Variational Imbedding for Coefficient Identification in Elliptic Partial Differential Equation
نویسندگان
چکیده
We consider the inverse problem for identification of the coefficient in an elliptic partial differential equation inside of the unit square D, with overposed boundary data. This problem is not investigated enough in the literature due to lack of results about the uniqueness of the inverse problem. Following the main idea of the so-called Method of Variational Imbedding (MVI), we embed the solution of the inverse problem into the elliptic boundary value problem stemming from the necessary conditions for minimization of the quadratic functional of the original equation. The system contains a well posed fourth-order boundary value problem for the sought function and an explicit equation for the unknown coefficient. We solve the imbedding b.v.p. numerically making use of operator-splitting for the forth-order boundary value problem. Convergence and stability of the numerical method are established. We introduce an effective way for finding the best approximation for the coefficient. Featuring examples are elaborated numerically.
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