Improperly Posed Problems and Their Numerical Treatment : Conference Held at the Mathematisches Forschungsinstitut, Oberwolfach, September 26–October 2, 1982

We develop and analyze a marching procedure for the numerical computation of backwards parabolic equations with variable coefficients and noisy initial data. The scheme is stable (but inconsistent) and leads to error bounds of logarithmic convexity type for t bounded away from the line t = T, where the solution is only of class L2. The scheme is a two step procedure where the solution is appropriately filtered in the frequency domain, at every alternate step. The procedure assumes a constraint on the class of solutions which is stronger than the usual L2 bound at t = T. This stronger constraint is equivalent to the usual constraint in the constant coefficient case. (2.3) u(x,O) g(x), 0.;; x';; 71. Here, a(x) and c(x) are smooth functions with (2.4) o 0 known, and we let u (x,t) be the unique solution of (2.1), (2.2), e corresponding to g (x). We assume further that u (x,t) satisfies an a-priori e e bound condition at t = T, i.e., we are given Q > 0 such that (2.6) lIu (. ,T)II .;; Q. e Given n, £, and g(x), there are infinitely many sets of exact data g (x) e satisfying (2.5) with corresponding solution u (x,t) satisfying (2.6). e 1 2 However, as is well known, [8], if u (x,t) and u (x,t) are any two solutions, e e then (2.7) lIu1(•,t) -u 2 (•,t)1I . Front Matter....Pages 1-10 A Stable Marching Scheme for an Ill-Posed Initial Value Problem....Pages 11-35 Stability of Parameters Estimated by the Output Least Square Method....Pages 37-45 Analytic and Numerical Methods in the Study of the Inverse Scattering Problem for Acoustic Waves....Pages 47-55 A Numerical Method for Solving the Inverse Stefan Problem in two Space Variables....Pages 57-63 On a Constrained Fourier Extrapolation Method for Numerical Deconvolution....Pages 65-80 On the Convergence of Regularization Methods for Ill-Posed Linear Operator Equations....Pages 81-95 Comments on Morozov’s Discrepancy Principle....Pages 97-104 Reconstruction of a Free Boundary....Pages 105-116 The Finite Element Method for a Class of Improperly Posed Integral Equations....Pages 117-131 Maxwell’s Equations with Incident Waves as a Field Source....Pages 133-148 The Identification of Kinetical Parameters in Medical Receptor Analysis....Pages 149-163 Regularizing the Cauchy Problem for the Heat Equation by Sign Restrictions....Pages 165-177 Numerical Solution of Fujita’s Equation....Pages 179-187 On the Order of Regularization Methods....Pages 189-203 Free Boundary Problem for a Non-Linear System of Parabolic Equations, Including One with Reversed Time....Pages 205-226 Regularisierungsverfahren für Gleichungen Erster Art mit Positiv Definiten Operatoren....Pages 227-236 Über Mathematische Methoden der Beobachtungsverarbeitung....Pages 237-256 Back Matter....Pages 257-264 Whilst improperly posed problems appear in several branches of applied and pure mathematics, this conference concentrated mainly on the practical treatment of ill posedness. The participants came from 12 countries. The interchange of ideas reflected the spectrum of questions arising in connection with the subject of the conference, where currently progresses in research are made. This volume contains 17 papers presented at the con ference. Focal points in the programme were: Problems of regularisation, parameter identification, free boundary and inverse problems in differential equations and inte gral equations of the first kind. Problems, which appear in science, in technical fields and in medicine are dis cussed as well as general operator equations. In a jOint discussion, several open problems have been worked out which are collected at the end of the volume. The editor's thanks go to all contributors and parti cipants who made the conference a success; to the manage ment of the institute with its unique atmosphere; to the Birkhauser Verlag for the possibility to publish the vo lume in the well-known ISNM series; to Dr. P. Jochum (Mlin chen) for assistance in organization and to Mrs. Chr. Rogg (Augsburg) for her excellent typing of several manuscripts