Spatiotemporal Orthogonal Polynomial Approximation for Partial Differential Equations

Starting with some fundamental concepts, in this article we present the essential aspects of spectral methods and their applications to the numerical solution of Partial Differential Equations (PDEs). We start by using Lagrange and Techbychef orthogonal polynomials for spatiotemporal approximation of PDEs as a weighted sum of polynomials. We use collocation at some clustered grid points to generate a system of equations to approximate the weights for the polynomials. We finish the study by demonstrating approximate solutions of some PDEs in one space dimension.Comment: 9 pages, 9 figure

Monte Carlo approximations of the Neumann problem

We introduce Monte Carlo methods to compute the solution of elliptic equations with pure Neumann boundary conditions. We first prove that the solution obtained by the stochastic representation has a zero mean value with respect to the invariant measure of the stochastic process associated to the equation. Pointwise approximations are computed by means of standard and new simulation schemes especially devised for local time approximation on the boundary of the domain. Global approximations are computed thanks to a stochastic spectral formulation taking into account the property of zero mean value of the solution. This stochastic formulation is asymptotically perfect in terms of conditioning. Numerical examples are given on the Laplace operator on a square domain with both pure Neumann and mixed Dirichlet-Neumann boundary conditions. A more general convection-diffusion equation is also numerically studied

Tchebychev Polynomial Approximations for mthm^{th} Order Boundary Value Problems

Higher order boundary value problems (BVPs) play an important role modeling various scientific and engineering problems. In this article we develop an efficient numerical scheme for linear $m^{th}$ order BVPs. First we convert the higher order BVP to a first order BVP. Then we use Tchebychev orthogonal polynomials to approximate the solution of the BVP as a weighted sum of polynomials. We collocate at Tchebychev clustered grid points to generate a system of equations to approximate the weights for the polynomials. The excellency of the numerical scheme is illustrated through some examples.Comment: 21 pages, 10 figure

Multiobjective centralized DEA approach to Tokyo 2020 Olympic Games

"Open Access: This article is licensed under a Creative Commons Attribution 4.0 International License...."There exist two types of Data Envelopment Analysis (DEA) approaches to the Olympic Games: conventional and fixed-sum outputs (FSO). The approach proposed in this paper belongs to the latter category as it takes into account the total number de medals of each type awarded. Imposing these constraints requires a centralized DEA perspective that projects all the countries simultaneously. In this paper, a multiobjective FSO approach is proposed, and the Weighted Tchebychef solution method is employed. This approach aims to set all output targets as close as possible to their ideal values. In order to choose between the alternative optima, a secondary goal has been considered that minimizes the sum of absolute changes in the number of medals, which also renders the computed targets to be as close to the observed values as possible. These targets represent the output levels that could be expected if all countries performed at their best level. For certain countries, the targets are higher than the actual number of medals won while, for other countries, these targets may be lower. The proposed approach has been applied to the results of the Tokyo 2020 Olympic Games and compared with both FSO and non-FSO DEA method

The Use of Combined Multicriteria Method for the Valuation of Real Estate

A considered problem is a valuation of real estate. It is important to specify their exact market value, which is the result of several factors. Valuation of property is made on the basis of information and transactions on the local market. Moreover, the valuation always is based on the data of the similar properties. A comprehensive set of data is needed for these reasons. It is quite confusing because the number of transactions on the local market often is not sufficient. The purpose of this paper is to present a method for multicriteria valuation of real estate. This procedure is based on the Analytic Hierarchy Process (AHP) and the Goal Programming (GP). It was designed especially for valuation in situation in which information are limited. The proposed method was used for the valuation of the real estate located on [email protected] of Economic Sciences, Warsaw University of Life Sciences5(71)20821

On the Sum of the Square of a Prime and a Square-Free Number

We prove that every integer $n \geq 10$ such that $n \not\equiv 1 \text{mod} 4$ can be written as the sum of the square of a prime and a square-free number. This makes explicit a theorem of Erd\H{o}s that every sufficiently large integer of this type may be written in such a way. Our proof requires us to construct new explicit results for primes in arithmetic progressions. As such, we use the second author's numerical computation regarding GRH to extend the explicit bounds of Ramar\'e-Rumely.Comment: 12 page

Robust adaptive numerical integration of irregular functions with applications to basket and other multi-dimensional exotic options

International audienceWe improve an adaptive integration algorithm proposed by two of the authors by introducing a new splitting strategy based on a geometrical criterion. This algorithm is tested especially on the pricing of multidimensional vanilla options in the Black–Scholes framework which emphasizes the numerical problems of integrating non-smooth functions. In high dimensions, this new algorithm is used as a control variate after a dimension reduction based on principal component analysis. Numerical tests are performed on the Genz package, on the pricing of basket, put on minimum and digital options in dimensions up to ten

Adaptive Integration and Approximation over hyper-rectangular regions with applications to basket options pricing

International audienceWe describe an adaptive algorithm to compute sparse polynomial approximations and the integral of a multivariate function over hyper-rectangular regions in medium dimensions. Numerical examples are given on functions taken from the Genz package and on basket options pricing in dimension up to 5