Sergey P. Shary

A New Technique in Systems Analysis under Interval Uncertainty and Ambiguity

The main subject of this work is mathematical and computational aspects of modeling of static systems under interval uncertainty and/or ambiguity. A cornerstone of the new approach we are advancing in the present paper is, first, the rigorous and consistent use of the logical quantifiers to characterize and distinguish different kinds of interval uncertainty that occur in the course of modeling, and, second, the systematic use of Kaucher complete interval arithmetic for the solution of problems that are minimax by their nature. As a formalization of the mathematical problem statement, concepts of generalized solution sets and AE-solution sets to an interval system of equations, inequalities, etc., are introduced. The major practical result of our paper is the development of a number of techniques for inner and outer estimation of the so-called AE-solution sets to interval systems of equations. We work out, among others, formal approach, generalized interval Gauss-Seidel iteration, generalized preconditioning and PPS-methods. Along with the general nonlinear case, the linear systems are treated more thoroughly.

Algebraic approach to the interval linear static identification, tolerance, and control problems, or one more application of kaucher arithmetic

In this paper, theidentification problem, thetolerance problem, and thecontrol problem are treated for the interval linear equation Ax=b. These problems require computing an inner approximation of theunited solution set Σ∃∃(A, b)={x ∈ ℝn | (∃A ∈ A)(Ax ∈ b)}, of thetolerable solution set Σ∀∃(A, b)={x ∈ ℝn | (∀A ∈ A)(Ax ∈ b)}, and of thecontrollable solution set Σ∃∀(A, b)={x ∈ ℝn | (∀b ∈ b)(Ax ∈b)} respectively. Analgebraic approach to their solution is developed in which the initial problem is replaced by that of finding analgebraic solution of some auxiliary interval linear system in Kaucher extended interval arithmetic. The algebraic approach is proved almost always to give inclusion-maximal inner interval estimates of the solutionsets considered. We investigate basic properties of the algebraic solutions to the interval linear systems and propose a number of numerical methods to compute them. In particular, we present the simple and fastsubdifferential Newton method, prove its convergence and discuss numerical experiments.AbstractБ этой работе рассматриваютсяэa¶rt;aчa u¶rt;eнmuфuxaцuu, эa¶rt;aчa o ¶rt;onyckax н эa¶rt;aчa o¶rt; ynpaвlenuu для интервальной линейной системы Ax=b, требующие нахожления внутренней оценки дляоб Σ∃∃(A, b)={x ∈ ℝn | (∃A ∈ A)(Ax ∈ b)}, Σ∀∃(A, b)={x ∈ ℝn | (∀A ∈ A)(Ax ∈ b)}, и Σ∃∀(A, b)={x ∈ ℝn | (∀b ∈ b)(Ax ∈b)} соответственно. Развиваетсян к их решению, при котором исходная задача заменяется задачей отысканиян для некоторой вспомогательной интервальной линейной системы в расширенной интервальной арифметике Каухера. Показано, что алгебраический нодход почти всегда дает максимальные по включению внутренние оценки для рассматриваемых множеств решений. Исследуются основные свойства алгебраическнх решений интервальных систем, обсужлаются численные метолы для их нахожления. Б частности, мы предалагаем простой и быстрыйн, доказываем его сходимость и приводим результаты численных экспериментов с ним.

On optimal solution of interval linear equations

For interval linear algebraic systems

Interval Gauss-Seidel Method for Generalized Solution Sets to Interval Linear Systems

{\bf A}x = {\bf b}

Outer Estimation of Generalized Solution Sets to Interval Linear Systems

, we consider the problem of component-wise evaluation of the set

A Surprising Approach in Interval Global Optimization; Dedicated to Prof. Dr. Gregory G. Menshikov on the occasion of his 70th anniversary

\Sigma _{\exists \exists } ({\bf A},{\bf b}) = \{ {A^{ - 1} b\mid A \in {\bf A},b \in {\bf b}} \}

On solvability recognition for interval linear systems of equations

formed by all solutions of

Algebraic Approach in the "Outer Problem" for Interval Linear Equations

Ax = b

Tolerable Solution Set for Interval Linear Systems with Constraints on Coefficients.

when A and b vary independently in

Interval Methods for Data Fitting under Uncertainty: A Probabilistic Treatment

{\bf A}