Numerical methods for multiscale inverse problems
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This dissertation focuses on inverse problems for partial differential equations with multiscale coefficients in which the goal is to determine the coefficients in the equation using solution data. Such problems pose a huge computational challenge, in particular when the coefficients are of multiscale form. When faced with balancing computational cost with accuracy, most approaches only deal with models of large scale behavior and, for example, account for microscopic processes by using effective or empirical equations of state on the continuum scale to simplify computations. Obtaining these models often results in the loss of the desired fine scale details. In this thesis we introduce ways to overcome this issue using a multiscale approach. The first part of the thesis establishes the close relation between computational grids in multiscale modeling and sampling strategies developed in information theory. The theory developed is based on the mathematical analysis of multiscale functions of the type that are studied in averaging and homogenization theory and in multiscale modeling. Typical examples are two-scale functions f (x, x/[epsilon]), (0 < [epsilon] ≪ 1) that are periodic in the second variable. We prove that under certain band limiting conditions these multiscale functions can be uniquely and stably recovered from nonuniform samples of optimal rate. In the second part, we present a new multiscale approach for inverse homogenization problems. We prove that in certain cases where the specific form of the multiscale coefficients is known a priori, imposing an additional constraint of a microscale parametrization results in a well-posed inverse problem. The mathematical analysis is based on homogenization theory for partial differential equations and classical theory of inverse problems. The numerical analysis involves the design of multiscale methods, such as the heterogeneous multiscale method (HMM). The use of HMM solvers for the forward model has unveiled theoretical and numerical results for microscale parameter recovery, including applications to inverse problems arising in exploration seismology and medical imaging.