Computational Complexity and Feasibility of Data Processing and Interval Computations

Targeted audience • Specialists in numerical computations, especially in numerical optimiza­ tion, who are interested in designing algorithms with automatie result ver­ ification, and who would therefore be interested in knowing how general their algorithms caIi in principle be. • Mathematicians and computer scientists who are interested in the theory 0/ computing and computational complexity, especially computational com­ plexity of numerical computations. • Students in applied mathematics and computer science who are interested in computational complexity of different numerical methods and in learning general techniques for estimating this computational complexity. The book is written with all explanations and definitions added, so that it can be used as a graduate level textbook. What this book .is about Data processing. In many real-life situations, we are interested in the value of a physical quantity y that is diflicult (or even impossible) to measure directly. For example, it is impossible to directly measure the amount of oil in an oil field or a distance to a star. Since we cannot measure such quantities directly, we measure them indirectly, by measuring some other quantities Xi and using the known relation between y and Xi'S to reconstruct y. The algorithm that transforms the results Xi of measuring Xi into an estimate fj for y is called data processing.

1 Informal Introduction: Data Processing, Interval Computations, and Computational Complexity.- 2 The Notions of Feasibility and NP-Hardness: Brief Introduction.- 3 In the General Case, the Basic Problem of Interval Computations is Intractable.- 4 Basic Problem of Interval Computations for Polynomials of a Fixed Number of Variables.- 5 Basic Problem of Interval Computations for Polynomials of Fixed Order.- 6 Basic Problem of Interval Computations for Polynomials with Bounded Coefficients.- 7 Fixed Data Processing Algorithms, Varying Data: Still NP-Hard.- 8 Fixed Data, Varying Data Processing Algorithms: Still Intractable.- 9 What if We only Allow some Arithmetic Operations in Data Processing?.- 10 For Fractionally-Linear Functions, a Feasible Algorithm Solves the Basic Problem of Interval Computations.- 11 Solving Interval Linear Systems is NP-Hard.- 12 Interval Linear Systems: Search for Feasible Classes.- 13 Physical Corollary: Prediction is not Always Possible, Even for Linear Systems with Known Dynamics.- 14 Engineering Corollary: Signal Processing is NP-Hard.- 15 Bright Sides of NP-Hardness of Interval Computations I: NP-Hard Means That Good Interval Heuristics can Solve other Hard Problems.- 16 If Input Intervals are Narrow Enough, Then Interval Computations are Almost Always Easy.- 17 Optimization — a First Example of a Numerical Problem in which Interval Methods are used: Computational Complexity and Feasibility.- 18 Solving Systems of Equations.- 19 Approximation of Interval Functions.- 20 Solving Differential Equations.- 21 Properties of Interval Matrices I: Main Results.- 22 Properties of Interval Matrices II: Proofs and Auxiliary Results.- 23 Non-Interval Uncertainty I: Ellipsoid Uncertainty and its Generalizations.- 24 Non-Interval Uncertainty II:Multi-Intervals and Their Generalizations.- 25 What if Quantities are Discrete?.- 26 Error Estimation for Indirect Measurements: Interval Computation Problem is (Slightly) Harder than a Similar Probabilistic Computational Problem.- A In Case of Interval (Or More General) Uncertainty, no Algorithm can Choose the Simplest Representative.- B Error Estimation for Indirect Measurements: Case of Approximately Known Functions.- C From Interval Computations to Modal Mathematics.- D Beyond NP: Two Roots Good, one Root Better.- E Does “NP-Hard”Really Mean “Intractable”?.- F Bright Sides of NP-Hardness of Interval Computations II: Freedom of Will?.- G The Worse, The Better: Paradoxical Computational Complexity of Interval Computations and Data Processing.- References.