  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.
