Inverse analysis method using spectral decomposition of Green's function

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, relates the sources to the responses. Since this function behaves as a compact operator, it admits the spectral decomposition and is expressed as a sum of distinct eigen values and eigen functions. A suitable inverse operator that maps the responses to the sources is then determined. The proposed method consists of the following three procedures: (1) numerically computing the spectral decomposition and determining the inverse operator; (2) estimating a response function using a set of measured data; and (3) predicting a source function from the response function using inverse operator. A simple inverse analysis method, which uses a pointwise discretization of Green's function and computes a generalized inverse matrix applying the singular-value decomposition, is regarded as an approximation to compute the inverse operator. The accuracy, however, is much lower than the proposed method, because of the pointwise discretization and the less accurate computation of the spectral decomposition. Illustrative examples are solved to demonstrate the usefulness of the proposed inverse analysis method. Errors due to improper calculation of the inverse operator of Green’s function are shown.