Inverse Problems

The present volume contains manuscripts of lectures or topics related to the lectures which were given at the conference on "Inverse Problems" at the mathematical Research Institute at Oberwolfach. The conference took place during the week of May 18-24, 1986, and was managed by the editors. Recalling Professor Joseph Keller's paper entitled Inverse Problems, American Mathematical Monthly, 83 (1976), we give two direct quotes. "We call two problems inverses of one another if the formulation of each involves all or part of the solution of the other. Often, for historical reasons, one of the two problems has been studied extensively for some time, while the other is newer and not so well understood. In such cases, the former is called the direct problem, while the latter is called the inverse problem. " "The main sources of inverse problems are science and engineering. Often these problems concern the determination of the properties of some inaccess­ ible regions from observations on the boundary of that region. " Often, inverse problems are not well posed. This increases the difficulty in their analysis and numerical solution. As can be seen from the table of content of this volume, the conference covered inverse problems in scattering theory, seismology, tomography, estimation of coefficients and source terms in parabolic and elliptic differential equations, the inverse Sturm-Liouville problem, and numerical methods. The editors wish to thank Professor M. Barner and his co-workers of the Mathematical Research Institute for their help in creating the conference.

The linear functional strategy for improperly posed problems.- Determination of a source term in a linear parabolic differential equation with mixed boundary conditions.- Inverse problems for parabolic partial differential equations.- Three-dimensional inverse scattering.- On isospectral gradient flows solving matrix eigenproblems using differential equations.- On an integral equation of the first kind in inverse acoustic scattering.- Approximation of discrete probability distributions in spherical stereology.- Parameter estimation for distributed systems arising in fluid flow problems via time series methods.- Analytic solutions to the inverse problem of the Newtonian potential.- A note on an inverse problem related to the 3-D heat equation.- Undetermined coefficient problems for nonlinear elliptic and parabolic equations.- The method of `generalized interpolation for approximate solution of ill-posed problems.- Gel`fand-Levitan s theory and related inverse problems.- Reconstruction ambiguities of inverse scattering on the line.