The research detailed in this monograph was originally motivated by our interest in control problems involving partial and delay differential equations. Our attempts to apply control theory techniques to such prob­ lems in several areas of science convinced us that in the need for better and more detailed models of distributed/ continuum processes in biology and mechanics lay a rich, interesting, and challenging class of fundamen­ tal questions. These questions, which involve science and mathematics, are typical of those arising in inverse or parameter estimation problems. Our efforts on inverse problems for distributed parameter systems, which are infinite dimensional in the most common realizations, began about seven years ago at a time when rapid advances in computing capabilities and availability held promise for significant progress in the development of a practically useful as well as theoretically sound methodology for such problems. Much of the research reported in our presentation was not begun when we outlined the plans for this monograph some years ago. By publishing this monograph now, when only a part of the originally intended topics are covered (see Chapter VII in this respect), we hope to stimulate the research and interest of others in an area of scientific en­ deavor which has exceeded even our optimistic expectations with respect to excitement, opportunity, and stimulation. The computer revolution alluded to above and the development of new codes allow one to solve rather routinely certain estimation problems that would have been out of the question ten years ago.

Inhalt

I Examples of Inverse Problems Arising in Applications.- I.1. Inverse Problems in Ecology.- I.2. Inverse Problems in Lake and Sea Sedimentation Analysis.- I.3. Inverse Problems in the Study of Flexible Structures.- I.4. Inverse Problems in Physiology.- II Operator Theory Preliminaries.- II.1. Linear Semigroups.- II.2. Galerkin Schemes.- III Parameter Estimation: Basic Concepts and Examples.- III.1. The Parameter Estimation Problem.- III.2. Application of the Theory to Special Schemes for Linear Parabolic Systems.- III.2.1. Modal Approximations.- III.2.2. Cubic Spline Approximations.- III.3. Parameter Dependent Approximation and the Nonlinear Variation of Constants Formula.- IV Identifiability and Stability.- IV.1. Generalities.- IV.2. Examples.- IV.3. Identifiability and Stability Concepts.- IV.4. A Sufficient Condition for Identifiability.- IV.5. Output Least Squares Identifiability.- IV.5.1. Theory.- IV.5.2. Applications.- IV.6. Output Least Squares Stability.- IV.6.1. Theory.- IV.6.2. An Example.- IV.7. Regularization.- IV.7.1. Tikhonov's Lemma and Its Application.- IV.7.2. Regularization Revisited.- IV.8. Concluding Remarks on Stability.- IV.8.1. A Summary of Possible Approaches.- IV.8.2. Remarks on Implementation.- V Parabolic Equations.- V.1. Modal Approximations: Discrete Fit-to-Data Criteria.- V.2. Quasimodal Approximations.- V.3. Operator Factorization: A = -C*C.- V.4. Operator Factorization: A = A1/2A1/2.- V.5. Numerical Considerations.- V.6. Numerical Test Examples.- V.7. Examples with Experimental Data.- VI Approximation of Unknown Coefficients in Linear Elliptic Equations.- VI.1. Parameter Estimation Convergence.- VI.2. Function Space Parameter Estimation Convergence.- VI.3. Rate of Convergence for a Special Case.- VI.4. Methods Other Than Output-Least-Squares.- VI.4.1. Method of Characteristics.- VI.4.2. Equation Error Method.- VI.4.3. A Variational Technique.- VI.4.4. Singular Perturbation Techniques.- VI.4.5. Adaptive Control Methods.- VI.4.6. An Augmented Lagrangian Technique.- VI.5. Numerical Test Examples.- VII An Annotated Bibliography.- Al) Preliminaries.- A2) Linear Splines.- A3) Cubic Hermite Splines.- A5) Polynomial Splines, Quasi- Interpolation.