Parallel Adaptive Interpolation Algorithm based on Sparse Grids for Modeling Dynamic Systems with Interval Parameters

The paper presents a parallel algorithm for adaptive interpolation based on sparse grids for modeling dynamic systems with interval parameters. The idea of the algorithm is to construct a piecewise polynomial function that interpolates the dependence of the solution to the problem on the point values of the interval parameters. In the classical version of the algorithm, polynomial interpolation on complete grids is used, and with a large number of uncertainties, the algorithm becomes difficult to apply due to the exponential growth of computational costs. The use of sparse grids can significantly reduce the computational costs, but nevertheless the complexity of the algorithm in the general case remains exponential with respect to the number of interval parameters. In this regard, the issue of accelerating the algorithm is relevant. The algorithm can be divided into several sets of independent subtasks: updating the values corresponding to the grid nodes; calculation of weighting factors; interpolation of values at new nodes. The last two sets imply parallelization of recursion, so here the techniques for traversing the width of the call graph are mainly used. The parallel implementation of the algorithm was tested on two ODE systems containing two and six interval parameters, respectively, using a different number of computing cores. The results obtained demonstrate the effectiveness of the approaches used.

Automatic approximation using asymptotically optimal adaptive interpolation

We present an asymptotic analysis of adaptive methods for Lp approximation of functions f ∈ Cr([a, b]), where $1\le p\le +\infty $ 1 ≤ p ≤ + ∞ . The methods rely on piecewise polynomial interpolation of degree r − 1 with adaptive strategy of selecting m subintervals. The optimal speed of convergence is in this case of order m−r and it is already achieved by the uniform (nonadaptive) subdivision of the initial interval; however, the asymptotic constant crucially depends on the chosen strategy. We derive asymptotically best adaptive strategies and show their applicability to automatic Lp approximation with a given accuracy ε.

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

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.