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.

A Novel Algorithm for Path Planning of the Mobile Robot in Obstacle Environment

—In this paper, a design method of smoothing the path generated by a novel algorithm is proposed, which makes the mobile robot can more rapidly and smoothly follow the path and reach the target point. No matter the attitude vector angle is an acute angle or obtuse angle, there is no doubt that we can find the right curve, including polar polynomial curves and piecewise polynomial functions, which makes the path length and the circular arc tend to be similar and guarantees the shorter path length. In the condition of meeting the dynamic characteristics of the mobile robot, the tracking speed and quality are improved. Therefore, the symmetric polynomial curve and the piecewise polynomial function curve are used to generate a smooth path. This novel algorithm improves the path tracking accuracy and the flexibility of the mobile robot. At the same time, it expands the application range of mobile robot in structured environment.

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.

The impact of time-varying risk on stock returns: an experiment of cubic piecewise polynomial function model and the Fourier Flexible Form model

<abstract> <p>With fast evolving econometric techniques being adopted in asset pricing, traditional linear asset pricing models have been criticized by their limited function on capturing the time-varying nature of data and risk, especially the absence of data smoothing is of concern. In this paper, the impact of data smoothing is explored by applying two asset pricing models with non-linear feature: cubic piecewise polynomial function (CPPF) model and the Fourier Flexible Form (FFF) model are performed on US stock returns as an experiment. The traditional beta coefficient is treated asymmetrically as downside beta and upside beta in order to capture corresponding risk, and further, to explore the risk premia attached in a cross-sectional context. It is found that both models show better goodness of fit comparing to classic linear asset pricing model cross-sectionally. When appropriate knots and orders are determined by Akaike Information Criteria (AIC), the goodness of fit is further improved, and the model with both CPPF and FFF betas employed showed the best fit among other models. The findings fill the gap in literature, specifically on both investigating and pricing the time variation and asymmetric nature of systematic risk. The methods and models proposed in this paper embed advanced mathematical techniques of data smoothing and widen the options of asset pricing models. The application of proposed models is proven to superiorly provide high degree of explanatory power to capture and price time-varying risk in stock market.</p> </abstract>

Comparative Analysis of Different Configuration Domestic Refrigerators: A Computational Fluid Dynamics Approach

To reduce the tremendous increase in the energy consumption in the residential sector, there is a continuous need to improve the cooling efficiency and reduce running cost in domestic refrigerators. In this regard, three domestic refrigerator configurations have been considered. These configurations, namely, top mounted freezer (TMF), bottom mounted freezer (BMF), and side mounted freezer (SMF), were numerically simulated using ansys fluent 14 code. The refrigerators considered in this paper are air cooled by natural convection mechanism. For improved accuracy, piecewise polynomial function was used to obtain the temperature dependent specific heat capacity, while the discrete ordinate (DO) model was used to account for the radiation energy exchange between the refrigerator walls and cooling air. The effect of refrigerator opening and refrigerator load on the performance of the model refrigerators was also studied. Results show that cabinets that have the same relative position from the base (ground level) in TMF, BMF, and SMF configuration was observed to have similar cooling effectiveness irrespective of the compartment (i.e., freezer or fresh food). Load in the lowest parts of the model refrigerator consistently maintains the highest cooling effectiveness with about 15% more than their respective topmost cabinet. Thus, consumer preference of highly efficient compartment (either freezer or refrigerator) should be considered. After 300 min cooling time, the TMF and BMF cooling load are more than that of SMF by about 8%. This suggests that SMF with better cooling effectiveness will consume less energy and would have a lower running cost.

Piecewise Polynomial Mapping Method and Corresponding WENO Scheme with Improved Resolution

Abstract. The method of mapping function was first proposed by Henrick et al. [J. Comput. Phys. 207:542-547 (2005)] to adjust nonlinear weights in [0,1] for the fifth order WENO scheme, and through which the requirement of convergence order is satisfied and the performance of the scheme is improved. Different from Henrick’s method, a concept of piecewise polynomial function is proposed in this study and corresponding WENO schemes are obtained. The advantage of the new method is that the function can have a gentle profile at the location of the linear weight (or the mapped nonlinear weight can be close to its linear counterpart), and therefore is favorable for the resolution enhancement. Besides, the function also has the flexibility of quick convergence to identity mapping near two endpoints of [0,1], which is favorable for improved numerical stability. The fourth-, fifth- and sixth-order polynomial functions are constructed correspondingly with different emphasis on aforementioned flatness and convergence. Among them, the fifth-order version has the flattest profile. To check the performance of the methods, the 1-D Shu-Osher problem, the 2-D Riemann problem and the double Mach reflection are tested with the comparison of WENO-M, WENO-Z and WENO-NS. The proposed new methods show the best resolution for describing shear-layer instability of the Riemann problem, and they also indicate high resolution in computations of double Mach reflection, where only these proposed schemes successfully resolved the vortex-pairing phenomenon. Other investigations have shown that the single polynomial mapping function has no advantage over the proposed piecewise one, and it is of no evident benefit to use the proposed method for the symmetric fifth-order WENO. Overall, the fifth-order piecewise polynomial and corresponding WENO scheme are suggested for resolution improvement.