Feasible Computation in Symbolic and Numeric Integration

Two central concerns in scientific computing are the reliability and efficiency of algorithms. We introduce the term feasible computation to describe algorithms that are reliable and efficient given the contextual constraints imposed in practice. The main focus of this dissertation then, is to bring greater clarity to the forms of error introduced in computation and modeling, and in the limited context of symbolic and numeric integration, to contribute to integration algorithms that better account for error while providing results efficiently. Chapter 2 considers the problem of spurious discontinuities in the symbolic integration problem, proposing a new method to restore continuity based on a pair of unwinding numbers. Computable conditions for the unwinding numbers are specified, allowing the computation of a variety of continuous integrals. Chapter 3 introduces two structure-preserving algorithms for the symbolic-numeric integration of rational functions on exact input. A structured backward and forward error analysis for the algorithms shows that they are a posteriori backward and forward stable, with both forms of error exhibiting tolerance proportionality. Chapter 4 identifies the basic logical structure of feasible inference by presenting a logical model of stable approximate inference, illustrated by examples of modeling and numerical integration. In terms of this model it is seen that a necessary condition for the feasibility of methods of abstraction in modeling and complexity reduction in computational mathematics is the preservation of inferential structure, in a sense that is made precise. Chapter 5 identifies a robust pattern in mathematical sciences of transforming problems to make solutions feasible. It is showed that computational complexity reduction methods in computational science involve chains of such transformations. It is argued that the structured and approximate nature of such strategies indicates the need for a higher-order model of computation and a new definition of computational complexity

A symbolic sensor for an Antilock brake system of a commercial aircraft

The design of a symbolic sensor that identifies thecondition of the runway surface (dry, wet, icy, etc.) during the braking of a commercial aircraft is discussed. The purpose of such a sensor is to generate a qualitative, real-time information about the runway surface to be integrated into a future aircraft Antilock Braking System (ABS). It can be expected that this information can significantly improve the performance of ABS. For the design of the symbolic sensor different classification techniques based upon fuzzy set theory and neural networks are proposed. To develop and to verify theses classification algorithms data recorded from recent braking tests have been used. The results show that the symbolic sensor is able to correctly identify the surface condition. Overall, the application example considered in this paper demonstrates that symbolic information processing using fuzzy logic and neural networks has the potential to provide new functions in control system design. This paper is part of a common research project between E.N.S.I.C.A. and Aerospatiale in France to study the role of the fuzzy set theory for potential applications in future aircraft control systems

Symbolic integration with respect to the Haar measure on the unitary group

We present IntU package for Mathematica computer algebra system. The presented package performs a symbolic integration of polynomial functions over the unitary group with respect to unique normalized Haar measure. We describe a number of special cases which can be used to optimize the calculation speed for some classes of integrals. We also provide some examples of usage of the presented package.Comment: 7 pages, two columns, published version, software available at: https://github.com/iitis/Int