Complexity theory has become an increasingly important theme in mathematical research. This book deals with an approximate solution of differential or integral equations by algorithms using incomplete information. This situation often arises for equations of the form Lu = f where f is some function defined on a domain and L is a differential operator. We do not have complete information about f. For instance, we might only know its value at a finite number of points in the domain, or the values of its inner products with a finite set of known functions. Consequently the best that can be hoped for is to solve the equation to within a given accuracy at minimal cost or complexity. In this book, the theory of the complexity of the solution to differential and integral equations is developed. The relationship between the worst case setting and other (sometimes more tractable) related settings, such as the average case, probabilistic, asymptotic, and randomized settings, is also discussed. The author determines the inherent complexity of the problem and finds optimal algorithms (in the sense of having minimal cost). Furthermore, he studies to what extent standard algorithms (such as finite element methods for elliptic problems) are optimal. This approach is discussed in depth in the context of two-point boundary value problems, linear elliptic partial differential equations, integral equations, ordinary differential equations, and ill-posed problems. As a result, this volume should appeal to mathematicians and numerical analysts working on the approximate solution of differential and integral equations, as well as to complexity theorists addressing related questions in this area.
'This book ... is a most welcome addition to the theoretical computer science and numerical analysis literature. Though it is intended as a summary of current research, it is of the quality that would make it an excellent textbook on the subject for advanced numerical analysis and computer science courses .. it reads easily and lucidly.'
'An excellent and accessible introduction to the complexity of basic arithmetic operations ... it adds an interesting new dimension to the study of numerical methods for the solution of PDEs.'
Notices of the A.M.S.
Introduction; EXAMPLE: A TWO-POINT BOUNDARY VALUE PROBLEM: Introduction; Error, cost, and complexity; A minimal error algorithm; Complexity bounds; Comparison with the finite element method; Standard information; Final remarks; GENERAL FORMULATION: Introduction; Problem formulation; Information; Model of computation; Algorithms, their errors, and their costs; Complexity; Randomized setting; Asymptotic setting; THE WORST CASE SETTING: GENERAL RESULTS:
Introduction; Radius and diameter; Complexity; Linear problems; The residual error criterion; ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS IN THE WORST CASE SETTING; Introduction; Variational elliptic boundary value
problems; Problem formulation; The normed case with arbitrary linear information; The normed case with standard information; The seminormed case; Can adaption ever help?; OTHER PROBLEMS IN THE WORST CASE SETTING: Introduction; Linear elliptic systems; Fredholm problems of the second kind; Ill-posed problems; Ordinary differential equations; THE AVERAGE CASE SETTING: Introduction; Some basic measure theory; General results for the average case setting; Complexity of shift-invariant problems;
Ill-posed problems; The probabilistic setting; COMPLEXITY IN THE ASYMPTOTIC AND RANDOMIZED SETTINGS: Introduction; Asymptotic setting; Randomized setting; Appendices; Bibliography.