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

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 int...

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Vydáno v:Mathematics (Basel) Ročník 9; číslo 4; s. 298
Hlavní autoři: Morozov, Alexander Yu, Zhuravlev, Andrey A., Reviznikov, Dmitry L.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Basel MDPI AG 01.02.2021
Témata:
Adaptation
Adaptive algorithms
adaptive interpolation algorithm
Adaptive systems
Algorithms
Cauchy problems
Complexity
Computational mathematics
Costs
Dynamical systems
hierarchical basis
high dimensions
Interpolation
interval ordinary differential equations (ODEs)
Materials science
Mathematical models
multidimensional interpolation
Nonlinear dynamics
Optimization
Ordinary differential equations
Parameter uncertainty
Partial differential equations
Polynomials
sparse grids
ISSN:2227-7390, 2227-7390
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Shrnutí: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.
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ISSN:2227-7390
2227-7390
DOI:10.3390/math9040298