Ill-posed inverse problems arise in many fields of science and engineering. Their solution, if it exists, is very sensitive to perturbations in the data. The challenge of working with linear discrete ill-posed problems comes from the ill-conditioning and the possible large dimension of the problems. Regularization methods aim to reduce the sensitivity by replacing the given problem with a nearby one, whose solution is less affected by perturbations. The methods in this talk are concerned with solving large scale problems by projecting them into a Krylov or generalized Krylov subspace of fairly small dimension. The first type of methods discussed are based on Bregman-type iterative methods that even though the high quality reconstruction that they deliver, they may require a large number of iterations and this reduces their attractiveness. We develop a computationally attractive linearized Bregman algorithm by projecting the problem to be solved into an appropriately chosen low-dimensional Krylov subspace. Recently, the use of a \(p\)-norm to measure the fidelity term and a \(q\)-norm to measure the regularization term has received considerable attention. For applications such as image reconstruction, where the pixel values are non-negative, we impose a non-negativity constraint to make sure the reconstructed solution lies in the non-negative orthant. We propose techniques to select the regularization parameter without any significant computational cost. This makes the proposed method more efficient and useful especially for large-scale problems. In addition, we explore how to estimate maximum a posteriori when the available data are perturbed with non-Gaussian noise. Near the end of the talk will be discussed current work in progress on solving time-dependent inverse problems with the goal to preserve edges and model small changes in time, solving separable nonlinear inverse problems as well as learning strategies based on optimal experimental design and Krylov subspaces. Numerical examples illustrate the performances of the approaches proposed in terms of both accuracy and efficiency. We consider two-dimensional problems, with a particular attention to the restoration of blurred and noisy images.
Time: April 23, 2021 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Zhu Wang