Abstract
The paper is concerned with the issues of modeling dynamic systems with interval parameters. In previous works, the authors proposed an adaptive interpolation algorithm for solving interval problems; the essence of the algorithm is the dynamic construction of a piecewise polynomial function that interpolates the solution of the problem with a given accuracy. The main problem of applying the algorithm is related to the curse of dimension, i.e., exponential complexity relative to the number of interval uncertainties in parameters. The main objective of this work is to apply the previously proposed adaptive interpolation algorithm to dynamic systems with a large number of interval parameters. In order to reduce the computational complexity of the algorithm, the authors propose using adaptive sparse grids. This article introduces a novelty approach of applying sparse grids to problems with interval uncertainties. The efficiency of the proposed approach has been demonstrated on representative interval problems of nonlinear dynamics and computational materials science.
Highlights
Federal Research Center “Computer Science and Control” of Russian Academy of Sciences (FRC CSC RAS), Abstract: The paper is concerned with the issues of modeling dynamic systems with interval parameters
The number of grid nodes depends exponentially on the number of interval parameters, which limits the scope of the algorithm
The paper presents an adaptive interpolation algorithm on sparse grids, which allows for reducing the exponential complexity when solving multidimensional problems in parameter space
Summary
Federal Research Center “Computer Science and Control” of Russian Academy of Sciences (FRC CSC RAS), Abstract: The paper is concerned with the issues of modeling dynamic systems with interval parameters. Methods based on classical interval arithmetic are subject to the so-called wrap effect [1], which manifests itself in an unlimited increase in the width of the obtained interval estimates of solutions. Existing methods that are not subject to this effect, or weakly susceptible to it, often have exponential complexity with respect to the number of interval parameters It concerns symbolic methods operating in series, Monte-Carlo methods, and the adaptive interpolation algorithm [11]. While solving a considered class of problems, the main idea is to construct an explicit dependence of the solution to the corresponding non-interval problem on the point values of the interval parameters If such dependence is available, finding an interval estimate would be reduced to solving a certain number of constrained optimization problems for explicitly given functions. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
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