Inversion Method Using Spectral Decomposition of Green’s Function

Abstract

An inverse analysis method using the spectral decomposition of Green’s function is proposed. For linear inverse problems of identifying inner sources from surface responses, Green’s function, which is a solution of the corresponding physical problem, relates the sources to the responses and hence behaves as a compact operator. A suitable inverse operator that maps the responses to the sources can be determined by using the spectral decomposition of Green’s function. The proposed method numerically computes the spectral decomposition and determines the inverse operator. It is possible to regard a simple inverse analysis method, which uses a point-wise discretization of Green’s function and computes a generalized inverse matrix using the singular-value decomposition, as an approximation to compute the inverse operator. The accuracy, however, could be much lower than the proposed method, because of the point-wise discretization and the less accurate computation of the spectral decomposition. Illustrative examples are solved to demonstrate the usefulness of the proposed inverse analysis method, as well as errors inherent to inverse analysis methods which do not compute the spectral decomposition of Green’s function properly.