The Computational Complexity of Differential and Integral Equations: An Information-Based Approach

ISBN-13: 9780198535898 / Angielski / Twarda / 1991 / 344 str.

Arthur G. Werschulz

Recenzje

'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.

Spis treści

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.