Synopses & Reviews
The twin themes of computational complexity and information pervade this book. It starts with an introduction to information-based complexity, that is, the computational complexity of continuous mathematical models. It then moves to a variety of topics, including breaking the curse of dimensionality, complexity of path integration, solvability of ill-posed problems, value of information in computation, assigning values to mathematical hypotheses, and mathematical finance. The style is informal, and the goal is motivation and insight. Precise statements and proofs can be found in the monographs and papers included in the comprehensive bibliography. The book will be essential reading for researchers in the many disciplines influenced by the computational complexity of continuous problems.
Review
"This short volume packs so much information into so small a space that it stretches the imagination as to know how the authors did it. Clearly written, filled with interesting examples, important theorems and tantalizing conjectures, this book should be taken as holiday reading by everyone concerned with using a computer to solve real problems. It's destined to be a classic." New Scientist
Review
"The exposition is straightforward and clear, with well-chosen diagrams to smooth the way...The broad expanse between technical and general themes may be one of the more challenging and appealing aspects of the book...This book provides worthwhile evidence that IBC (information based complexity) is an important, albeit highly specialized, approach to understanding and applying information in modeling the complexity of a variety of mathematically formulated problems." Complexity
Review
"...gives a good grounding in the essential issues of complexity and information. For an overview of the larger issues, this small book is excellent." Bulletin of the AMS
Review
"Really a jewel...written with loving care and in an elegant style." Institute of Discrete Mathematics
Review
"Highly recommended." Choice
Review
"Anyone interested in numerical analysis and scientific computing can profit from the book and its many insights." Siam Review
Synopsis
First introductory book to subject; also illustrated with applications.
Synopsis
The twin themes of of computational complexity and information pervade this book. Information-based complexity is used to illustrate a variety of topics, including breaking the curse of dimension, 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 inf ormal, and the goals are exposition, insight and motivation. As the first introductory book on the subject it will invaluable as a guide to the area.
Table of 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.