Sparse Grid Adaptive Interpolation in Problems of Modeling Dynamic Systems with Interval Parameters

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.

Interval Model of a Memristor Crossbar Network

“…This section provides a brief description of the adaptive interpolation algorithm for modeling dynamic systems with interval parameters in accordance with. [23,26,27] Let us describe the algorithm using the example of the ODEs system considered in the previous section…”

“…This section provides a brief description of the adaptive interpolation algorithm for modeling dynamic systems with interval parameters in accordance with. [ 23,26,27 ]…”

“…The algorithm has been successfully applied to stiff systems of ordinary differential equations, systems with dynamic chaos and bifurcations, [ 26 ] applied problems of chemical kinetics and molecular dynamics. [ 27 ] The article [ 27 ] shows the possibility of using sparse grids [ 28–30 ] in the adaptive interpolation algorithm for modeling dynamic systems containing a large number of interval parameters and demonstrates the effectiveness of this approach.…”

Preface to the Special Issue on “Control, Optimization, and Mathematical Modeling of Complex Systems”

“…Morozov et al [19] are 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.…”

“…Numerical simulation in physical, social, and life sciences [1][2][3][4]; • Modeling and analysis of complex systems based on mathematical methods and AI/ML approaches [5,6]; • Control problems in robotics [3,[7][8][9][10][11][12]]; • Design optimization of complex systems [13]; • Modeling in economics and social sciences [4,14]; • Stochastic models in physics and engineering [1,[15][16][17][18]; • Mathematical models in material science [19]; • High-performance computing for mathematical modeling [20].…”