In this talk we outline the challenges facing typical linear inverse problems from physical sciences: limited amount of data compared to the number of unknowns in the linear system, noise in the right hand side data, poor conditioning of the matrices and shortage of computational resources for the large data sets that are typically encountered. We discuss algorithms and practical strategies to address these problems and illustrate with examples from an inverse problem in geotomography. We start with classical $\\ell_2$ based minimization and go on to discuss different classes of algorithms (convex and non-convex) for sparsity constrained minimization motivated by theoretical results in compressive sensing. We then discuss approaches for mixed penalty regularization and the use of wavelet based constraints.

Finally, we turn our attention to computation and discuss practical strategies for implementing the algorithms discussed with very large matrices using techniques such as wavelet based compression and randomized low rank singular value decompositions. We conclude by discussing additional applications and how the mentioned techniques can take advantage of today's multiprocessor/core architectures.