MSC: 65R32 65N21 35J05 45A05 Keywords: Corrosion detection Robin inverse problem Boundary integral equation a b s t r a c t Based on an integral equation formulation, we present numerical methods for the inverse problem of recovering part of the domain boundary from boundary measurements of solutions to the Laplace equation on an accessible part of the boundary.

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We study the inverse problem of recovering an interior interface from a boundary measurement in an elliptic boundary value problem arising from a semiconductor transistor model. We set up a nonlinear least-squares formulation for solving the inverse problem, and establish the necessary derivatives with respect to the interface. We then propose both the GaussNewton iterative method and the conjugate gradient method for the leastsquares problem, and present implementation of these methods using integral equations.

In this dissertation, we study boundary inverse problems for some elliptic partial differential equations. These are problems arising from quantitative analysis of various nondestructive testing techniques in applications. In such a problem, we are interested in using boundary measurements of the solution to recover either an unknown coefficient function in the boundary condition, or a portion of the boundary, or an unknown interior interface. We first introduce formulations of the boundary value problems into integral equations, then design numerical algorithms for solving each of these inverse problems. Numerical implementation and examples are presented to illustrate the feasibility and effectiveness of the numerical methods.