Nathan King | Searcharxiv

arXiv2024

Projected Walk on Spheres: A Monte Carlo Closest Point Method for Surface PDEs

We present projected walk on spheres (PWoS), a novel pointwise and discretization-free Monte Carlo solver for surface PDEs with Dirichlet boundaries, as a generalization of the walk on spheres method (WoS) [Muller 1956; Sawhney and Crane 2020]. We adapt the recursive relationship of WoS designed for PDEs in volumetric domains to a volumetric neighborhood around the surface, and at the end of each recursion step, we project the sample point on the sphere back to the surface. We motivate this simple modification to WoS with the theory of the closest point extension used in the closest point method. To define the valid volumetric neighborhood domain for PWoS, we develop strategies to estimate the local feature size of the surface and to compute the distance to the Dirichlet boundaries on the surface extended in their normal directions. We also design a mean value filtering method for PWoS to improve the method's efficiency when the surface is represented as a polygonal mesh or a point cloud. Finally, we study the convergence of PWoS and demonstrate its application to graphics tasks, including diffusion curves, geodesic distance computation, and wave propagation animation. We show that our method works with various types of surfaces, including a surface of mixed codimension.

math.NA↗

arXiv2023

A Closest Point Method for PDEs on Manifolds with Interior Boundary Conditions for Geometry Processing

Many geometry processing techniques require the solution of partial differential equations (PDEs) on manifolds embedded in $\mathbb{R}^2$ or $\mathbb{R}^3$, such as curves or surfaces. Such manifold PDEs often involve boundary conditions (e.g., Dirichlet or Neumann) prescribed at points or curves on the manifold's interior or along the geometric (exterior) boundary of an open manifold. However, input manifolds can take many forms (e.g., triangle meshes, parametrizations, point clouds, implicit functions, etc.). Typically, one must generate a mesh to apply finite element-type techniques or derive specialized discretization procedures for each distinct manifold representation. We propose instead to address such problems in a unified manner through a novel extension of the closest point method (CPM) to handle interior boundary conditions. CPM solves the manifold PDE by solving a volumetric PDE defined over the Cartesian embedding space containing the manifold, and requires only a closest point representation of the manifold. Hence, CPM supports objects that are open or closed, orientable or not, and of any codimension. To enable support for interior boundary conditions we derive a method that implicitly partitions the embedding space across interior boundaries. CPM's finite difference and interpolation stencils are adapted to respect this partition while preserving second-order accuracy. Additionally, we develop an efficient sparse-grid implementation and numerical solver. We demonstrate our method's convergence behaviour on selected model PDEs and explore several geometry processing problems: diffusion curves on surfaces, geodesic distance, tangent vector field design, harmonic map construction, and reaction-diffusion textures. Our proposed approach thus offers a powerful and flexible new tool for a range of geometry processing tasks on general manifold representations.

cs.GR↗