An abstract framework for elliptic inverse problems: Part 1. An output least-squares approach

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The solution of an elliptic boundary value problem is an infinitely differentiable function of the coefficient in the partial differential equation. When the (coefficient-dependent) energy norm is used, the result is a smooth, convex output least-squares functional. Using total variation regularization, it is possible to estimate discontinuous coefficients from interior measurements. The minimization problem is guaranteed to have a solution, which can be obtained in the limit from finite-dimensional discretizations of the problem. These properties hold in an abstract framework that encompasses several interesting problems: the standard (scalar) elliptic BVP in divergence form, the system of isotropic elasticity, and others.

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Copyright 2007 SAGE Publications. Publisher's version of record: https://doi.org/10.1177/1081286505055758

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Mathematics and Mechanics of Solids