Compressed sensing for inverse problems

Inverse problems are fundamental in many areas of science and engineering, yet their theoretical analysis often assumes access to an infinite number of measurements. While some studies address scenarios with finite measurements, they frequently neglect sparsity constraints, which are crucial in practical applications where only a limited number of noisy measurements can be obtained. The principles of compressed sensing (CS) suggest that leveraging sparsity can significantly enhance reconstruction performance. This highlights the need for a comprehensive sample complexity theory for ill-posed inverse problems to determine the minimal measurement requirements for stable and accurate reconstruction. The aim of this thesis is to take the first steps toward developing an abstract framework for compressed sensing in inverse problems. Our framework enables improved sample complexity estimates under specific structural conditions of the forward map and measurement operators. The setting is further extended to address practical challenges, including the need to balance truncation errors and optimize sampling strategies. By incorporating these refinements, the theory gains broader applicability while maintaining rigorous recovery guarantees. The proposed methodology is demonstrated through various applications, including sparse angle tomography, Fourier sampling for MRI, deconvolution problems, inverse source recovery for elliptic PDEs, and photoacoustic tomography. Each case study illustrates how the abstract framework translates into concrete sample complexity bounds and stability results, confirming its versatility across diverse inverse problem settings. By advancing the theoretical foundations of compressed sensing for inverse problems, this work lays the groundwork for further research in efficient and stable subsampling strategies in inverse problems.