al 2022]. Our focus in this paper is to further push the envelope along the second thrust: we develop WoSt, the first Monte Carlo estimator that can solve Neumann and mixed-boundary problems on general, non-convex domains while providing a performance-bias trade-off comparable to WoS.…”

“…There is a non-zero probability for a random walk to wander off to infinity when using WoSt in an open domain or in the exterior of a closed domain; this corresponds to the non-recurrent behavior of Brownian motion in 2D and 3D [Borodin and Salminen 2015, Chapter 2]. Nabizadeh et al [2021] use WoS to solve exterior problems with pure Dirichlet boundary conditions outside a closed domain by performing a spherical inversion of the domain. We leave extending their approach to mixed boundary-value problems to future work.…”

Solving nonlinear Klein-Gordon equations on unbounded domains via the finite element method

“…From there, several techniques can be found in the literature as early as the late 1970s: infinite elements introduced by Refs. [52][53][54] and implemented in the proprietary software comsol, combination of FEM and BEM (Boundary Element Method) [55], and several methods building on the Kelvin inversion [38,[56][57][58] or similar transformations [59,60].…”

Solving nonlinear Klein-Gordon equations on unbounded domains via the Finite Element Method

“…From there, several techniques can be found in the literature as early as the late 1970s: infinite elements introduced by Refs. [53][54][55] and implemented in the proprietary software comsol, combination of FEM and BEM (Boundary Element Method) [56], and several methods building on the Kelvin inversion [39,[57][58][59] or similar transformations [60,61].…”