In this talk we discuss sparse spectral methods capable of rapidly and automatically determining a set of Fourier basis functions whose span is guaranteed to contain an accurate approximation of the solution of a given PDE on a (potentially very high-dimensional) periodic domain. This small, near-optimal Fourier basis is then used to efficiently solve the given PDE in a runtime which only depends on the PDE's data compressibility properties, while breaking the curse of dimensionality and relieving linear dependence on any multiscale structure in the original problem. Convergence analysis in the Sobolev norm for a general class of non-constant diffusion equations will be discussed in the elliptic setting, as well as initial attempts to extend the methods to related parabolic PDE. Numerical experiments will demonstrate good empirical performance on several multiscale and high-dimensional example problems, showcasing the promise of the proposed methods in practice.

This talk will draw on joint work with various subsets of Craig Gross/Grosch (MSU) and Tanvi Mahajan (MSU).

Bio: Mark A. Iwen is a professor at Michigan State University whose research lies at the intersection of computational harmonic analysis, mathematical data science, signal processing, sparse approximation, and randomized numerical algorithms. His work focuses particularly on developing fast and memory-efficient methods for large-scale and high-dimensional computational problems.

He received B.S. degrees in Computer Science and Mathematics from the University of Wisconsin–Milwaukee in 2002 and a Ph.D. in Applied and Interdisciplinary Mathematics from the University of Michigan in 2008.

His research includes sparse Fourier and spectral methods, dimensionality reduction and Johnson–Lindenstrauss embeddings, compressive sensing, phase retrieval, low-rank matrix and tensor approximation, and numerical methods for high-dimensional partial differential equations.

Iwen has led and participated in multiple NSF-funded research projects. His honors include the Michigan State University College of Natural Science Early Career Research Award and a CRM–Simons Visiting Professorship. He has also helped organize the One World Mathematics of INformation, Data, and Signals (MINDS) Seminar.