Chapter 19
Adaptive Interpolation,
TT-Decomposition and Sparse Grids
for Modeling Dynamic Systems
with Interval Parameters
Alexander Yu. Morozov and Dmitry L. Reviznikov
Abstract Problems with uncertainties arise in many practical fields and traditionally
are formulated as dynamic systems with interval parameters. Often the complexity of
existing methods is exponential in relation to the number of interval parameters. The
adaptive interpolation algorithm and approaches directed to reducing the curse of
dimensionality are considered. The main assumption on which these approaches are
based is that not all interval parameters make a significant contribution to the solution
of the problem. The use of tensor train decomposition and sparse grids allows us to
take into account these features and expand the scope of the algorithm for the case of a
large number of interval parameters. The effectiveness of the considered approaches
is confirmed on several model problems.
19.1 Introduction
Problems with inaccurate data appear in many important areas of modern science. In
particular, when solving various applied problems of the aerospace industry, problems of mechanics, and others, the situations often occur when some parameters are
not exactly known, but there is information about the ranges, in which their values
are located. Since most problems are formulated as a system of Ordinary Differential
Equations (ODEs), it becomes necessary to solve the Cauchy problem with interval
initial conditions or parameters [1].
A. Yu. Morozov · D. L. Reviznikov (B)
Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow
125993, Russian Federation
e-mail: reviznikov@mai.ru
A. Yu. Morozov
e-mail: morozov@infway.ru
Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences,
44, Vavilov ul., Moscow 119333, Russian Federation
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
L. C. Jain et al. (eds.), Applied Mathematics and Computational Mechanics for Smart
Applications, Smart Innovation, Systems and Technologies 217,
https://doi.org/10.1007/978-981-33-4826-4_19
271
Adaptive Interpolation,
TT-Decomposition and Sparse Grids
for Modeling Dynamic Systems
with Interval Parameters
Alexander Yu. Morozov and Dmitry L. Reviznikov
Abstract Problems with uncertainties arise in many practical fields and traditionally
are formulated as dynamic systems with interval parameters. Often the complexity of
existing methods is exponential in relation to the number of interval parameters. The
adaptive interpolation algorithm and approaches directed to reducing the curse of
dimensionality are considered. The main assumption on which these approaches are
based is that not all interval parameters make a significant contribution to the solution
of the problem. The use of tensor train decomposition and sparse grids allows us to
take into account these features and expand the scope of the algorithm for the case of a
large number of interval parameters. The effectiveness of the considered approaches
is confirmed on several model problems.
19.1 Introduction
Problems with inaccurate data appear in many important areas of modern science. In
particular, when solving various applied problems of the aerospace industry, problems of mechanics, and others, the situations often occur when some parameters are
not exactly known, but there is information about the ranges, in which their values
are located. Since most problems are formulated as a system of Ordinary Differential
Equations (ODEs), it becomes necessary to solve the Cauchy problem with interval
initial conditions or parameters [1].
A. Yu. Morozov · D. L. Reviznikov (B)
Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow
125993, Russian Federation
e-mail: reviznikov@mai.ru
A. Yu. Morozov
e-mail: morozov@infway.ru
Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences,
44, Vavilov ul., Moscow 119333, Russian Federation
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
L. C. Jain et al. (eds.), Applied Mathematics and Computational Mechanics for Smart
Applications, Smart Innovation, Systems and Technologies 217,
https://doi.org/10.1007/978-981-33-4826-4_19
271