Infinite dimensional compressed sensing from anisotropic measurements and applications to inverse problems in PDE

TL;DR

This paper extends compressed sensing theory to infinite-dimensional settings with non-tight frames, providing explicit measurement bounds and demonstrating applications to inverse PDE problems like electrical impedance tomography.

Contribution

It introduces a framework for compressed sensing with frames in infinite-dimensional spaces, including explicit measurement bounds and applications to PDE inverse problems.

Findings

Explicit bounds on measurements for stable recovery

Effective nonuniform sampling strategies for asymptotically incoherent systems

Application to inverse PDE problems such as electrical impedance tomography

Abstract

We consider a compressed sensing problem in which both the measurement and the sparsifying systems are assumed to be frames (not necessarily tight) of the underlying Hilbert space of signals, which may be finite or infinite dimensional. The main result gives explicit bounds on the number of measurements in order to achieve stable recovery, which depends on the mutual coherence of the two systems. As a simple corollary, we prove the efficiency of nonuniform sampling strategies in cases when the two systems are not incoherent, but only asymptotically incoherent, as with the recovery of wavelet coefficients from Fourier samples. This general framework finds applications to inverse problems in partial differential equations, where the standard assumptions of compressed sensing are often not satisfied. Several examples are discussed, with a special focus on electrical impedance tomography.