Monte Carlo simulations in zeolites

In this short review, the applications of Monte Carlo simulations to the study of the adsorption and diffusion of hydrocarbons in zeolites are discussed. We focus on those systems for which the conventional molecular simulation techniques, molecular dynamics and Monte Carlo, are not sufficiently efficient. In particular, to simulate the adsorption and diffusion of long-chain hydrocarbons, novel Monte Carlo techniques have been developed. Here we discuss configurational-bias Monte Carlo (CBMC) and kinetic Monte Carlo (KMC). CBMC was developed to compute the thermodynamic properties. KMC is applied to compute transport properties. The use of these methods is illustrated with examples of technological importance.     

A nonintrusive stochastic multiscale solver

In this paper, we describe a practical nonintrusive multiscale solver that permits consideration of uncertainties in heterogeneous materials without exhausting the available computational resources. The computational complexity of analyzing heterogeneous material systems is governed by the physical and probability spaces at multiple scales. To deal with these large spaces, we employ reduced order homogenization approach in combination with the Karhunen–Loeve expansion and stochastic collocation method based on sparse grid. The resulting nonintrusive multiscale solver, which is aimed at providing practical solutions for complex multiscale stochastic problems, has been verified against the Latin Hypercube Monte–Carlo method. Copyright © 2011 John Wiley & Sons, Ltd.     

Stochastic analysis on flexural behavior of reinforced concrete beams based on piecewise response surface scheme

To save the computational efforts of Monte Carlo simulation together with nonlinear finite element method, the analysis framework combining the response surface method and Monte Carlo simulation is usually adopted to investigate the stochastic nonlinear behavior of structures. It is found that the traditional response surface method could not describe stochastic behavior of cracked concrete beams. In order to overcome the discontinuity caused by cracking nonlinearity, a scheme by introducing a piecewise response surface is proposed in this paper. This scheme is evaluated by the stochastic analysis of several concrete beams. The comparison between the proposed method and some traditional methods, including Monte Carlo simulation and Monte Carlo simulation with traditional response surface, shows that the proposed method could well depict the stochastic behavior of concrete beams. Finally, the probability density evolution of concrete beams from short-term deflection to a long-term deflection is analyzed.     

Multiscale stochastic finite element modeling of random elastic heterogeneous materials

In Xu et al. (Comput Struct 87:1416–1426, 2009) a novel Green-function-based multiscale stochastic finite element method (MSFEM) was proposed to model boundary value problems involving random heterogeneous materials. In this paper, we describe in detail computational aspects of the MSFEM explicitly across macro–meso–micro scales. Different numerical algorithms are introduced and compared in terms of numerical accuracy and convergence.     

Stochastic multiscale fracture analysis of three-dimensional functionally graded composites

A new moment-modified polynomial dimensional decomposition (PDD) method is presented for stochastic multiscale fracture analysis of three-dimensional, particle-matrix, functionally graded materials (FGMs) subject to arbitrary boundary conditions. The method involves Fourier-polynomial expansions of component functions by orthonormal polynomial bases, an additive control variate in conjunction with Monte Carlo simulation for calculating the expansion coefficients, and a moment-modified random output to account for the effects of particle locations and geometry. A numerical verification conducted on a two-dimensional FGM reveals that the new method, notably the univariate PDD method, produces the same crude Monte Carlo results with a five-fold reduction in the computational effort. The numerical results from a three-dimensional, edge-cracked, FGM specimen under a mixed-mode deformation demonstrate that the statistical moments or probability distributions of crack-driving forces and the conditional probability of fracture initiation can be efficiently generated by the univariate PDD method. There exist significant variations in the probabilistic characteristics of the stress-intensity factors and fracture-initiation probability along the crack front. Furthermore, the results are insensitive to the subdomain size from concurrent multiscale analysis, which, if selected judiciously, leads to computationally efficient estimates of the probabilistic solutions.     

Parallel processing in computational stochastic dynamics

Studying large complex problems that often arise in computational stochastic dynamics (CSD) demands significant computer power and data storage. Parallel processing can help meet these requirements by exploiting the computational and storage capabilities of multiprocessing computational environments. The challenge is to develop parallel algorithms and computational strategies that can take full advantage of parallel machines. This paper reviews some of the characteristics of parallel computing and the techniques used to parallelize computational algorithms in CSD. The characteristics of parallel processor environments are discussed, including parallelization through the use of message passing and parallelizing compilers. Several applications of parallel processing in CSD are then developed: solutions of the Fokker–Planck equation, Monte Carlo simulation of dynamical systems, and random eigenvector problems. In these examples, parallel processing is seen to be a promising approach through which to resolve some of the computational issues pertinent to CSD.     

Computational strategy for the crash design analysis using an uncertain computational mechanical model

The framework of this paper is the robust crash analysis of a motor vehicle. The crash analysis is carried out with an uncertain computational model for which uncertainties are taken into account with the parametric probabilistic approach and for which the stochastic solver is the Monte Carlo method. During the design process, different configurations of the motor vehicle are analyzed. Usual interpolation methods cannot be used to predict if the current configuration is similar or not to one of the previous configurations already analyzed and for which a complete stochastic computation has been carried out. In this paper, we propose a new indicator that allows to decide if the current configuration is similar to one of the previous analyzed configurations while the Monte Carlo simulation is not finished and therefore, to stop the Monte Carlo simulation before the end of computation.     

Assessment of atomistic coarse-graining methods

This paper reviews classical theories of coarse-graining and gives a short introduction to representative coarse-grained atomistic models that were developed based on structure reduction, an assumption of homogenous deformation, and field representation. The applicability and limitations of these coarse-grained models are analyzed on the basis of their theoretical frameworks as well as the coarse-graining methods they employ.     

Error estimation and model adaptation for a stochastic‐deterministic coupling method based on the Arlequin framework

The paper deals with the issue of accuracy for multiscale methods applied to solve stochastic problems. It more precisely focuses on the control of a coupling, performed using the Arlequin framework, between a deterministic continuum model and a stochastic continuum one. By using residual‐type estimates and adjoint‐based techniques, a strategy for goal‐oriented error estimation is presented for this coupling and contributions of various error sources (modeling, space discretization, and Monte Carlo approximation) are assessed. Furthermore, an adaptive strategy is proposed to enhance the quality of outputs of interest obtained by the coupled stochastic‐deterministic model. Performance of the proposed approach is illustrated on 1D and 2D numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

Multiscale modeling and uncertainty quantification in nanoparticle-mediated drug/gene delivery

Nanoparticle (NP)-mediated drug/gene delivery involves phenomena at broad range spatial and temporal scales. The interplay between these phenomena makes the NP-mediated drug/gene delivery process very complex. In this paper, we have identified four key steps in the NP-mediated drug/gene delivery: (i) design and synthesis of delivery vehicle/platform; (ii) microcirculation of drug carriers (NPs) in the blood flow; (iii) adhesion of NPs to vessel wall during the microcirculation and (iv) endocytosis and exocytosis of NPs. To elucidate the underlying physical mechanisms behind these four key steps, we have developed a multiscale computational framework, by combining all-atomistic simulation, coarse-grained molecular dynamics and the immersed molecular electrokinetic finite element method (IMEFEM). The multiscale computational framework has been demonstrated to successfully capture the binding between nanodiamond, polyethylenimine and small inference RNA, margination of NP in the microcirculation, adhesion of NP to vessel wall under shear flow, as well as the receptor-mediated endocytosis of NPs. Moreover, the uncertainties in the microcirculation of NPs has also been quantified through IMEFEM with a Bayesian updating algorithm. The paper ends with a critical discussion of future opportunities and key challenges in the multiscale modeling of NP-mediated drug/gene delivery. The present multiscale modeling framework can help us to optimize and design more efficient drug carriers in the future.     

Monte Carlo methods for pricing financial options

Pricing financial options is amongst the most important and challenging problems in the modern financial industry. Except in the simplest cases, the prices of options do not have a simple closed form solution and efficient computational methods are needed to determine them. Monte Carlo methods have increasingly become a popular computational tool to price complex financial options, especially when the underlying space of assets has a large dimensionality, as the performance of other numerical methods typically suffer from the ‘curse of dimensionality’. However, even Monte-Carlo techniques can be quite slow as the problem-size increases, motivating research in variance reduction techniques to increase the efficiency of the simulations. In this paper, we review some of the popular variance reduction techniques and their application to pricing options. We particularly focus on the recent Monte-Carlo techniques proposed to tackle the difficult problem of pricing American options. These include: regression-based methods, random tree methods and stochastic mesh methods. Further, we show how importance sampling, a popular variance reduction technique, may be combined with these methods to enhance their effectiveness. We also briefly review the evolving options market in India.     

Stochastic projection schemes for deterministic linear elliptic partial differential equations on random domains

We present stochastic projection schemes for approximating the solution of a class of deterministic linear elliptic partial differential equations defined on random domains. The key idea is to carry out spatial discretization using a combination of finite element methods and stochastic mesh representations. We prove a result to establish the conditions that the input uncertainty model must satisfy to ensure the validity of the stochastic mesh representation and hence the well posedness of the problem. Finite element spatial discretization of the governing equations using a stochastic mesh representation results in a linear random algebraic system of equations in a polynomial chaos basis whose coefficients of expansion can be non‐intrusively computed either at the element or the global level. The resulting randomly parametrized algebraic equations are solved using stochastic projection schemes to approximate the response statistics. The proposed approach is demonstrated for modeling diffusion in a square domain with a rough wall and heat transfer analysis of a three‐dimensional gas turbine blade model with uncertainty in the cooling core geometry. The numerical results are compared against Monte–Carlo simulations, and it is shown that the proposed approach provides high‐quality approximations for the first two statistical moments at modest computational effort. Copyright © 2010 John Wiley & Sons, Ltd.     

Kinetic Monte Carlo method for dislocation glide in silicon

A kinetic Monte Carlo (KMC) approach to the mesoscale simulation of dislocation glide via the kink mechanism is developed. In this paper we present the details of the KMC methodology, highlighting three features: (1) inclusion of dislocation dissociation; (2) efficient method of sampling the double-kink nucleation process; and (3) exact calculation of dislocation segment interactions. This revised version was published online in July 2006 with corrections to the Cover Date.     

A posteriori stochastic correction of reduced models in delayed-acceptance MCMC,with application to multiphase subsurface inverse problems

Sample-based Bayesian inference provides a route to uncertainty quantification in the geosciences and inverse problems in general but is very computationally demanding in the naïve form, which requires simulating an accurate computer model at each iteration. We present a new approach that constructs a stochastic correction to the error induced by a reduced model, with the correction improving as the algorithm proceeds. This enables sampling from the correct target distribution at reduced computational cost per iteration, as in existing delayed-acceptance schemes, while avoiding appreciable loss of statistical efficiency that necessarily occurs when using a reduced model. Use of the stochastic correction significantly reduces the computational cost of estimating quantities of interest within desired uncertainty bounds. In contrast, existing schemes that use a reduced model directly as a surrogate do not actually improve computational efficiency in our target applications. We build on recent simplified conditions for adaptive Markov chain Monte Carlo algorithms to give practical approximation schemes and algorithms with guaranteed convergence. The efficacy of this new approach is demonstrated in two computational examples, including calibration of a large-scale numerical model of a real geothermal reservoir, that show good computational and statistical efficiencies on both synthetic and measured data sets.     

TiN薄膜生长的无格点Kinetic Monte Carlo模拟

本文介绍了采用无格点Kinetic Monte Carlo（KMC）方法，模拟TiN薄膜在TiN（001）基底表面上外延生长的仿真结果。在此无格点KMC方法中，使用了Dimer算法在势能面中搜索鞍点和低能盆底。仿真的结果证实，此无格点KMC方法对于仿真二元薄膜外延生长是有效的。本文中还讨论了无格点KMC的计算量问题、运算过程中Dimer的初始位置问题和势能面中浅盆底问题。     

A polynomial chaos efficient global optimization approach for Bayesian optimal experimental design

This paper proposes a global optimization framework to address the high computational cost and non convexity of Optimal Experimental Design (OED) problems. To reduce the computational burden and the presence of noise in the evaluation of the Shannon expected information gain (SEIG), this framework proposes the coupling of Laplace approximation and polynomial chaos expansions (PCE). The advantage of this procedure is that PCE allows large samples to be employed for the SEIG estimation, practically vanishing the noisy introduced by the sampling procedure. Consequently, the resulting optimization problem may be treated as deterministic. Then, an optimization approach based on Kriging surrogates is employed as the optimization engine to search for the global solution with limited computational budget. Four numerical examples are investigated and their results are compared to state-of-the-art stochastic gradient descent algorithms. The proposed approach obtained better results than the stochastic gradient algorithms in all situations, indicating its efficiency and robustness in the solution of OED problems.     

Coarse-Grained Finite-Temperature Theory for the Bose Condensate in Optical Lattices

In this paper, we derive a coarse-grained finite-temperature theory for a Bose condensate in a one-dimensional optical lattice, in addition to a confining harmonic trap potential. We start with a two-particle irreducible (2PI) effective action on the Schwinger-Keldysh closed-time contour path. In principle, this action involves all information of equilibrium and non-equilibrium properties of the condensate and noncondensate atoms. In constructing a theory for the condensate and noncondensate in a periodic lattice potential, a difficulty arises from the rapid spatial variation due to a lattice potential, compared to the length scale of the harmonic potential. We employ a coarse-graining procedure to eliminate this rapid variation. By introducing a variational ansatz for the condensate order parameter in an effective action, we derive a coarse-grained effective action, which describes the dynamics on the length scale much longer than a lattice constant. Using the variational principle, coarse-grained equations of motion for condensate variables are obtained. These equations include a dissipative term due to collisions between condensate and noncondensate atoms, as well as noncondensate mean-field. As a result of a coarse-graining procedure, the effects of a lattice potential are incorporated into equations of motion for the condensate by an effective mass, a renormalized coupling constant, and an umklapp scattering process. To illustrate the usefulness of our formalism, we discuss a Landau instability of the condensate moving in optical lattices by using the coarse-grained generalized Gross-Pitaevskii hydrodynamics. We find that the collisional damping rate due to collisions between the condensate and noncondensate atoms changes its sign when the condensate velocity exceeds a renormalized sound velocity, leading to a Landau instability consistent with the Landau criterion. Our results in this work give an insight into the microscopic origin of the Landau instability.      

Efficient robust design via Monte Carlo sample reweighting

A novel probabilistic method for the optimization of robust design problems is presented. The approach is based on an efficient variation of the Monte Carlo simulation method. By shifting most of the computational burden to outside of the optimization loop, optimum designs can be achieved efficiently and accurately. Furthermore by reweighting an initial set of samples the objective function and constraints become smooth functions of changes in the probability distribution of the parameters, rather than the stochastic functions obtained using a standard Monte Carlo method. The approach is demonstrated on a beam truss example, and the optimum designs are verified with regular Monte Carlo simulation. Copyright © 2006 John Wiley & Sons, Ltd.     

The reduction of computational time of the Monte Carlo method,applied to radiative heat transfer with variable properties,by using a fixed properties loop

The Monte Carlo method has the excellent feature of easy adaptation to problemens such as radiative heat transfer with variable properties, and problems of system of heat transfer including radiation, with complex geometries. However, this method has a deficiency that it requires a large computational time when the energy equation is non-linear. In this paper, a modified Monte Carlo method, named the DPEF method, is suggested to overcome this deficiency, by adding an iterative loop of fixed properties to the conventional method, without sacrificing the advantageous of the Monte Carlo method. An analytical example of this new method, as applied to a model of radiative heat transfer in a furnace with variable properties, is given. It is found that the computational time is reduced by half of that of the conventional Monte Carlo method, and moreover, the stability in iteration process is found to be improved.