A major goal of this talk is to connect with SCAN community on some specific computational questions from us, a group of data physicists and engineers. We have been working on solving the tissue magnetism (susceptibility, highly valuable in biomedicine) inverse problem, which has an ill-posed dipole kernel connecting susceptibility to a field (integral solution, IS), or a wave operator on susceptibility equal tp a Laplacian on a field (PDE). We have some good success (quantitative susceptibility mapping, QSM), and we hope to discuss with the audience a question on data fitting with IS vs with PDE: they seem to be equivalent to each other, but numerically they seem to give different results. We have very rich data experience, so we would like to share our engineering experience on preconditioning, which seems to be very important in practice, and we would also like to share our engineering experience with various gradient based solvers, which seems not to matter much. Then the second question is why so regarding preconditioning and solver. We seem to be comfortable in dealing with a single regularization parameter using the discrepancy principle (L curve), but we are struggling with selection of multiple regularization parameters. So the third question is, any good advice on selecting multiple regularization strengths? Lastly, we will touch on an exciting new development on solving transport inverse problem (critically important in biomedicine).