Some approaches to the analysis of computation optimization problems

Abstract

In [1] we summarized some issues of optimization of computations that had been discussed and debated at 25 schoolseminars and symposia on optimization of computations organized since 1969 by the V. M. Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine. In the same article we also formulated some computation optimization problems that follow from the presented results.In this article we review and analyze the new results relating to the computation optimization problems formulated in [1]. A more detailed information about these results is contained in other articles in this issue of the journal. These additional articles are based on the papers presented at the 26th school-seminar of the V. M. Glushkov Institute of Cybernetics held in December 1995 in memory of its late scientific leader, Academician V. S. Mikha.~evich.Optimization of computations is a very broad concept. Its main components are the objects being optimized, the criteria used for optimization, and also methods and techniques to achieve optimality. The objects of optimization in this area are classes of problems, classes of numerical methods or algorithms, as well as classes of application software implementing these methods and algorithms on computers. Optimization criteria typically involve attaining the best or acceptable value of a chosen measure of accuracy, speed, or memory space as the main characteristics of computer algorithms. The techniques for achieving these goals, i.e., construction of algorithms optimal in a certain sense, include theoretical, experimental, and mixed approaches. In theoretical approaches, optimality of the algorithm is proved analytically on the basis of theorems. In the experimental approach, the best algorithm is identified by testing the corresponding program. In the mixed approach, testing results provide feedback to theoretical analysis of the algorithm, or vice versa.The main classes of problems investigated in the framework of this school include statistical data processing, reconstruction of functions and functionals, solution of various classes of equations, minimization of functions, and mathematical programming.The main computation optimization problems formulated in [1] can be divided into five groups. 1. Problems concerning development of new methods and extension of existing methods to more complex large problems with essential nonlinearities of single-criterion optimization. This includes decomposition methods, simulation methods, solution of nonlinear equations with one or many solutions, etc.2. Problems of ensuring desired characteristics of computer algorithms (accuracy, speed, memory requirements) and constructing algorithms optimal by various criteria. The component parts of this problem include extraction and refinement of prior information, construction of optimal information operators, derivation of lower bounds on the complexity of the algorithm for single-and multi-processor computers, development of accuracy and time efficient algorithms for high-precision compu...