A RBF partition of unity collocation method based on finite difference for initial-boundary value problems

Meshfree radial basis function (RBF) methods are popular tools used to numerically solve partial differential equations (PDEs). They take advantage of being flexible with respect to geometry, easy to implement in higher dimensions, and can also provide high order convergence. Since one of the main disadvantages of global RBF-based methods is generally the computational cost associated with the solution of large linear systems, in this paper we focus on a localizing RBF partition of unity method (RBF-PUM) based on a finite difference (FD) scheme. Specifically, we propose a new RBF-PUM-FD collocation method, which can successfully be applied to solve time-dependent PDEs. This approach allows to significantly decrease ill-conditioning of traditional RBF-based methods. Moreover, the RBF-PUM-FD scheme results in a sparse matrix system, reducing the computational effort but maintaining at the same time a high level of accuracy. Numerical experiments show performances of our collocation scheme on two benchmark problems, involving unsteady convection-diffusion and pseudo-parabolic equations

A low cost and highly accurate technique for big data spatial-temporal interpolation

The high velocity, variety and volume of data generation by today's systems have necessitated Big Data (BD) analytic techniques. This has penetrated a wide range of industries; BD as a notion has various types and characteristics, and therefore a variety of analytic techniques would be required. The traditional analysis methods are typically unable to analyse spatial-temporal BD. Interpolation is required to approximate the values between the already existing data points, yet since there exist both location and time dimensions, only a multivariate interpolation would be appropriate. Nevertheless, existing software are unable to perform such complex interpolations. To overcome this challenge, this paper presents a layer by layer interpolation approach for spatial-temporal BD. Developing this layered structure provides the opportunity for working with much smaller linear system of equations. Consequently, this structure increases the accuracy and stability of numerical structure of the considered BD interpolation. To construct this layer by layer interpolation, we have used the good properties of Radial Basis Functions (RBFs). The proposed new approach is applied to numerical examples in spatial-temporal big data and the obtained results confirm the high accuracy and low computational cost. Finally, our approach is applied to explore one of the air pollution indices, i.e. daily PM2.5 concentration, based on different stations in the contiguous United States, and it is evaluated by leave-one-out cross validation

Copula-based probabilistic assessment of intensity and duration of cold episodes: A case study of Malayer vineyard region

Frost, particularly during the spring, is one of the most damaging weather phenomena for vineyards, causing significant economic losses to vineyards around the world each year. The risk of tardive frost damage in vine-yards due to changing climate is considered as an important threat to the sustainable production of grapes. Therefore, the cold monitoring strategies is one of the criteria with significant impacts on the yields and prosperity of horticulture and raisin factories. Frost events can be characterized by duration and severity. This paper investigates the risk and impacts of frost phenomenon in the vineyards by modeling the joint distribution of duration and severity factors and analyzing the influential parameter’s dependency structure using capabilities of copula functions. A novel mathematical framework is developed within this study to understand the risk and uncertainties associate with frost events and the impacts on yields of vineyards by analyzing the non-linear dependency structure using copula functions as an efficient tool. The developed model was successfully vali-dated for the case study of vineyard in Malayer city of Iran. The copula model developed in this study was shown to be a robust tool for predicting the return period of the frost events