Mathematical and Analytical Methods in the Physical Sciences: Inverse Problems and Singular Perturbation Theory

The importance of mathematics in the study of problems arising from the real world, and the increasing success with which it has been used to model situations ranging from the purely deterministic to the stochastic, in all areas of today's Physical Sciences and Engineering, is well established. The purpose of the sets of volumes, the present one being the first in a planned series of sequential sets, is to make available authoritative, up to date, and self-contained accounts of some of the most important and useful of these analytical approaches and techniques. Each volume in each set will provide a detailed introduction to a specific subject area of current importance, and then goes beyond this by reviewing recent contributions, thereby serving as a valuable reference source. The progress in applicable mathematics has been brought about by the extension and development of many important analytical approaches and techniques, in areas both old and new, frequently aided by the use of computers without which the solution of realistic problems in modern Physical Sciences and Engineering would otherwise have been impossible. A case in point is the analytical technique of singular perturbation theory (Volume 3), which has a long history. In recent years it has been used in many different ways, and its importance has been enhanced by its having been used in various fields to derive sequences of asymptotic approximations, each with a higher order of accuracy than its predecessor. These approximations have, in turn, provided a better understanding of the subject and stimulated the development of new methods for the numerical solution of the higher order approximations. A typical example of this type is to be found in the general study of nonlinear wave propagation phenomena as typified by the study of water waves. Elsewhere, as with the identification and emergence of the study of inverse problems (volumes 1 and 2), new analytical approaches have stimulated the development of numerical techniques for the solution of this major class of problems. Such work divides naturally into two parts, the first being the identification and formulation of inverse problems, the theory of ill-posed problems and the class of one-dimensional inverse problems, and the second being the study and theory of multidimensional inverse problems. Volume 1: Inverse Problems 1: Volume 2: Inverse Problems 2: These volumes present the theory of inverse spectral and scattering problems and of many other inverse problems for differential equations in an essentially self-contained way. Highlights of these volumes include novel presentation of the classical theories (Gel'fand-Levitan's and Marchenko's), analysis of the invertibility of the inversion steps in these theories, study of some new inverse problems in one-and multi-dimensional cases; I-function and applications to classical and new inverse scattering and spectral problems, study of inverse problems with "incomplete data", study of some new inverse problems for parabolic and hyperbolic equations, discussion of some non-overdetermined inverse problems, a study of inverse problems arising in the theory of ground-penetrating radars, development of DSM (dynamical systems method) for solving ill-posed nonlinear operator equations, comparison of the Ramm's inversion method for solving fixed-energy inverse scattering problem with the method based on the Dirichlet-to-Neumann map, derivation of the range of applicability and error estimates for Born's inversion, a study of some integral geometry problems, including tomography, inversion formulas for the spherical means, proof of the invertibility of the steps in the Gel'fand-Levitan and Marchenko inversion procedures, derivation of the inversion formulas and stability estimates for the multidimensional inverse scattering problems with fixed-energy noisy discrete data, new uniqueness and stability results in obstacle inverse scattering, formulation and a solution of an inverse problem of radiomeasurements, and methods for finding small inhomogeneities from surface scattering data. Several open problems are formulated. Volume 3: Singular Perturbation Theory: The theory of singular perturbations has been with us, in one form or another, for a little over a century (although the term 'singular perturbation' dates from the 1940s). The subject, and the techniques associated with it, have evolved over this period as a response to the need to find approximate solutions (in an analytical form) to complex problems. Typically, such problems are expressed in terms of differential equations which contain at least one small parameter, and they can arise in many fields: fluid mechanics, particle physics and combustion processes, to name but three. The essential hallmark of a singular perturbation problem is that a simple and straightforward approximation (based on the smallness of the parameter) does not give an accurate solution throughout the domain of that solution. Perforce, this leads to different approximations being valid in different parts of the domain (usually requiring a 'scaling' of the variables with respect to the parameter). This in turn has led to the important concepts of breakdown, matching, and so on. This volume has been written in a form that should enable the relatively inexperienced (or new) worker in the field of singular perturbation theory to learn and apply all the essential ideas. To this end, the text has been designed as a learning tool.

Table of Contents:
Volume One:- 1: Introduction. 1.1. Why are inverse problems interesting and practically important? 1.2. Examples of inverse problems. 1.3. Ill-posed problems. 1.4. Examples of Ill-posed problems. 2: Methods of solving ill-posed problems. 2.1. Variational regularization. 2.2. Quasisolutions, quasinversion, and Backus-Gilbert method. 2.3. Iterative methods. 2.4. Dynamical system method (DSM). 2.5. Examples of solutions of ill-posed problems. 2.6. Projection methods for ill-posed problems. 3: One-dimensional inverse scattering and spectral problems. 3.1. Introduction. 3.2. Property C for ODE. 3.3. Inverse problem with I-function as the data. 3.4. lnverse spectral problem. 3.5. Inverse scattering on half-line. 3.6. Inverse scattering problem with fixed-energy. 3.7. Inverse scattering with "incomplete data". 3.8. Recovery of quarkonium systems. 3.9. Krein's method in inverse scattering. 3.10. Inverse problems for the heat and wave equations. 3.11. Inverse problem for an inhomogeneous Schrodinger equation. 3.12. An inverse problem of ocean acoustics. 3.13. Theory of ground-penetrating radars. Volume Two:- 4: Inverse obstacle scattering. 4.1. Statement of the problem. 4.2. Inverse obstacle scattering problems. 4.3. Stability estimates for the solution to IOSP. 4.4. High-frequency asymptotics. 4.5. Remarks about numerical methods. 4.6. Analysis of a method for identification of obstacles. 5: Inverse scattering problem. 5.1. Introduction. 5.2. Inverse potential scattering problem with fixed-energy data. 5.3. Inverse geophysical scattering with fixed-frequency data. 5.4. Proofs of some estimates. 5.5. Construction of the Dirichlet-to-Neumann map. 5.6. Property C. 5.7. Necessary and sufficient condition for scatterers. 5.8. The Born inversion. 5.9. Uniqueness theorems for inverse spectral problems. 6: Non-uniqueness and uniqueness results. 6.1. Examples of nonuniqueness for an inverse problem of geophysics. 6.2. A uniqueness theorem for an inverse problem. 6.3. Property C and an inverse problem for a hyperbolic equation. 6.4. Continuation of the data. 7: Inverse problems of potential theory. 7.1. Inverse problem of potential theory. 7.2. Antenna synthesis problems. 7.3. Inverse source problem for hyperbolic equations. 8: Non-overdetermined inverse problems. 8.1. Introduction. 8.2. Assumptions. 8.3. The problem and the result. 8.4. Finding lpj(s) from lp](s). 8.5. Appendix. 9: Low-frequency inversion. 9.1. Derivation of the basic equation. Uniqueness results. 9.2. Analytical solution of the basic equation. 9.3. Characterization of the low-frequency data. 9.4. Problems of numerical implementation. 9.5. Half-spaces with different properties. 9.6. Inversion of the data given on a sphere. 9.7. Inversion of the data given on a cylinder. 9.8. Two-dimensional inverse problems. 9.9. One-dimensional inversion. 9.10. Inversion of the backscattering data. 9.11. Inversion of the well-to-well data. 9.12. Induction logging problems. 9.13. Examples of non-uniqueness of the solution. 9.14. Scattering in absorptive medium. 9.15. A geometrical inverse problem. 9.16. An inverse problem for a biharmonic equation. 9.17. Inverse scattering when the background is variable. 9.18. Remarks concerning the basic equation. 10: Wave scattering by small bodies of arbitrary shapes. 10.1. Wave scattering by small bodies. 10.2. Equations for the self-consistent field. 10.3. Finding small subsurface inhomogeneities from scattering data. 10.4. Inverse problem of radiomeasurements. 11: The Pompeiu problem. 11.1. The Pompeiu problem. 11.2. Necessary and sufficient condition. Bibliographical Notes. References. Volume 3:- 1: Mathematical preliminaries. 1.1. Some introductory examples. 1.2. Notation. 1.3. Asymptotic sequences and asymptotic expansions. 1.4. Convergent series versus divergent series. 1.5. Asymptotic expansions with a parameter. 1.6. Uniformity or breakdown. 1.7. Intermediate variables and the overlap region. 1.8. The matching principle. 1.9. Matching with logarithmic terms. 1.10. Composite expansions. 2: Introductory applications. 2.1. Roots of equations. 2.2. Integration of functions represented by asymptotic expansions. 2.3. Ordinary differential equations: regular problems. 2.4. Ordinary differential equations: simple singular problems. 2.5. Scaling of differential equations. 2.6. Equations which exhibit a boundary-layer behaviour. 2.7. Where is the boundary layer? 2.8. Boundary layers and transition layers. 3: Further applications. 3.1. A regular problem. 3.2. Singular problems I. 3.3. Singular problems II. 3.4. Further applications to ordinary differential equations. 4: The method of multiple scales. 4.1. Nearly linear oscillations. 4.2. Nonlinear oscillations. 4.3. Applications to classical ordinary differential equations. 4.4. Applications to partial differential equations. 4.5. A limitation on the use of the method of multiple scales. 4.6. Boundary-layer problems. 5: Some worked examples arising from physical problems. 5.1. Mechanical & electrical systems. 5.2. Celestial mechanics. 5.3. Physics of particles & light. 5.4. Semi- and superconductors. 5.5. Fluid mechanics. 5.6. Extreme thermal processes. 5.7. Chemical & biochemical reactions. Further Readings and Exercises are provided at the end of each chapter. Appendix:- The Jacobian elliptic functions. Answers and Hints.