Integral equations of the first kind, inverse problems and regularization: a crash course

This paper is an expository survey of the basic theory of regularization for Fredholm integral equations of the first kind and related background material on inverse problems. We begin with an historical introduction to the field of integral equations of the first kind, with special emphasis on model inverse problems that lead to such equations. The basic theory of linear Fredholm equations of the first kind, paying particular attention to E. Schmidt’s singular function analysis, Picard’s existence criterion, and the Moore-Penrose theory of generalized inverses is outlined. The fundamentals of the theory of Tikhonov regularization are then treated and a collection of exercises and a bibliography are provided. 1. Historical Background Most of history is guessing, and the rest is prejudice. W. & A. Durant The recasting of Ivar Fredholm’s theory of linear integral equations of the second kind by David Hilbert and Erhardt Schmidt in the first decade of the last century has had an enormous influence on modern mathematics. Indeed, the Hilbert-Schmidt geometrization of analysis, which is an outgrowth of Fredholm’s theory of integral equations of the second kind, is one of the great triumphs of twentieth century mathematics. On the other hand, it seems that integral equations of the first kind are much less familiar, even exotic. But, as the terminology suggests, integral equations of the first kind came first (the term “integral equation” seems to have been coined by du Bois-Reymond in 1888 [15]). As remarked by one of the early chroniclers of the theory of integral equations, Maxime Bôcher [14]: The theory of integral equations may be regarded as dating back at least as far as the discovery by Fourier of the theorem concerning integrals which bears his name; for, though this was not the point of view of Fourier, this theorem may be regarded as a statement of the solution of a certain integral equation of the first kind. Here Bôcher is referring to Fourier’s “inversion” formula for the integral equation of the first kind g(x) = √ 2 π ∫ ∞ 0 cos (xξ)f(ξ)dξ namely, f(x) = √ 2 π ∫ ∞ 0 cos (xξ)g(ξ)dξ. Inverse Problems in Applied Sciences—towards breakthrough IOP Publishing Journal of Physics: Conference Series 73 (2007) 012001 doi:10.1088/1742-6596/73/1/012001 c © 2007 IOP Publishing Ltd 1 A more generally accepted view is that the theory of integral equations began with an inverse problem – Abel’s analysis of the mechanical problem of finding the curve of descent, given the time of descent as a function of the vertical distance of fall [1]. The problem is to find the unknown path in the plane along which a particle will fall, under the influence of gravity alone, so that at each instant the time of fall is a known function of the distance fallen. Suppose that the particle falls from height z and that the path of descent is parameterized by arclength s, that is, at time t the length of arc traversed is s(t) (s(0) = 0). Assuming that the particle starts from rest, we find by equating the gain in kinetic energy to the loss in potential energy that: 1 2 ( ds dt )2 = a(z − y), where a is the gravitational acceleration. Integrating this, we find that the time of descent from z to the base line y = 0, g(z) is given by