Numerical Methods for Partial Differential Equations Seminar

Time-harmonic acoustic scattering in variable media can often be modeled by variable-coefficient Helmholtz equations. Such equations pose several numerical challenges. For example, they can be intrinsically ill-conditioned, necessitate the imposition of radiation conditions, and produce pollution errors when discretized with standard finite difference or finite element methods.

To avoid these issues, it is often convenient to reformulate the Helmholtz equation as an integral equation known as the Lippmann-Schwinger equation. The tradeoffs are that an integral operator with a singular kernel must be discretized and that the resulting linear system that must be inverted is dense. In this talk, I will focus on the latter issue and present a new direct solver for this purpose.

For a problem with fixed wavenumber and N degrees of freedom, the number of operations required for the solver scales as O(N^(3/2)) with very favorable constants. Moreover, the solver is highly stable in practice. The solver is particularly effective when used to compute a preconditioner for an iterative solver. In this regime, I will show that problems up to 500 by 500 wavelengths with large amounts of back scattering can be solved in mere hours on a workstation. Time permitting, I will also discuss some work in progress that extends this solver to Maxwell's equations.

This is joint work with Gunnar Martinsson (UT Austin) and the extension to Maxwell's equation is joint work with Hanwen Zhang (Yale).

Atomic systems (molecules, crystals, proteins, etc.) are naturally represented by a set of coordinates in 3D space labeled by atom type. This is a challenging representation to use for machine learning because the coordinates are sensitive to 3D rotations, translations, and inversions (the symmetries of 3D Euclidean space). In this talk I’ll give an overview of Euclidean invariance and equivariance in machine learning for atomic systems. Then, I’ll share some recent applications of these methods on a variety of atomistic modeling tasks (ab initio molecular dynamics, prediction of crystal properties, and scaling of electron density predictions). Finally, I’ll explore open questions in expressivity, data-efficiency, and trainability of methods leveraging invariance and equivariance.