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

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