Log In Sign Up

Kernel-based methods for Solving Time-Dependent Advection-Diffusion Equations on Manifolds

by   Qile Yan, et al.

In this paper, we extend the class of kernel methods, the so-called diffusion maps (DM) and ghost point diffusion maps (GPDM), to solve the time-dependent advection-diffusion PDE on unknown smooth manifolds without and with boundaries. The core idea is to directly approximate the spatial components of the differential operator on the manifold with a local integral operator and combine it with the standard implicit time difference scheme. When the manifold has a boundary, a simplified version of the GPDM approach is used to overcome the bias of the integral approximation near the boundary. The Monte-Carlo discretization of the integral operator over the point cloud data gives rise to a mesh-free formulation that is natural for randomly distributed points, even when the manifold is embedded in high-dimensional ambient space. Here, we establish the convergence of the proposed solver on appropriate topologies, depending on the distribution of point cloud data and boundary type. We provide numerical results to validate the convergence results on various examples that involve simple geometry and an unknown manifold. Additionally, we also found positive results in solving the one-dimensional viscous Burger's equation where GPDM is adopted with a pseudo-spectral Galerkin framework to approximate nonlinear advection term.


page 15

page 17


Ghost Point Diffusion Maps for solving elliptic PDE's on Manifolds with Classical Boundary Conditions

In this paper, we extend the class of kernel methods, the so-called diff...

Kernel Methods for Bayesian Elliptic Inverse Problems on Manifolds

This paper investigates the formulation and implementation of Bayesian i...

Spectral methods for solving elliptic PDEs on unknown manifolds

In this paper, we propose a mesh-free numerical method for solving ellip...

Radial basis approximation of tensor fields on manifolds: From operator estimation to manifold learning

We study the Radial Basis Function (RBF) approximation to differential o...

Efficient numerical valuation of European options under the two-asset Kou jump-diffusion model

This paper concerns the numerical solution of the two-dimensional time-d...

Diffusion Curvature for Estimating Local Curvature in High Dimensional Data

We introduce a new intrinsic measure of local curvature on point-cloud d...