The investigators aim to reduce drastically the costs of numerical inversion (as occurs, for example, in medical imaging) by blending new parametric level-set approaches and nonlinear least squares methods together with innovations in linear solvers, preconditioning, and model reduction of parameterized systems. Four strategies are combined to reduce the computation necessary while not degrading accuracy of the solution. First, the dimension of the inverse problem is drastically reduced by developing low-order parametric inversion methods replacing the usual voxel-based inversion. Second, the optimization underlying parametric inversion incorporates novel nonlinear least-squares solvers specifically designed to deal with ill-conditioned Jacobians. Third, the high cost of solving many large forward problems is reduced through new model reduction techniques that are particularly well-suited to the structure of the inverse problems under consideration. Fourth, the high costs of computing reduced models and solving forward problems is reduced by innovations in Krylov subspace recycling and efficient reuse of preconditioners for parameterized linear systems.
The inverse problems studied here involve recovery of images describing how unknown quantities of diagnostic interest (such as electrical conductivity) are distributed throughout a given medium (such as human tissue or soil). These images can reveal the presence or absence of anomalies, such as tumors in human tissue or contaminant plumes in soil. The computational extraction of high quality images from noisy surface measurements in reasonable time is a very difficult task. As rapid advances in technology make it possible to take vastly more measurements, computational bottlenecks become ever more acute, impeding innovation in medical and other areas of imaging. This project aims to combine innovations in diverse fields within computational linear algebra, systems theory, and optimization to create dramatically improved strategies for image extraction.
