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

Abstract

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.