Adaptive Interpolation Algorithm Using TT-Decomposition for Modeling Dynamical Systems with Interval Parameters

Problems with inexactly specified parameters arise in many applications, and they are often formulated as dynamical systems with interval parameters. For such systems, it is required to know how to obtain an interval estimate of solution given the interval values of the parameters. The main idea und...

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Veröffentlicht in:Computational mathematics and mathematical physics Jg. 61; H. 9; S. 1387 - 1400
Hauptverfasser: Gidaspov, V. Yu, Morozov, A. Yu, Reviznikov, D. L.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Moscow Pleiades Publishing 01.09.2021
Springer Nature B.V
Schlagworte:
Adaptive algorithms
Algorithms
Arrays
Chemical reactions
Computational Mathematics and Numerical Analysis
Decomposition
Dynamical systems
General Numerical Methods
Initial conditions
Interpolation
Mathematical models
Mathematics
Mathematics and Statistics
Nodes
Parameters
Polynomials
tensor train
adaptive interpolation algorithm
chemical kinetics
kd-tree
TT-decomposition
interval ODEs
ISSN:0965-5425, 1555-6662
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Zusammenfassung:Problems with inexactly specified parameters arise in many applications, and they are often formulated as dynamical systems with interval parameters. For such systems, it is required to know how to obtain an interval estimate of solution given the interval values of the parameters. The main idea underlying the adaptive interpolation algorithm is to construct an adaptive hierarchical grid based on the kd-tree over the set formed by the interval initial conditions and parameters of the problem. In this grid, each cell contains an interpolation grid. For each point in time, depending on the features of solution, the decomposition is adaptively rearranged. At each step, the algorithm produces a piecewise polynomial function that interpolates the dependence of the problem solution on the parameter values with prescribed accuracy. As the number of interval parameters grows, the number of nodes in the interpolation grid grows exponentially, which restricts the algorithm scope. To mitigate this drawback, it is proposed to use the TT-decomposition of the multidimensional arrays in which the values of nodes of the interpolation grids are stored instead of these arrays. The efficiency of the proposed modification of the algorithm is demonstrated on model problems. In particular, the combustion of hydrogen–oxygen mixture in the presence of indeterminacy in the constants of chemical reaction rates is simulated.
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ISSN:0965-5425
1555-6662
DOI:10.1134/S0965542521090098