Analysis & Applied Math

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)