ANALYTICAL METHODS FOR EXTRACTING DISCONTINUITY IN INVERSE PROBLEMS: THE PROBE METHOD AFTER 10 YEARS

摘要

In this paper we consider inverse problems for partial differential equations. Here we mean by inverse problems, the problem of how to extract information about unknown objects from observation data consisting of several physical quantities. Many important inverse problems are formulated as inverse problems for partial differential equations, the observation data are described by using the solutions and there is much research from the mathematics side [47]. Especially, almost ten years ago, the author, who was seeking a direct method in inverse problems originating from noninvasive or nondestructive testings of the living body or other material, discovered the Probe Method [18] and the Enclosure Method [22] which aim to extract the discontinuities such as cavities, inclusions, cracks and obstacles in the medium. Mysteriously, almost simultaneously, other methods such as (1) the Linear Sampling Method of Colton-Kirsch [8], (2) the Factorization Method of Kirsch [49, 50], and (3) the Singular Sources Method of Potthast [56] had also appeared. 1 The various modifications of the methods and attempts at expanding the range of applications to other inverse problems are still performed at the same time as studying them. However, to our regret, it is a current state that one cannot say easily that the research of this direction is recognized enough in the community of functional equations in Japan. Then, in this paper, we introduce the ideas of both the Probe Method and the Enclosure Method, again by applying some prototype inverse problems for the Laplace equation, the Helmholtz equation, and the heat equation, and discuss the current state of the research and the problems in the future.