The twin themes of computational complexity and information pervade this book. It starts with an introduction to the computational complexity of continuous mathematical models, that is, information-based complexity. This is then used to illustrate a variety of topics, including breaking the curse of dimensionality, complexity of path integration, solvability of ill-posed problems, the value of information in computation, assigning values to mathematical hypotheses, and new, improved methods for mathematical finance. The style is informal, and the goals are exposition, insight and motivation. A comprehensive bibliography is provided, to which readers are referred for precise statements of results and their proofs. As the first introductory book on the subject it will be invaluable as a guide to the area for the many students and researchers whose disciplines, ranging from physics to finance, are influenced by the computational complexity of continuous problems.
• First book for a general audience on the computational complexity of continuous models of science • Coverage of new, improved methods for mathematical finance • Comprehensive bibliography covering the last decade’s progress in this exciting area
Contents
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.
Review
‘Clearly written, filled with interesting examples, important theorems and tantalising conjectures, this is destined to be a classic.’ New Scientist
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