Mathematical Structure of Numerical Analysis and Regularization of an Inverse Boundary Value Problem in Laplace Field

Abstract

Inverse boundary value problems deal with the estimation of boundary values on incompletely prescribed boundaries. Numerical analysis of inverse boundary value problems using discretization methods can be reduced to solving simultaneous equations, which are ill-conditioned due to the ill-posedness of the problems. Errors included in prescribed boundary values are therefore magnified remarkably by inverse calculation without regularization. In this study, the mathematical structure of this error magnification behavior was studied theoretically for inverse boundary value problems of the Laplace field. Singular value decomposition was applied to evaluate the magnification amplitude called the condition number, which was given as the ratio of the maximum singular value to the nonzero minimum singular value of the coefficient matrix. The condition number was found to represent the magnification amplitude of the highest frequency fluctuation mode of variables. A regularization method using effective pseudoinverse was introduced, in which the rank of the coefficient matrix was reduced to effective rank and therefore small singular values were ignored. The effectiveness of the use of the effective pseudoinverse was explained using the singular values and the right singular vectors. An equation was proposed for evaluating the condition number. The admissible condition number method was proposed for determining the effective rank. Numerical simulations showed that the proposed equation and method were useful for estimating the condition number and the effective rank, and obtaining good estimates of boundary values.