Monte Carlo and quasi-Monte Carlo methods

Caflisch, Russel E.

doi:10.1017/s0962492900002804

Cited by 1,231 publications

(912 citation statements)

References 43 publications

(42 reference statements)

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“…, x i,max ]. The method converges slowly with the rate K −1/2 but independently of m and the regularity of ωp 0 [2,12].…”

Section: Approximation Of the Summationmentioning

confidence: 99%

“…This PDF satisfies a Fokker-Planck equation and is solved by a finite volume scheme in [22] but the dimension of the stochastic problem is then restricted to, say, five. Here, we apply SSA to the stochastic part and compute the coupling to the deterministic part using Monte Carlo and Quasi Monte Carlo summation [2,12]. The equations for the expected values are integrated in time by an unconditionally stable, implict method.…”

Section: Introductionmentioning

confidence: 99%

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Hybrid method for the chemical master equation

Hellander

Lötstedt

2007

Journal of Computational Physics

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The chemical master equation is solved by a hybrid method coupling a macroscopic, deterministic description with a mesoscopic, stochastic model. The molecular species are divided into one subset where the expected values of the number of molecules are computed and one subset with species with a stochastic variation in the number of molecules. The macroscopic equations resemble the reaction rate equations and the probability distribution for the stochastic variables satisfy a master equation. The probability distribution is obtained by the Stochastic Simulation Algorithm due to Gillespie. The equations are coupled via a summation over the mesoscale variables. This summation is approximated by Monte Carlo and Quasi Monte Carlo methods. The error in the approximations is analyzed. The hybrid method is applied to three chemical systems from molecular cell biology.

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“…, x i,max ]. The method converges slowly with the rate K −1/2 but independently of m and the regularity of ωp 0 [2,12].…”

Section: Approximation Of the Summationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Hybrid method for the chemical master equation

Hellander

Lötstedt

2007

Journal of Computational Physics

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“…Dealing with the computations of statistics for some quantities of interest, the Monte-Carlo fundamental principle is to estimate the integral -i.e the expectation -of this functional on some measure space through its evaluation for a finite number of realizations which are randomly chosen (Metropolis and Ulam, 1949;Caflisch, 1998). This idea appears in (2), where Ψ denotes the functional that needs to be evaluated and which depends on the solution u(ω).…”

Section: Monte-carlo Methodsmentioning

confidence: 99%

“…Those approaches are usually divided into two classes (Spanos and Ghanem, 2002). On the one hand, the direct integration methods, which are mostly gathered into the so-called "Monte-Carlo" simulations (Metropolis and Ulam, 1949;Caflisch, 1998), are best viewed as numerical integration techniques. They require to solve many realizations of the deterministic problem and usually require a high computational effort.…”

Section: Due To the Always Remaining Uncertainties It Is Usually Notmentioning

confidence: 99%

Effect of the Young modulus variability on the mechanical behaviour of a nuclear containment vessel

Larrard¹,

Colliat²,

Benboudjema³

et al. 2010

Nuclear Engineering and Design

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This study aims at investigating the influence of the Young modulus variability on the mechanical behaviour of a nuclear containment vessel in case of a loss of cooling agent accident and under the assumption of an elastic behaviour. To achieve this investigation, the Monte-Carlo Method is carried out thanks to a middleware which encapsulates the different components (random field generation, FE simulations) and enables calculations parallelization. The main goal is to quantify the uncertainty propagation by comparing the maximal values of outputs of interest (orthoradial stress and Mazars equivalent strain) with the ones obtained from a deterministic calculation. The Young modulus is supposed to be accurately represented by a weakly homogeneous random field and realizations are provided through its -troncated -Karhunen-Loève expansion. This study reveals a significant increase of the maximal equivalent strain if the Young modulus variability is considered (compared to a deterministic approach). The influence of the correlation length is investigated too. Finally it is shown that there is no correlation between the maximal values location of equivalent strain and the ones where the Young modulus extreme values are observed.

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“…The solution of the mathematical problem can be determined by the stochastic numerical material model, where the system response of the model renders a distribution with statistical mean and variance. A ubiquitous strategy for its solution is the Monte Carlo method [2,3]. An alternative to reduce the computational effort, is the spectral method by Ghanem and Spanos [4], which is considered in this paper.…”

Section: Introductionmentioning

confidence: 99%

Multidimensional Stochastic Material Modeling at Large Deformations Considering Parameter Correlations

Penner¹,

Caylak²,

Mahnken³

2017

Proceedings of the 2nd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECO

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Abstract. In many engineering applications the material behavior, e.g. hyper-elasticity and plasticity, is described by appropriate mathematical models. However, these become uncertain due to different types of uncertainty such as variation in the manufacturing process, measurement errors and missing or incomplete information on material properties. This contribution presents a framework for nonlinear elastic stochastic material model at large deformations. As a key idea uncertainty of the material is described by parameters, which are modeled as stochastic variables.To this end, 150 specimens for three different rubber materials are experimentally investigated in tensile tests. Based on experimental results 1000 artificial data are generated by aid of an ARMA process [1]. The artificial data are used for parameter identification of an Ogden material model for rubber materials . Furthermore, statistical analysis of material parameters including their correlations is studied. The number of material parameters define the dimension of the stochastic space. Usually, the stochastic material parameters are considered as stochastically independent. However, in our work we consider the dependency including the correlation obtained from experimental data.The hyperelastic stochastic material parameters are expanded with the multivariate PCE. In this context, the stresses depend on stochastic variables. To determine the corresponding PC coefficients for non-independent stochastic material parameters we use a Cholesky decomposition. As a numerical example we consider the static problem for uniaxial tension of the rectangular plate. This structure is investigated in order to represent the experimental setup conditions.

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Monte Carlo and quasi-Monte Carlo methods

Cited by 1,231 publications

References 43 publications

Hybrid method for the chemical master equation

Hybrid method for the chemical master equation

Effect of the Young modulus variability on the mechanical behaviour of a nuclear containment vessel

Multidimensional Stochastic Material Modeling at Large Deformations Considering Parameter Correlations

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