Evgenija D. Popova

Explicit Description of AE Solution Sets for Parametric Linear Systems

Consider linear systems whose input data are linear functions of uncertain parameters varying within given intervals. We are interested in an explicit description of the so-called AE parametric solution sets (where all universally quantified parameters precede all existentially quantified ones) by a set of inequalities not involving the parameters. This work presents how to obtain explicit description of AE parametric solution sets by combining a modified Fourier--Motzkin type elimination of existentially quantified parameters with the elimination of the universally quantified parameters. Some necessary (and sufficient) conditions for existence of nonempty AE parametric solution sets are discussed, as well as some properties of the parametric AE solution sets, e.g., shape of the solution set and some inclusion relations. Explicit descriptions of particular classes of AE parametric solution sets (tolerable, controllable, any two-dimensional) are given. Numerical examples illustrate the solution sets and th...

On the Solution of Parametrised Linear Systems

Considered are parametrised linear systems which parameters are subject to tolerances. Rump’s fixed-point iteration method for finding outer and inner approximations of the hull of the solution set is studied and applied to an electrical circuit problem. Interval Gauss-Seidel iteration for parametrised linear systems is introduced and used for improving the enclosures, obtained by the fixed-point method, whenever they are not good enough. Generalised interval arithmetic (on proper and improper intervals) is considered as a computational tool for efficient handling of proper interval problems (to obtain inner interval estimations without inward rounding and to eliminate the dependency problem in parametrised Gauss-Seidel iteration). Numerical results from the application of the above methods to an electrical circuit problem are discussed.

Multiplication Distributivity of Proper and Improper Intervals

The arithmetic on an extended set of proper and improper intervals presents algebraic completion of the conventional interval arithmetic allowing thus efficient solution of some interval algebraic problems. In this paper we summarize and present all distributive relations, known by now, on multiplication and addition of generalized (proper and improper) intervals.

Parametric Interval Linear Solver

IntervalComputations‘LinearSystems’ is a Mathematica package supporting tools for solving parametric and nonparametric linear systems involving uncertainties. It includes a variety of functions, implementing different interval techniques, that help in producing sharp and rigorous results in validated interval arithmetic. The package is designed to be easy to use, versatile, to provide a necessary background for further exploration, comparisons and prototyping, and to provide some indispensable tools for solving parametric interval linear systems. This paper presents the functionality, provided by the current version of the package, and briefly discusses the underlying methodology. A new hybrid approach for sharp parametric enclosures, that combines parametric residual iteration, exact bounds, based on monotonicity properties, and refinement by interval subdivision, is outlined.

Outer enclosures to the parametric AE solution set

We consider systems of linear equations, where the elements of the matrix and of the right-hand side vector are linear functions of interval parameters. We study parametric AE solution sets, which are defined by universally and existentially quantified parameters, and the former precede the latter. Based on a recently obtained explicit description of such solution sets, we present three approaches for obtaining outer estimations of parametric AE solution sets. The first approach intersects inclusions of parametric united solution sets for all combinations of the end-points of the universally quantified parameters. Polynomially computable outer bounds for parametric AE solution sets are obtained by parametric AE generalization of a single-step Bauer–Skeel method. In the special case of parametric tolerable solution sets, we derive an enclosure based on linear programming approach; this enclosure is optimal under some assumption. The application of these approaches to parametric tolerable and controllable solution sets is discussed. Numerical examples accompanied by graphic representations illustrate the solution sets and properties of the methods.

Computer-Assisted Proofs in Solving Linear Parametric Problems

Consider a linear system A(p)x = b(p) whose input data depend on a number of uncertain parameters p = (p1,...,pk) varying within given intervals [p]. The objective is to verify by numerical computations monotonic (and convexity/concavity) dependence of a solution component xi(p) with respect to a parameter pj over the interval box [p], or more general, to prove if some boundary inf / sup xi(p) for all p isin [p] is attained at the end-points of [p]. Such knowledge is useful in many applications in order to facilitate the solution of some underlying linear parametric problem involving uncertainties. In this paper we present a technique, for proving the desired properties of the parametric solution, which is alternative to the approaches based on extreme point computations. The proposed computer-aided proof is based on guaranteed interval enclosures for the partial derivatives of the parametric solution for all p isin [p]. The availability of self-validated methods providing guaranteed enclosure of a parametric solution set by floating-point computations is a key for the efficiency and the expanded scope of applicability of the proposed approach. Linear systems involving nonlinear parameter dependencies, and dependencies between A(p) and b(p), as well as non-square linear parametric systems can be handled successfully. Presented are details of the algorithm design and mathematica tools implementing the proposed approach. Numerical examples from structural mechanics illustrate its application.

Solving linear systems whose input data are rational functions of interval parameters

The paper proposes an approach for self-verified solving of linear systems involving rational dependencies between interval parameters. A general inclusion method is combined with an interval arithmetic technique providing inner and outer bounds for the range of monotone rational functions. The arithmetic on proper and improper intervals is used as an intermediate computational tool for eliminating the dependency problem in range computation and for obtaining inner estimations by outwardly rounded interval arithmetic. Supporting software tools with result verification, developed in the environment of CAS Mathematica, are reported.

Improved solution enclosures for over- and underdetermined interval linear systems

In this paper we discuss an inclusion method for solving rectangular (over- and under-determined) dense linear systems where the input data are uncertain and vary within given intervals. An improvement of the quality of the solution enclosures is described for both independent and parameter dependent input intervals. A fixed-point algorithm with result verification that exploits the structure of the problems to be solved is given. Mathematica functions for solving the discussed rectangular problems are developed and presented. Numerical examples illustrate the advantages of the proposed improved approach.

Improved enclosure for some parametric solution sets with linear shape

Consider linear algebraic systems where the elements of the matrix and the right-hand side vector depend linearly on a number of interval parameters. We prove some sufficient conditions for the united parametric solution set of such a system to have linear boundary. These conditions imply an equivalent representation of the parametric system where each parameter appears once in a diagonal matrix. The latter representation allows us to expand the scope of applicability of the best known so far interval method, developed by A. Neumaier and A. Pownuk, for enclosing the parametric solution set and to generalize the method for systems where the parameter dependencies connect the matrix and the right-hand side vector. Some examples demonstrate that: parametric solution sets with linear boundary appear in various application domains, the generalized method improves the solution enclosure and the proven sufficient conditions can be helpful for solving various other interval problems.

Quality of the solution sets of parameter-dependent interval linear systems

Consider linear systems whose matrix and right-hand side vector depend affine-linearly on parameters varying within prescribed bounds. We present some sufficient conditions under which the interval hull (or some bounds) of the solution set of a parametrised interval linear system coincides with the interval hull (or bounds) of the non-parametric interval linear system corresponding to the parametric one.