Graph-theoretic algorithms for Kolmogorov operators: Approximating solutions and their gradients in elliptic and parabolic problems on manifolds
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Publication:6366542
DOI10.1007/S10092-022-00495-0arXiv2104.15124MaRDI QIDQ6366542
Dimitrios Giannakis, Andrew D. Davis
Publication date: 30 April 2021
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 - tree algorithm to compute the sparse kernel matrix, which is a computational bottleneck.
Computational methods for problems pertaining to statistics (62-08) Density estimation (62G07) Statistics on manifolds (62R30) Learning and adaptive systems in artificial intelligence (68T05) Graphs and linear algebra (matrices, eigenvalues, etc.) (05C50)
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