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Title: Elliptic Inverse Problems of Identifying Nonlinear Parameters
Inverse problems of identifying parameters in partial differential equations (PDEs) is an important class of problems with many real-world applications. Inverse problems are commonly studied in optimization setting with various known approaches having their advantages and disadvantages. Although a non-convex output least-squares (OLS) objective has often been used, a convex modified output least-squares (MOLS) attracted quite an attention in recent years. However, the convexity of the MOLS has only been established for parameters appearing linearly in the PDEs. The primary objective of this work is to introduce and analyze a variant of the MOLS for the inverse problem of identifying parameters that appear nonlinearly in variational problems. Besides giving an existence result for the inverse problem, we derive the first-order and second-order derivative formulas for the new functional and use them to identify the conditions under which the new functional is convex. We give a discretization scheme for the continuous inverse problem and prove its convergence. We also obtain discrete formulas for the new MOLS functional and present detailed numerical examples.
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Pure and Applied Functional Analysis
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National Science Foundation
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