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We employ kernel-based approaches that use samples from a probabilitydistribution to approximate a Kolmogorov operator on a manifold. Theself-tuning variable-bandwidth kernel method [Berry & Harlim, Appl. Comput.Harmon. Anal., 40(1):68--96, 2016] computes a large, sparse matrix thatapproximates the differential operator. Here, we use the eigendecomposition ofthe discretization to (i) invert the operator, solving a differential equation,and (ii) represent gradient vector fields on the manifold. These methods onlyrequire samples from the underlying distribution and, therefore, can be appliedin high dimensions or on geometrically complex manifolds when spatialdiscretizations are not available. We also employ an efficient $k$-$d$ treealgorithm to compute the sparse kernel matrix, which is a computationalbottleneck.

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