Rank-deficient and discrete ill-posed problems. Numerical aspects of linear inversion.

*(English)*Zbl 0890.65037
SIAM Monographs on Mathematical Modeling and Computation, 4. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. 247 p. (1997).

Unlike a number of books on regularization and stabilization of inverse problems that have appeared in recent years, this book is exclusively devoted to practical methods for solving the finite-dimensional problem. The tone and context throughout is that of computational linear algebra. While the book assumes a knowledge of the basic issues raised by inverse problems, an introductory chapter lays the groundwork with a lucid discussion of smoothing, ill-posedness, and regularization of mathematical models of inverse problems and sets up several numerical test problems that recur throughout the book. The second chapter deals with the stock-in-trade of numerical linear algebra: decompositions. Included are detailed discussions of the singular value decomposition (SVD) and its generalizations, rank-revealing QR and LU decompositions, and specialized decompositions for regularization problems in standard form.

The remainder of the book is concerned with the computational solution of linear algebraic systems which the author classifies as rank deficient problems or discretized ill-posed problems. For rank deficient problems, that is, systems with (numerically) rank deficient coefficient matrix in which a clear gap exists between the large and small singular values, the concept of numerical rank is developed and stable solution methods based on the SVD and rank revealing decomposition are discussed. In the fourth chapter discrete ill-posed problems, that is, linear systems in which the singular values of the coefficient matrix decay steadily to zero with no notable gap between the large and small singular values, are treated. For such problems regularization is the strategy of choice.

Direct regularization methods, including Tikhonov regularization, truncated SVD, total least squares, mollifier methods, and the Backus-Gilbert method are the subject of chapter five. Next, iterative regularization methods, specifically Landweber iteration, iterated Tikhonov regularization, the conjugate gradient method, and hybrid methods are discussed. The topic of chapter seven, parameter choice strategies, is the key ingredient of effective solution methods for discrete ill-posed problems. Methods surveyed include the discrepancy principle and its extensions generalized cross-validation, and the \(L\)-curve criterion. The final chapter is a brief introduction and user’s guide to Regularization Tools, the author’s package of fifty three MATLAB programs for the analysis and solution of discrete ill-posed problems. The book concludes with a very extensive and useful bibliography consisting of nearly 400 items on the theory of, and methods for, discrete ill-posed problems.

This book is written with the practitioner in mind. Each of the main chapters on the principal methods contains a “methods in action” section containing numerical illustrations, test runs, and computational advice. It provides the nonexpert with a readable survey of modern computational methods for discrete ill-posed problems and the practitioner with an authoritative reference work on the latest methods. The numerical analysis community will certainly welcome this book.

The remainder of the book is concerned with the computational solution of linear algebraic systems which the author classifies as rank deficient problems or discretized ill-posed problems. For rank deficient problems, that is, systems with (numerically) rank deficient coefficient matrix in which a clear gap exists between the large and small singular values, the concept of numerical rank is developed and stable solution methods based on the SVD and rank revealing decomposition are discussed. In the fourth chapter discrete ill-posed problems, that is, linear systems in which the singular values of the coefficient matrix decay steadily to zero with no notable gap between the large and small singular values, are treated. For such problems regularization is the strategy of choice.

Direct regularization methods, including Tikhonov regularization, truncated SVD, total least squares, mollifier methods, and the Backus-Gilbert method are the subject of chapter five. Next, iterative regularization methods, specifically Landweber iteration, iterated Tikhonov regularization, the conjugate gradient method, and hybrid methods are discussed. The topic of chapter seven, parameter choice strategies, is the key ingredient of effective solution methods for discrete ill-posed problems. Methods surveyed include the discrepancy principle and its extensions generalized cross-validation, and the \(L\)-curve criterion. The final chapter is a brief introduction and user’s guide to Regularization Tools, the author’s package of fifty three MATLAB programs for the analysis and solution of discrete ill-posed problems. The book concludes with a very extensive and useful bibliography consisting of nearly 400 items on the theory of, and methods for, discrete ill-posed problems.

This book is written with the practitioner in mind. Each of the main chapters on the principal methods contains a “methods in action” section containing numerical illustrations, test runs, and computational advice. It provides the nonexpert with a readable survey of modern computational methods for discrete ill-posed problems and the practitioner with an authoritative reference work on the latest methods. The numerical analysis community will certainly welcome this book.

Reviewer: C.W.Groetsch (Cincinnati)

##### MSC:

65F20 | Numerical solutions to overdetermined systems, pseudoinverses |

65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |

47A50 | Equations and inequalities involving linear operators, with vector unknowns |

15A09 | Theory of matrix inversion and generalized inverses |

65R30 | Numerical methods for ill-posed problems for integral equations |

45B05 | Fredholm integral equations |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |