Application of spectral decomposition of Green’s function to linear inverse problem Muneo Hori * Earthquake Research Institute, University of Tokyo, Yayoi 1-1-1, Bunkyo-ku, Tokyo 113-0032, Japan Received 14 January 2002; revised 11 March 2002; accepted 2 April 2002 Abstract An inverse analysis method using the spectral decomposition of Green’s function is proposed for linear inverse problems of identifying inner sources from data of surface responses. It is assumed that Green’s function of the corresponding physical problem is given. Applying the spectral decomposition, Green’s function is discretized as a sum of eigen-values and eigen-functions. From the comparison of the measurement accuracy with the eigen-values, it is shown that responses which can be actually measured are given as a linear combination of eigen-functions corresponding to larger eigen-values. Such responses are found by determining coefficients of the eigen-functions from the measured data, and then sources which are predictable are determined just by calculating their coefficients for the eigen-functions. Without any ambiguity, the proposed method can determine the predictable inner sources from the data which are measured with the limited accuracy. A numerical simulation of solving a simple example problem is carried out to demonstrate the usefulness of the proposed inverse analysis method, and the results are discussed. q 2003 Elsevier Ltd. All rights reserved. Keywords: Green’s function; Inverse analysis method; Function analysis theory; Spectral decomposition; Eigen-value problem for operator 1. Introduction In many fields of engineering and sciences, solving a linear inverse problem of finding sources within a body from measured data is a common practice. The data are usually measured on points on the surfaces of the body. Thus, this problem often becomes ill-posed, as a small change in the data induces a large change in the predicted source. It is essentially important to develop a reliable method which can solve this ill-posedness; see, for instance, Refs. [1–4] for a list of references related to research in the geophysics; see also Refs. [5–8]. While there are a variety of pairs of sources and responses, a linear relation that corresponds to a certain physical process holds between them and the linear relation is mathematically expressed in terms of Green’s function. Hence, the authors are seeking a rational treatment of Green’s function to find such a reliable inverse analysis method. It should be emphasized that a closed analytic form is not necessarily given to Green’s function considered here; the proposed method is applicable to cases when Green’s function is numerically computed. The inverse problem of finding the source from the data has two elements, the physical process and the measure- ment. The physical process relates the source to the response, and the measurement gives the data as the values of the response at some points. Fig. 1(a) schematically shows these two elements; an arrow from the source to the response represents the physical process, and an arrow from the response to the data the measurement. While ordinary inversion analysis methods seek to directly find the source from the data, it appears more natural to follow the two arrows backwards. We take this methodology, first to predict the response from the data and then to determine the source from the response. These two procedures are called the estimation and the backward mapping; see Fig. 1(b). Although the above methodology appears appealing, it is not easy to carry out the estimation and the backward mapping since, for instance, the estimation is to find a function which represents the response from a set of discrete data. The actual measurement always has some noises, and we cannot expect that a response function is perfectly retrieved from the data. The inverse analysis method must 0955-7997/03/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0955-7997(03)00049-3 Engineering Analysis with Boundary Elements 28 (2004) 183–193 www.elsevier.com/locate/enganabound * Tel.: þ81-3-3812-2111; fax: þ 81-3-3816-1159. E-mail address: hori@eri.u-tokyo.ac.jp (M. Hori).