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

**Alexander Yu Morozov \*, Andrey A. Zhuravlev and Dmitry L. Reviznikov**

Federal Research Center "Computer Science and Control" of Russian Academy of Sciences (FRC CSC RAS), 119333 Moscow, Russia; zhuravlyow.andrei@yandex.ru (A.A.Z.); reviznikov@mai.ru (D.L.R.)

**\*** Correspondence: morozov@infway.ru

**Abstract:** 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.

**Keywords:** adaptive interpolation algorithm; interval ordinary differential equations (ODEs); sparse grids; hierarchical basis; multidimensional interpolation; high dimensions; molecular dynamics modeling
