Adaptive Interpolation Algorithm on Sparse Meshes for Numerical Integration of Systems of Ordinary Differential Equations with Interval Uncertainties
A. Yu. Morozov, Dmitry Leonidovich Reviznikov · Differential Equations · 2021
Abstract We consider the theoretical aspects of generalization of the adaptive interpolation algorithm to the case of a large number of interval uncertainties with the use of sparse grids. The classical adaptive interpolation algorithm essentially constructs an adaptive hierarchical partition of the uncertainty domain into subdomains each of which corresponds to a polynomial function of some degree interpolating the dependence of the solution of the problem on the point values of interval parameters with a given accuracy. The main disadvantage of the algorithm is its exponential dependence on the number of interval parameters. Already in the presence of five to six interval uncertainties, the algorithm becomes practically inapplicable. In this regard, it is proposed to use interpolation on adaptive sparse meshes instead of interpolation on a regular mesh; in some cases, this permits considerably expanding the scope of the algorithm. An estimate is produced for the global error of the algorithm on sparse meshes. The linear dependence of the global algorithm error on the mesh level is shown. The efficiency of our approach is demonstrated on a representative series of problems.