scholarly journals Dealing with Dependent Uncertainty in Modelling: A Comparative Study Case through the Airy Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
J.-C. Cortés ◽  
J.-V. Romero ◽  
M.-D. Roselló ◽  
R.-J. Villanueva
Keyword(s):  
Differential Equations ◽  
Random Variables ◽  
Monte Carlo Sampling ◽  
Statistical Dependence ◽  
Mathematical Treatment ◽  
Multivariate Gaussian Distribution ◽  
Random Differential Equations ◽  
Highly Oscillatory ◽  
Airy Equation ◽  
Study Case

The consideration of uncertainty in differential equations leads to the emergent area of random differential equations. Under this approach, inputs become random variables and/or stochastic processes. Often one assumes that inputs are independent, a hypothesis that simplifies the mathematical treatment although it could not be met in applications. In this paper, we analyse, through the Airy equation, the influence of statistical dependence of inputs on the output, computing its expectation and standard deviation by Fröbenius and Polynomial Chaos methods. The results are compared with Monte Carlo sampling. The analysis is conducted by the Airy equation since, as in the deterministic scenario its solutions are highly oscillatory, it is expected that differences will be better highlighted. To illustrate our study, and motivated by the ubiquity of Gaussian random variables in numerous practical problems, we assume that inputs follow a multivariate Gaussian distribution throughout the paper. The application of Fröbenius method to solve Airy equation is based on an extension of the method to the case where inputs are dependent. The numerical results show that the existence of statistical dependence among the inputs and its magnitude entails changes on the variability of the output.

scholarly journals Mean Square Convergent Non-Standard Numerical Schemes for Linear Random Differential Equations with Delay

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1417 ◽  
Author(s):  
Julia Calatayud ◽  
Juan Carlos Cortés ◽  
Marc Jornet ◽  
Francisco Rodríguez
Keyword(s):  
Stochastic Process ◽  
Differential Equations ◽  
Random Variables ◽  
Mean Square ◽  
Discrete Delay ◽  
Numerical Schemes ◽  
Recent Contribution ◽  
Order Of Accuracy ◽  
Euler’S Method ◽  
Random Differential Equations

In this paper, we are concerned with the construction of numerical schemes for linear random differential equations with discrete delay. For the linear deterministic differential equation with discrete delay, a recent contribution proposed a family of non-standard finite difference (NSFD) methods from an exact numerical scheme on the whole domain. The family of NSFD schemes had increasing order of accuracy, was dynamically consistent, and possessed simple computational properties compared to the exact scheme. In the random setting, when the two equation coefficients are bounded random variables and the initial condition is a regular stochastic process, we prove that the randomized NSFD schemes converge in the mean square (m.s.) sense. M.s. convergence allows for approximating the expectation and the variance of the solution stochastic process. In practice, the NSFD scheme is applied with symbolic inputs, and afterward the statistics are explicitly computed by using the linearity of the expectation. This procedure permits retaining the increasing order of accuracy of the deterministic counterpart. Some numerical examples illustrate the approach. The theoretical m.s. convergence rate is supported numerically, even when the two equation coefficients are unbounded random variables. M.s. dynamic consistency is assessed numerically. A comparison with Euler’s method is performed. Finally, an example dealing with the time evolution of a photosynthetic bacterial population is presented.

scholarly journals Shapley effects for sensitivity analysis with dependent inputs: bootstrap and kriging-based algorithms

2019 ◽  
Vol 65 ◽  
pp. 266-293 ◽  
Author(s):  
Nazih Benoumechiara ◽  
Kevin Elie-Dit-Cosaque
Keyword(s):  
Sensitivity Analysis ◽  
Confidence Intervals ◽  
Global Sensitivity Analysis ◽  
Estimation Error ◽  
Random Variables ◽  
Monte Carlo Sampling ◽  
Estimation Methods ◽  
Statistical Dependence ◽  
Global Sensitivity ◽  
Sensitivity Indices

In global sensitivity analysis, the well-known Sobol’ sensitivity indices aim to quantify how the variance in the output of a mathematical model can be apportioned to the different variances of its input random variables. These indices are based on the functional variance decomposition and their interpretation becomes difficult in the presence of statistical dependence between the inputs. However, as there are dependencies in many application studies, this drawback enhances the development of interpretable sensitivity indices. Recently, the Shapley values that were developed in the field of cooperative games theory have been connected to global sensitivity analysis and present good properties in the presence of dependencies. Nevertheless, the available estimation methods do not always provide confidence intervals and require a large number of model evaluations. In this paper, a bootstrap resampling is implemented in existing algorithms to assess confidence intervals. We also propose to consider a metamodel in substitution of a costly numerical model. The estimation error from the Monte-Carlo sampling is combined with the metamodel error in order to have confidence intervals on the Shapley effects. Furthermore, we compare the Shapley effects with existing extensions of the Sobol’ indices in different examples of dependent random variables.

Statistical Dependence of Input Variables in Doproc Method / Statistická Závislost Vstupních Veličin V Metodě Popv

2012 ◽  
Vol 12 (2) ◽  
pp. 48-58 ◽  
Author(s):  
Petr Janas ◽  
Krejsa Martin
Keyword(s):  
Reliability Assessment ◽  
Random Variables ◽  
Statistical Dependence ◽  
Software System ◽  
Dependent Variables ◽  
Computational Procedures ◽  
Input Variables

Abstract In probabilistic tasks, input random variables are often statistically dependent. This fact should be considered in correct computational procedures. In case of the newly developed Direct Optimized Probabilistic Calculation (DOProC), the statistically dependent variables can be expressed by the socalled multidimensional histograms, which can be used e.g. for probabilistic calculations and reliability assessment in the software system ProbCalc.

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