Analysis and Optimization of an Adaptive Interpolation Algorithm for the Numerical Solution of a System of Ordinary Differential Equations with Interval Parameters

Morozov, A. Yu.; Zhuravlev, A.A.; Ревизников, Д. Л.

doi:10.1134/s0012266120070125

Cited by 5 publications

(7 citation statements)

References 11 publications

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“…For ODE systems given by Equations ( 11) and (12), it was possible to obtain a solution in a reasonable time only using the approach proposed in the paper since the number of interval uncertainties is quite large. To solve system given by Equation (11), the equivalent number of sampling points was about 80 thousand, and in the case of using the classical algorithm with a degree of polynomial 4, the value would be about 10 million.…”

Section: Discussionmentioning

confidence: 99%

“…Let us consider an applied problem of computational materials science, within the framework of which the stresses arising during the deformation of an ideal crystal are calculated [12]. Different angles are possible between the orientation of the crystal lattice and the direction of stretching with a fixed stretch value.…”

Section: Computation Of An Interval Stress Tensor For Materials With a Covalent Chemical Bondmentioning

confidence: 99%

“…Papers [11][12][13][14] describe the adaptive interpolation algorithm in detail. 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.…”

Section: Introductionmentioning

confidence: 99%

“…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 theoretical basis of the algorithm is given in References [11,12]. The algorithm has a number of essential features: it is not subject to the wrapping effect [11]; efficiently parallelizes on GPUs; able to simulate rigid systems [13]; determines the presence of bifurcations and chaos in the system [14].…”

Section: Introductionmentioning

confidence: 99%

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

2021

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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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Section: Discussionmentioning

confidence: 99%

Section: Computation Of An Interval Stress Tensor For Materials With a Covalent Chemical Bondmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2021

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“…Its essence is to construct a piecewise polynomial function that interpolates the dependence of the solution to the problem on point values of interval parameters. The algorithm has a theoretical justification [13,17] and has been successfully applied to rigid systems of ordinary differential equations (ODEs) [18], systems with dynamic chaos and bifurcations [19], applied problems of chemical kinetics [18], gas and molecular dynamics [20].…”

Section: Introductionmentioning

confidence: 99%

Adaptive Sparse Grids with Nonlinear Basis in Interval Problems for Dynamical Systems

Morozov,

Reviznikov

2023

Computation

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Problems with interval uncertainties arise in many applied fields. The authors have earlier developed, tested, and proved an adaptive interpolation algorithm for solving this class of problems. The algorithm’s idea consists of constructing a piecewise polynomial function that interpolates the dependence of the problem solution on point values of interval parameters. The classical version of the algorithm uses polynomial full grid interpolation and, with a large number of uncertainties, the algorithm becomes difficult to apply due to the exponential growth of computational costs. Sparse grid interpolation requires significantly less computational resources than interpolation on full grids, so their use seems promising. A representative number of examples have previously confirmed the effectiveness of using adaptive sparse grids with a linear basis in the adaptive interpolation algorithm. The purpose of this paper is to apply adaptive sparse grids with a nonlinear basis for modeling dynamic systems with interval parameters. The corresponding interpolation polynomials on the quadratic basis and the fourth-degree basis are constructed. The efficiency, performance, and robustness of the proposed approach are demonstrated on a representative set of problems.

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Adaptive Interpolation Algorithm on Sparse Meshes for Numerical Integration of Systems of Ordinary Differential Equations with Interval Uncertainties

Morozov

Ревизников

2021

Diff Equat

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Analysis and Optimization of an Adaptive Interpolation Algorithm for the Numerical Solution of a System of Ordinary Differential Equations with Interval Parameters

Cited by 5 publications

References 11 publications

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

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

Adaptive Sparse Grids with Nonlinear Basis in Interval Problems for Dynamical Systems

Adaptive Interpolation Algorithm on Sparse Meshes for Numerical Integration of Systems of Ordinary Differential Equations with Interval Uncertainties

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