Graph-theoretic algorithms for Kolmogorov operators: Approximating solutions and their gradients in elliptic and parabolic problems on manifolds

TL;DR

This paper introduces kernel-based graph algorithms to approximate Kolmogorov operators on manifolds, enabling solutions and gradients computation in high-dimensional or complex geometric settings without spatial discretization.

Contribution

It develops a method using eigendecomposition of kernel matrices to invert operators and compute gradients, applicable directly from samples on manifolds.

Findings

Efficient kernel-based algorithms approximate differential operators on manifolds.

The methods work in high dimensions without requiring spatial discretization.

Use of $k$-$d$ tree accelerates kernel matrix computation.

Abstract

We employ kernel-based approaches that use samples from a probability distribution to approximate a Kolmogorov operator on a manifold. The self-tuning variable-bandwidth kernel method [Berry & Harlim, Appl. Comput. Harmon. Anal., 40(1):68--96, 2016] computes a large, sparse matrix that approximates the differential operator. Here, we use the eigendecomposition of the discretization to (i) invert the operator, solving a differential equation, and (ii) represent gradient vector fields on the manifold. These methods only require samples from the underlying distribution and, therefore, can be applied in high dimensions or on geometrically complex manifolds when spatial discretizations are not available. We also employ an efficient $k$-$d$ tree algorithm to compute the sparse kernel matrix, which is a computational bottleneck.