Inverse problems that are governed by partial differential equations arise in many applications in computational science and engineering. Solving these large scale problems is a real challenge to the existing numerical methods, as they are generally highly ill-posed and non-convex. These difficulties are usually handled by introducing statistical Bayesian estimation methods that promote a-priori knowledge to the problems. Such methods address the uncertainties in the inverse problems, such as the noise and unknown parameters, so that the solution of the inverse problems is meaningful and realistic. Additionally, numerically solving these large-scale problems requires highly efficient and scalable numerical methods, which are still missing or inadequate for many applications.
Multiscale and multigrid methods are extremely efficient and scalable for some applications, but there are other problems that still pose severe challenges. In this research I plan to study two problems: one is the challenging inverse wave equation, which appears in many applications such as seismic exploration of reservoirs, and medical imaging. The other problem is the rather unexplored 4D imaging of flow in porous media, used for monitoring of reservoirs, carbon sequestration among other applications. I plan to develop efficient multiscale (and multigrid) methods for some of the key ingredients of the numerical solution of these problems. Such methods may enable a scalable and efficient solution of these inverse problems. In particular, to solve the inverse wave equation, one needs to efficiently solve the Helmholtz equation, which is still considered an open question. Another example is a multiscale approach for efficiently estimating an inverse of a covariance matrix given a few measurements. This problem lies in the heart of statistical Bayesian estimation methods, and it can be addressed if one considers the structure of the covariance for the problems of interest.
Fields of science
- natural sciencesmathematicsapplied mathematicsnumerical analysis
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equations
- engineering and technologymedical engineeringdiagnostic imaging
- natural sciencescomputer and information sciencescomputational science
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