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 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.