Multiscale PDEs & inverse problems

For physical problems governed by PDEs, I use structural insights to guide computation: reduced representations for multiscale forward problems, and scaling limits for inverse problems.

Numerical homogenization and feature learning

Resolving fine-scale oscillations directly can make a PDE solver prohibitively expensive. Homogenization suggests that the essential behavior may admit a much smaller representation. I connect this observation with low-rank approximation and manifold learning to develop efficient multiscale methods.

Inverse problems through scaling limits

Recovering an unknown medium from scattered waves is often ill-conditioned. I use high-frequency limits to connect wave-based inverse problems with particle descriptions, and to identify the information that measurements should retain.

For inverse Helmholtz scattering, we probe with concentrated Gaussian beams and use the Husimi transform to record position and direction. We prove that the resulting scattering map approaches its Liouville counterpart at high frequencies and verify the connection numerically. The same connection can be drawn for time-independent problems.

Top row of Figure 4: three real Helmholtz wave fields at frequencies k equals 512, 1024, and 2048, concentrating around blue Liouville trajectories as frequency increases. — From waves to rays: the real Helmholtz wave field concentrates around the blue Liouville trajectory as frequency increases (k = 512, 1024, 2048, left to right). Figure 4, top row. Paper ↗

For papers, preprints, and related work, see Publications. Research code is available on GitHub ↗.