On preconditioning and solving an extended class of interval parametric linear systems

TL;DR

This work tackles interval parametric linear systems (IPLS) by seeking tight enclosures of their united parametric solution sets. It advances a novel approach that combines right and double preconditioning with revised affine forms to produce a parametric solution $x(p)=Fp+a$ that preserves dependencies and expands the solvable problem class. The paper analyzes various preconditioning strategies (including LU, SVD, and QR decompositions of the midpoint inverse) and demonstrates, through numerical experiments, that double LU preconditioning often yields the tightest enclosures and enables solving problems inaccessible to existing IPLS methods. The findings highlight the practical impact of improved preconditioning on accuracy and tractability for IPLS across benchmarking examples and real-world-like circuits and structures.

Abstract

We deal with interval parametric systems of linear equations and the goal is to solve such systems, which basically comes down to finding an enclosure for a parametric solution set. Obviously we want this enclosure to be as tight as possible. The review of the available literature shows that in order to make a system more tractable most of the solution methods use left preconditioning of the system by the midpoint inverse. Surprisingly, and in contrast to standard interval linear systems, our investigations have shown that double preconditioning can be more efficient than a single one, both in terms of checking the regularity of the system matrix and enclosing the solution set. Consequently, right (which was hitherto mentioned in the context of checking regularity of interval parametric matrices) and double preconditioning together with the p-solution concept enable us to solve a larger class of interval parametric linear systems than most of existing methods. The applicability of the proposed approach to solving interval parametric linear systems is illustrated by several numerical examples.

Figures (9)

• Figure 1: Comparison of the $p$-solution (dark gray region) and the interval solution (light gray region) for a given two dimensional IPLS with two parameters
• Figure 2: Results for Example \ref{['ex:romatrices_1']}: boxplot of ratios $\rho({A}^\Delta)/\rho({H}^\Delta)$ obtained from 500 repetitions for first variant; ${\textrm{\boldmath $u$}}=[-0.5,1.0]$, ${\textrm{\boldmath $v$}}=[2,2.5]$
• Figure 3: Results for Example \ref{['ex:hrmatrices']}: boxplot of ratio $\rho({A}^\Delta)/\rho({H}^\Delta)$ obtained from 500 repetitions for first variant; ${\textrm{\boldmath $u$}}=[-0.5,1.0]$, ${\textrm{\boldmath $v$}}=[2,2.5]$
• Figure 4: Results for Example \ref{['ex:randm_nonidmid']}: boxplot of ratio $\rho({A}^\Delta)/\rho({H}^\Delta)$; first variant (left), second variant (right); ${\textrm{\boldmath $u$}}=[-1,2]$, ${\textrm{\boldmath $v$}}=[2,3]$
• Figure 5: The elements of the solution set of $(\ref{['eq:ex1_org']})$ as functions of parameter $p$ (since $x_1$ and $x_2$ are monotone with respect to $p$, their extremal values are attained at respective endpoints of ${\textrm{\boldmath $p$}}$) and the solution set of system $(\ref{['eq:ex1_org']})$; Example \ref{['ex:ex1']}

Theorems & Definitions (16)

• proposition 1
• definition 1
• proposition 2
• proposition 3
• proposition 4
• proof
• proposition 5
• proof
• remark 1: Order of preconditioning and relaxation
• definition 2