Theory of Inverse ProblemsThis research was funded by NSF grants DMS-1616340 and NRT-HDR 1922658. Separable nonlinear inverse problemsCompressed-sensing theory does not explain the success of convex-programming methods for deterministic inverse problems where the measurement operator is highly locally coherent. Examples include spike deconvolution, heat-source localization and estimation of brain activity from electroencephalography data. We build a general theory of sparse recovery adapted to such settings.
Sampling theorems for deconvolutionEstimation of spikes from samples of their convolution with a smooth kernel is an important inverse problem in imaging and reflection seismography. Here we establish a sampling theorem, showing that exact and robust recovery via convex programming is possible from any sampling pattern that contains two samples close to each spike, as long as the signal satisfies a minimum-separation condition.
Super-resolution of point sourcesWe consider the problem of super-resolving point sources from low-pass data via convex programming and the related problem of super-resolving the spectrum of a multisinusoidal signal from a finite number of samples. We establish exact-recovery and stability guarantees under a minimum separation condition, as well as robustness to outliers.
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