On the Identication of Coecient and Source Parameters in Elliptic Systems Modelled with Many Boundary Values Problems
Nilson C. Roberty *
Nuclear Engineering Program-COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil.
*Author to whom correspondence should be addressed.
Abstract
The inverse problem for determination of parameters related to the support and/or functions describing the intensity of coefficient and sources in models based strongly elliptic second order systems is posed with Cauchy data over specification at boundary. This stablish a set of various boundary value problems associated with the same group of unknown parameters. A Lipschitz boundary dissection is used for decomposing each Cauchy data into pairs of complementary mixed boundary values problems. The concept of Calderon projector is introduced as a tool to check the consistency of the Cauchy data and to demonstrate the equivalence of these two problems. This lets you define a discrepancy function to measure the distance between the solutions of problems obtained by dissecting Lipschitz Cauchy data. This discrepancy appears as a consequence of inadequate parameters values in the constitutive relations. For Cauchy noisy data, the difference between these solutions would be small if the parameters used in the solution are correct. The methodology we propose explores concepts as Lipschitz Boundary Dissection, Complementary Mixed Problems with trial parameters and Internal Discrepancy fields. Differentiable and non-differentiable optimizations algorithms can then be used in the reconstruction of these parameters simultaneously. Numerical experiments are presented.
Keywords: Inverse problems, coecients and source reconstruction, elliptic systems, Lipschitz Dissection, Calderon projector, internal discrepancy.