## Research Highlights

**When is superresolution of sparse signals possible?
** We quantify regimes of stable super-resolved recovery of sparse signals
from bandlimited measurements. In the case of adversarial deterministic noise of magnitude sigma, for a system of low-frequency
sinusoids with large super-resolution factor SRF (inverse bandwidth over grid spacing), we show that recovery of any k-sparse signal
holds in the regime shown in the picture, without additional spike spacing assumption. This rate is tight in the minimax sense: no
method can recover all k-sparse signals with higher noise levels. Compressed sensing (ell_1 minimization) generally operates at the
Shannon-Nyquist rate (SRF less than 1), hence is suboptimal for this task. Paper with Nam Nguyen.

**A scalable solver for the Helmholtz equation.
** We present a numerical method for the 2D high-frequency Helmholtz equation with online parallel complexity that scales
sublinearly as O(N/L), where N is the number of volume unknowns, and L is the number of processors, as long as L is a small fractional
power of N. The solver decomposes the domain into L layers, and uses transmission conditions in boundary integral form to explicitly
define "polarized traces", i.e., directional waves sampled at interfaces. Local direct solvers are used in each layer to precompute
traces of local Green's functions in an embarrassingly parallel way (the offline part), and incomplete Green's formulas are used to
propagate interface data in a sweeping fashion, as a preconditioner inside a GMRES loop (the online part). The favorable scalability
owes to the availability of fast algorithms for the integral kernels. Paper with Leonardo Zepeda-Nunez.

**Convex recovery from interferometric measurements.
** We show a deterministic stability result for the recovery of vectors from interferometric measurements, which have important
applications in model-robust imaging from scattered waves. The connectivity of the underlying graph determines the stability constant via the
spectral gap of the graph Laplacian. Paper 1 and
Paper 2, with Vincent Jugnon.

**Matrix probing: randomized
fitting for the
wave-equation Hessian.** What can be determined about the
pseudoinverse *pinv(A)* of a matrix A from one
application of A to a vector of random entries? A surprising lot, provided
pinv(A) is known to belong to a "good" vector subspace. This approach
offers a compelling preconditioner for the wave-equation Hessian (normal
operator) in seismic imaging. The performance guarantees follow from large
deviation estimates of independent interest. Paper with P. Letourneau, N.
Boumal, H. Calandra, J. Chiu, and S. Snelson.

**A butterfly algorithm for
synthetic aperture radar imaging.** We propose what is perhaps the
first O(N log N) controlled-accuracy algorithm for SAR imaging. We use the
butterfly scheme, an alternative to the FFT which works for much more
general oscillatory integrals than the Fourier transform. With M. Ferrara,
N. Maxwell, J. Poulson, and L. Ying. The paper is here,
and a version of the software is available here.

**Wave computation with
Fourier integral operators.** We propose a new time upscaling
method to avoid the CFL condition for acoustic wave propagation in a
smooth heterogeneous medium, by numerically representing the Green's
function as a sum of Fourier integral operators. The method hinges on
handling pseudodifferential symbols via discrete symbol calculus, and
Fourier integrals using any of the fast methods that have recently been
developed for their fast computation. With L. Ying. The paper is here.