Preconditioning techniques for large-scale sparsity-promoting inverse problems

Speaker: Jonathan Lindbloom (Dartmouth)

Date: 4/29/25

Abstract: Hybrid projection methods have proven to be a powerful technique for solving large-scale linear inverse problems with ℓ2 (smoothing) regularization, enabling efficient regularization parameter selection via a reduced basis while also reducing the number of matrix-vector products with the forward model. However, a major obstacle arises when applying these methods to sparsity-promoting inverse problems: the prescribed reduced basis must be very large to adequately capture the sparsity in the solution, which greatly hinders the computational efficiency of the hybrid projection method. To address this, in this talk we introduce a new preconditioning technique that constructs a reduced basis in a transformed space which drastically reduces the number of basis vectors required to appropriately capture the sparsity. We use tools from matrix analysis to develop a theory characterizing the conditions under which our method is expected to outperform conventional methods and illustrate the efficacy of our technique using comprehensive numerical tests that compare our approach to existing methods based on generalized Krylov subspaces (GKS) or flexible Golub-Kahan bidiagonalizations (GKB).