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.

The Analysis and the Measurement of Poverty: An Interval-Based Composite Indicator Approach

The study of poverty and its quantification is a critical yet unresolved problem in social science. This work seeks to use a new composite indicator to assess poverty as a multidimensional concept. However, subjective decisions, such as various weighting systems on the indicator’s creation, may affect its perception. In order to solve this issue, we propose to use random different composite indicators based on simulated weightings and specifications to get a comprehensive interval-based composite indicator. Our method generates robust and trustworthy measurements based on a meaningful conceptual model of poverty. Furthermore, we use some interval parameters such as the upper bound, center, and lower bound to compare the different intervals related to the different statistical units and rankings to aid in analyzing extreme situations and policy scenarios. In Sicily, Calabria, Campania, and Puglia, we identify urgent circumstances. The findings reveal a consistent indicator measurement and the shadow sector’s influence on the final measurements.

Computational Method and Simulation of Reliability for Series, Parallel, and k-Out-of-n Systems with Interval Parameters

For series, parallel, and k-out-of-n voting system reliability calculation methods, the six σ principles have been proposed in this study to derive the interchange relationship between interval parameters and random parameters. The interval reliability index can be expressed in the function of the random reliability index. The interval reliability index can then be transformed into a random reliability index. The computational method of the reliability for series, parallel, and k-out-of-n voting systems with interval parameters is established. Finally, it has been shown that the proposed method is rational, practical, and applicable with two engineering practical simulations.

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.