scholarly journals Analytical-Numerical Solution of a Parabolic Diffusion Equation Under Uncertainty Conditions Using DTM with Monte Carlo Simulations

2015 ◽  
Vol 11 (22) ◽  
pp. 49-72 ◽  
Author(s):  
Gilberto González Parra ◽  
Abraham J Arenas ◽  
Miladys Cogollo
Keyword(s):  
Monte Carlo ◽  
Diffusion Coefficient ◽  
Hybrid Method ◽  
Source Term ◽  
Closed Form Solution ◽  
Random Variables ◽  
Form Solution ◽  
Mean Values ◽  
The Monte Carlo Method ◽  
Condi Tions

A numerical method to solve a general random linear parabolic equationwhere the diffusion coefficient, source term, boundary and initial condi-tions include uncertainty, is developed. Diffusion equations arise in manyfields of science and engineering, and, in many cases, there are uncertaintiesdue to data that cannot be known, or due to errors in measurements andintrinsic variability. In order to model these uncertainties the correspon-ding parameters, diffusion coefficient, source term, boundary and initialconditions, are assumed to be random variables with certainprobabilitydistributions functions. The proposed method includes finite differenceschemes on the space variable and the differential transformation methodfor the time. In addition, the Monte Carlo method is used to deal withthe random variables. The accuracy of the hybrid method is investigatednumerically using the closed form solution of the deterministic associated equation. Based on the numerical results, confidence intervals and ex-pected mean values for the solution are obtained. Furthermore, with theproposed hybrid method numerical-analytical solutions are obtained.

scholarly journals Approximation of continuous random variables for the evaluation of the reliability parameter of complex stress–strength models

2021 ◽  
Author(s):  
Alessandro Barbiero ◽  
Asmerilda Hitaj
Keyword(s):  
Order Approximation ◽  
Linear Optimization ◽  
Closed Form Solution ◽  
Random Variables ◽  
Discrete Distribution ◽  
Computation Time ◽  
Form Solution ◽  
Engineering Problem ◽  
Approximation Technique ◽  
Reliability Parameter

AbstractIn many management science or economic applications, it is common to represent the key uncertain inputs as continuous random variables. However, when analytic techniques fail to provide a closed-form solution to a problem or when one needs to reduce the computational load, it is often necessary to resort to some problem-specific approximation technique or approximate each given continuous probability distribution by a discrete distribution. Many discretization methods have been proposed so far; in this work, we revise the most popular techniques, highlighting their strengths and weaknesses, and empirically investigate their performance through a comparative study applied to a well-known engineering problem, formulated as a stress–strength model, with the aim of weighting up their feasibility and accuracy in recovering the value of the reliability parameter, also with reference to the number of discrete points. The results overall reward a recently introduced method as the best performer, which derives the discrete approximation as the numerical solution of a constrained non-linear optimization, preserving the first two moments of the original distribution. This method provides more accurate results than an ad-hoc first-order approximation technique. However, it is the most computationally demanding as well and the computation time can get even larger than that required by Monte Carlo approximation if the number of discrete points exceeds a certain threshold.

scholarly journals SOLUTION OF THE MULTIDIMENSIONAL DIFFUSION EQUATION USING LIE SYMMETRIES: SIMULATION OF POLLUTANT DISPERSION IN THE ATMOSPHERE

2005 ◽  
Vol 4 (2) ◽  
Author(s):  
J. R. Zabadal ◽  
C. A. Poffal
Keyword(s):  
Differential Operators ◽  
Hybrid Methods ◽  
Formal Solution ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Pollutant Dispersion ◽  
Form Solution ◽  
Lie Symmetries ◽  
Advection Equation ◽  
Mean Values

Several analytical, numerical and hybrid methods are being used to solve diffusion and diffusion advection problems. In this work, a closed form solution of the three-dimensional diffusion advection equation in a Cartesian coordinate system is obtained by applying rules, based on the Lie symmetries, to manipulate the exponential of the differential operators that appear in its formal solution. There are many advantages of applying these rules: the increase in processing velocity so that the solution may be obtained in real time, the reduction in the amount of memory required to perform the necessary tasks in order to obtain the solution, since the analytical expressions can be easily manipulated in post-processing and also the discretization of the domain may not be necessary in some cases, avoiding the use of mean values for some parameters involved. These rules yield good results when applied to obtain solutions for problems in fluid mechanics and in quantum mechanics. In order to show the performance of the method, a one-dimensional scenario of the pollutant dispersion in a stable boundary layer is simulated, considering that the horizontal component of the velocity field is dominant and constant, disregarding the other components. The results are compared with data available in the literature.

scholarly journals SOLUTION OF THE MULTIDIMENSIONAL DIFFUSION EQUATION USING LIE SYMMETRIES: SIMULATION OF POLLUTANT DISPERSION IN THE ATMOSPHERE

2005 ◽  
Vol 4 (2) ◽  
pp. 197
Author(s):  
J. R. Zabadal ◽  
C. A. Poffal
Keyword(s):  
Differential Operators ◽  
Hybrid Methods ◽  
Formal Solution ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Pollutant Dispersion ◽  
Form Solution ◽  
Lie Symmetries ◽  
Advection Equation ◽  
Mean Values

Several analytical, numerical and hybrid methods are being used to solve diffusion and diffusion advection problems. In this work, a closed form solution of the three-dimensional diffusion advection equation in a Cartesian coordinate system is obtained by applying rules, based on the Lie symmetries, to manipulate the exponential of the differential operators that appear in its formal solution. There are many advantages of applying these rules: the increase in processing velocity so that the solution may be obtained in real time, the reduction in the amount of memory required to perform the necessary tasks in order to obtain the solution, since the analytical expressions can be easily manipulated in post-processing and also the discretization of the domain may not be necessary in some cases, avoiding the use of mean values for some parameters involved. These rules yield good results when applied to obtain solutions for problems in fluid mechanics and in quantum mechanics. In order to show the performance of the method, a one-dimensional scenario of the pollutant dispersion in a stable boundary layer is simulated, considering that the horizontal component of the velocity field is dominant and constant, disregarding the other components. The results are compared with data available in the literature.

scholarly journals A Hybrid Method for Evaluating of Lightning Performance of Overhead Lines based on Monte Carlo Procedure

2016 ◽  
Vol 67 (4) ◽  
pp. 246-252
Author(s):  
Reza Shariatinasab ◽  
Pooya Tadayon ◽  
Akihiro Ametani
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Risk Analysis ◽  
Hybrid Method ◽  
Overhead Lines ◽  
Monte Carlo Procedure ◽  
Acceptable Accuracy ◽  
The Monte Carlo Method ◽  
Analytical Relations

Abstract This paper proposes a hybrid method for calculating lightning performance of overhead lines caused by direct strokes by combining Lattice diagram together with the Monte Carlo method. In order to go through this, firstly, the proper analytical relations for overvoltages calculation are established based on Lattice diagram. Then, the Monte Carlo procedure is applied to the obtained analytical relations. The aim of the presented method that will be called ‘ML method’ is simply estimation of the lightning performance of the overhead lines and performing the risk analysis of power apparatus with retaining the acceptable accuracy. To confirm the accuracy, the calculated results of the presented ML method are compared with those calculated by the EMTP/ATP simulation.

scholarly journals Monte Carlo simulation of climate-weather change process at maritime ferry operating area

2017 ◽  
Vol 1 (21) ◽  
pp. 5-17
Author(s):  
Ewa Kuligowska
Keyword(s):  
Monte Carlo ◽  
Change Process ◽  
Fixed Time ◽  
Limit Values ◽  
Mean Values ◽  
Sojourn Times ◽  
Process Characteristics ◽  
The Monte Carlo Method ◽  
The Mean ◽  
Weather Change

The paper presents a computer simulation technique applied to generating the climate-weather change process at Baltic Sea restricted waters and its characteristics evaluation. The Monte Carlo method is used under the assumption of semi-Markov model of this process. A procedure and an algorithm of climate-weather change process’ realizations generating and its characteristics evaluation are proposed to be applied in C# program preparation. Using this program, the climate-weather change process’ characteristics are predicted for the maritime ferry operating area. Namely, the mean values and standard deviations of the unconditional sojourn times, the limit values of transient probabilities and the mean values of total sojourn times for the fixed time at the climate-weather states are determined.

Putting Statistics Into the Statistical Energy Analysis of Automotive Vehicles

1993 ◽  
Author(s):  
Clark J. Radcliffe ◽  
Xian Li Huang
Keyword(s):  
Monte Carlo ◽  
Energy Analysis ◽  
Computational Effort ◽  
Design Tool ◽  
Physical Parameters ◽  
Statistical Energy Analysis ◽  
Mean Values ◽  
Interior Spaces ◽  
The Monte Carlo Method ◽  
Design Values

Abstract Sound and vibration transmission modeling methods are important to the design process for high quality automotive vehicles. Statistical Energy Analysis (SEA) is an emerging design tool for the automotive industry that was initially developed in the 1960’s to estimate root-mean-square sound and vibration levels in structures and interior spaces. Although developed to estimate statistical mean values, automotive design application of SEA needs the additional ability to predict statistical variances of the predicted mean values of sound and vibration. This analytical ability would allow analysis of vehicle sound and vibration response sensitivity to changes in vehicle design specifications and their statistical distributions. This paper will present an algorithm to extend the design application of the SEA method through prediction of the variances of RMS responses of vibro-acoustic automobile structures and interior spaces from variances in SEA automotive model physical parameters. The variance analysis is applied to both a simple, complete illustrative example and a more complex automotive vehicle example. Example variance results are verified through comparison with a Monte Carlo test of 2,000 SEA responses whose physical parameters were given Gaussian distributions with means at design values. Analytical predictions of the response statistics agree with the statistics generated by the Monte Carlo method but only require about 1/300 of the computational effort.

Putting Statistics into the Statistical Energy Analysis of Automotive Vehicles

1997 ◽  
Vol 119 (4) ◽  
pp. 629-634 ◽  
Author(s):  
C. J. Radcliffe ◽  
X. L. Huang
Keyword(s):  
Monte Carlo ◽  
Energy Analysis ◽  
Computational Effort ◽  
Design Tool ◽  
Physical Parameters ◽  
Statistical Energy Analysis ◽  
Mean Values ◽  
Interior Spaces ◽  
The Monte Carlo Method ◽  
Design Values

Sound and vibration transmission modeling methods are important to the design process for high quality automotive vehicles. Statistical Energy Analysis (SEA) is an emerging design tool for the automotive industry that was initially developed in the 1960’s to estimate root-mean-square sound and vibration levels in structures and interior spaces. Although developed to estimate statistical mean values, automotive design application of SEA needs the additional ability to predict statistical variances of the predicted mean values of sound and vibration. This analytical ability would allow analysis of vehicle sound and vibration response sensitivity to changes in vehicle design specifications and their statistical distributions. This paper will present an algorithm to extend the design application of the SEA method through prediction of the variances of RMS. responses of vibro-acoustic automobile structures and interior spaces from variances in SEA automotive model physical parameters. The variance analysis is applied to both a simple, complete illustrative example and a more complex automotive vehicle example. Example variance results are verified through comparison with a Monte Carlo test of 2,000 SEA responses whose physical parameters were given Gaussian distributions with means at design values. Analytical predictions of the response statistics agree with the statistics generated by the Monte Carlo method but only require about 1/300 of the computational effort.

Generation of k-wise independent random variables with small randomness

2019 ◽  
Vol 25 (3) ◽  
pp. 259-270
Author(s):  
Taku Achiha ◽  
Hiroshi Sugita ◽  
Kenta Tonohiro ◽  
Yuto Yamamoto
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Random Variables ◽  
Independent Random Variables ◽  
The Monte Carlo Method

Abstract A quick generation method of k-wise independent uniformly distributed m-bit random variables with small randomness is proposed with applications to the Monte Carlo method.

scholarly journals On distribution of a maximum of random variables

2020 ◽  
pp. 124-136
Author(s):  
Степан Алексеевич Рогонов ◽  
Илья Сергеевич Солдатенко
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Random Variables ◽  
Fuzzy Random Variables ◽  
Maximum Function ◽  
The Monte Carlo Method ◽  
Weighted Random Variables ◽  
Fuzzy Random

В работе исследуется способ идентификации методом Монте-Карло распределений нечетких случайных величин, в параметрическом задании которых присутствует функция максимума от взвешенных случайных величин. In this paper, we study a method for identifying by the Monte Carlo method distributions of fuzzy random variables, in the parametric setting of which there is a maximum function of weighted random variables.

Sign in / Sign up

Export Citation Format

Share Document