ACMD Seminar: Monotone discretizations of levelset convex geometric PDEs

Wonjun Lee
NIST-IMA Postdoctoral fellow, Institute for Mathematics and its Applications, University of Minnesota

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Abstract: We present a new algorithm to compute level set convex viscosity solutions of Hamiton-Jacobi equations. The algorithm can be used to solve a wide class of curvature motion PDEs, as well as a recent Hamilton-Jacobi equation for the Tukey depth, which is a statistical depth measure of data points. The algorithm is based on monotone schemes that involve partial derivatives in directions orthogonal to the gradient. We provide the convergence analysis of the algorithm on regular Cartesian grids and unstructured point clouds in any dimensions, and numerical experiments that demonstrate approximated solutions of affine flows in 2D and Tukey depth measure of high dimensional data.

Bio: Wonjun Lee is a joint NIST-IMA Postdoctoral fellow in Analysis of Machine Learning at the Institute for Mathematics and its Applications (IMA) at the University of Minnesota (UMN). He completed his Ph.D. at the University of California, Los Angeles in mathematics in 2022 under the guidance of Professor Stan Osher. Throughout his Ph.D., he has developed optimal transport-based algorithms to solve nonlinear partial differential equations (PDEs) such as Darcy’s law, tumor growth model, and mean field games. As a Postdoc, he is working with Professor Jeff Calder, Gilad Lerman, and Li Wang at UMN to develop PDE-based algorithms to solve high-dimensional machine learning problems and analyze the theoretical properties of the algorithms.

Host: Andrew Dienstfrey

Note: This talk will be recorded to provide access to NIST staff and associates who could not be present to the time of the seminar. The recording will be made available in the Math channel on NISTube, which is accessible only on the NIST internal network. This recording could be released to the public through a Freedom of Information Act (FOIA) request. Do not discuss or visually present any sensitive (CUI/PII/BII) material. Ensure that no inappropriate material or any minors are contained within the background of any recording. (To facilitate this, we request that cameras of attendees are muted except when asking questions.)

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