Information-based complexity

Abstract: Information-based complexity seeks to develop general results about the intrinsic difficulty of solving problems where available information is partial or approximate and to apply these results to specific problems. This allows one to determine what is meant by an optimal algorithm in many practical situations, and offers a variety of interesting and sometimes surprising theoretical results.

Abstract: Many of the mathematical models used in fields such as the physical sciences, engineering, economics, and mathematical finance use continuous mathematical models. These models typically require the numerical solution of multivariate problems (often in a very large number of variables) such as integrals, ordinary and partial differential equations (q.v.), optimization, approximation, integral equations, and nonlinear equations. The study of the computational complexity of continuous mathematical problems is called information-based complexity (IBC). This is a branch of computational complexity (q.v.) which studies the minimal computer resources (typically time or space) needed to solve mathematically posed problems.

Abstract: Part I. Fundamentals: 1. Introduction 2. Information-based complexity 3. Breaking the curse of dimensionality Part II. Some Interesting Topics: 4. Very high-dimensional integration and mathematical finance 5. Complexity of path integration 6. Are ill-posed problems solvable? 7. Complexity of nonlinear problems 8. What model of computation should be used by scientists? 9. Do impossibility theorems from formal models limit scientific knowledge? 10. Complexity of linear programming 11. Complexity of verification 12. Complexity of implementation testing 13. Noisy information 14. Value of information in computation 15. Assigning values to mathematical hypotheses 16. Open problems 17. A brief history of information-based complexity Part III. References: 18. A guide to the literature Bibliography Subject index Author index.

Abstract: We introduce a near-linear complexity (geometric and meshless/algebraic) multigrid/multiresolution method for PDEs with rough ($L^\infty$) coefficients with rigorous a priori accuracy and performance estimates. The method is discovered through a decision/game theory formulation of the problems of (1) identifying restriction and interpolation operators, (2) recovering a signal from incomplete measurements based on norm constraints on its image under a linear operator, and (3) gambling on the value of the solution of the PDE based on a hierarchy of nested measurements of its solution or source term. The resulting elementary gambles form a hierarchy of (deterministic) basis functions of $H^1_0(\Omega)$ (gamblets) that (1) are orthogonal across subscales/subbands with respect to the scalar product induced by the energy norm of the PDE, (2) enable sparse compression of the solution space in $H^1_0(\Omega)$, and (3) induce an orthogonal multiresolution operator decomposition. The operating diagram of the multig...