Analysis of Nested Multilevel Monte Carlo Using Approximate Normal Random Variables

This documents describes several studies undertaken to assess the applicability of MultiLevel Monte Carlo (MLMC) methods to problems of interest; namely in turbulent fluid flow over civil engineering structures. Several numerical experiments are presented wherein the convergence of quantities of interest with mesh parameters are studied at different Reynolds’ numbers and geometries. It was found that MLMC methods could be used successfully for low Reynolds’ number flows when combined with appropriate Adaptive Mesh Refinement (AMR) strategies. However, the hypotheses for optimal MLMC performance were found to not be satisfied at higher turbulent Reynolds’ numbers despite the use of AMR strategies. Recommendations are made for future research directions based on these studies. A tentative outline for an MLMC algorithm with adapted meshes is made, as well as recommendations for alternatives to MLMC methods for cases where the underlying assumptions for optimal MLMC performance are not satisfied.

Multilevel sequential Monte Carlo samplers

In this report, we study the use of Multi-Level Monte Carlo (MLMC) methods for time dependent problems. It was found that the usability of MLMC methods depends strongly on whether or not the underlying time dependent problem is chaotic in nature. Numerical experiments are conducted on both simple problems, as well as fluid flow problems of practical interest to the ExaQUte project, to demonstrate this. For the non-chaotic cases, the hypotheses that enable the use of MLMC methods were found to be satisfied. For the chaotic cases, especially the case of high Reynolds’ number fluid flow, the hypotheses were not satisfied. However, it was found that correlations between the different levels were high enough to merit the use of multi-fidelity or control-variate approaches. It was also noted that MLMC methods could work for chaotic problems if the time window of analysis were chosen to be small enough. Future studies are proposed to examine this possibility.

Abstract: Uncertainty quantification for multi‐physics problems like fluid‐structure interaction (FSI) is challenging in terms of the required computational effort and complexity of physics involved. A Multilevel Monte Carlo (MLMC) approach for stochastic analysis of FSI problems is presented in the current study. An overall reduction in computational cost is achieved by variance reduction in MLMC method. The efficacy of MLMC algorithm for expensive problems like FSI is compared with classical Monte Carlo(MC) method and MLMC is found to achieve the required accuracy at a significantly reduced computational cost. An FSI benchmark test case with uncertain inputs is presented here. The possibility of uncertainty quantification for FSI systems with a plausible computational cost using MLMC is demonstrated.

In this paper, the application of the multilevel Monte Carlo (MLMC) method to numerical simulations of turbulent flows with uncertain parameters is investigated. Several strategies for setting up the MLMC method are presented, and the advantages and disadvantages of each strategy are also discussed. A numerical experiment is carried out using an idealized model for the Antarctic Circumpolar Current (ACC) with uncertain, small-scale bottom topographic features. It is demonstrated that unlike the pointwise solutions, the averaged volume transports are correlated across grid resolutions, and the MLMC method can increase simulation efficiency without losing accuracy in uncertainty assessments.

Reservoir heterogeneity can be detrimental to the success of chemical enhanced oil recovery (EOR) processes. Therefore, it is important to evaluate the effect of uncertainty in reservoir heterogeneity on the performance of chemical EOR. Usually, a Monte Carlo sampling approach is followed were a number of stochastic reservoir model realizations are generated and then numerical simulation is performed to obtain a certain objective function, such as the recovery factor; however, Monte Carlo simulation (MCS) has a slow convergence and requires a large number of samples to produce accurate results. This is computationally expensive when using reservoir simulators. This study applies an extension to MCS using a multi-scale approach. The applied method is known as the multilevel Monte Carlo (MLMC) method and has been only recently applied to problems of flow in porous media. This method is based on running a small number of expensive simulations on the finer scale model and a large number of less expensive simulations on coarser scale models — these are upscaled models of the fine scale model — and then combining the results to produce the quantities of interest. The purpose of this method is to reduce computational cost while maintaining the accuracy of the finer scale model. The results of this approach are compared with reference MCS, assuming a large number of simulations on the fine scale model. This study used MLMC to efficiently quantify the effect of uncertainty in heterogeneity on the recovery factor of different chemical EOR processes. The permeability field was assumed to be the random input. This approach was implemented by writing a MATLAB code to generate the stochastic realizations for the permeability field and also performing the coarsening processes. The code is then coupled with ECLIPSE, which was used as the numerical simulator for the chemical EOR processes to obtain the recovery factor. The code then combines the results obtained from the different scale models to produce the statistical moments for the recovery factor, such as the mean and variance. This method was applied for two-dimensional (2D) and three-dimensional (3D) stylized reservoir models using Gaussian randomly generated permeability fields. Different coarsening algorithms were used and compared, such as the renormalization and pressure solver methods, and polymer and surfactant-polymer (SP) flooding processes where the chemical EOR processes were considered. The results were compared with running Monte Carlo for the fine scale model while equating the computational cost for the multilevel Monte Carlo method. Both of these results were then compared with the reference case, which uses a large number of runs of the fine scale model. The results show that it is possible to robustly quantify spatial uncertainty for chemical EOR processes while greatly reducing the computational requirement, up to two orders of magnitude compared to traditional Monte Carlo. The method can be easily extendable to other EOR processes to quantify spatial uncertainty such as carbon dioxide (CO2) EOR.

Heterogeneity and a paucity of measurements of key material properties undermine the veracity of quantitative predictions of subsurface flow and transport. For such model forecasts to be useful as a management tool, they must be accompanied by computationally expensive uncertainty quantification, which yields confidence intervals, probability of exceedance, and so forth. We design and implement novel multilevel Monte Carlo (MLMC) algorithms that accelerate estimation of the cumulative distribution functions (CDFs) of quantities of interest, for example, water breakthrough time or oil production rate. Compared to standard non‐smoothed MLMC, the new estimators achieve a significant variance reduction at each discretization level by smoothing the indicator function with a Gaussian kernel or replacing standard Monte Carlo (MC) with the recently developed hierarchical Latinized stratified sampling (HLSS). After validating the kernel‐smoothed MLMC and HLSS‐enhanced MLMC methods on a single‐phase flow test bed, we demonstrate that they are orders of magnitude faster than standard MC for estimating the CDF of breakthrough times in multiphase flow problems.

This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis as an alternative to strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy.

Determining optimal multilevel Monte Carlo parameters with application to fault tolerance

Multilevel Monte Carlo method for ergodic SDEs without contractivity

Multilevel Monte Carlo Using Approximate Distribution of the CIR Process

Determining distributions of the functions of random variables is a very important problem with a wide range of applications in Risk Management, Finance, Economics, Science, and many other areas. This paper develops the theory on both density and distribution functions for the quotient Y = X 1 X 2 and the ratio of one variable over the sum of two variables Z = X 1 X 1 + X 2 of two dependent or independent random variables X 1 and X 2 by using copulas to capture the structures between X 1 and X 2 . Thereafter, we extend the theory by establishing the density and distribution functions for the quotients Y = X 1 X 2 and Z = X 1 X 1 + X 2 of two dependent normal random variables X 1 and X 2 in the case of Gaussian copulas. We then develop the theory on the median for the ratios of both Y and Z on two normal random variables X 1 and X 2 . Furthermore, we extend the result of median for Z to a larger family of symmetric distributions and symmetric copulas of X 1 and X 2 . Our results are the foundation of any further study that relies on the density and cumulative probability functions of ratios for two dependent or independent random variables. Since the densities and distributions of the ratios of both Y and Z are in terms of integrals and are very complicated, their exact forms cannot be obtained. To circumvent the difficulty, this paper introduces the Monte Carlo algorithm, numerical analysis, and graphical approach to efficiently compute the complicated integrals and study the behaviors of density and distribution. We illustrate our proposed approaches by using a simulation study with ratios of normal random variables on several different copulas, including Gaussian, Student-t, Clayton, Gumbel, Frank, and Joe Copulas. We find that copulas make big impacts from different Copulas on behavior of distributions, especially on median, spread, scale and skewness effects. In addition, we also discuss the behaviors via all copulas above with the same Kendall’s coefficient. The approaches developed in this paper are flexible and have a wide range of applications for both symmetric and non-symmetric distributions and also for both skewed and non-skewed copulas with absolutely continuous random variables that could contain a negative range, for instance, generalized skewed-t distribution and skewed-t Copulas. Thus, our findings are useful for academics, practitioners, and policy makers.

Multilevel Monte Carlo for noise estimation in stochastic multiscale systems

This paper provides a fortran algorithm that can be used to compute the cdf of the product of two normal distribution random variables. We also give references that provide mathematical properties, tables, and applications of this distribution

In this article we consider computing expectations w.r.t.~probability laws associated to a certain class of stochastic systems. In order to achieve such a task, one must not only resort to numerical approximation of the expectation, but also to a biased discretization of the associated probability. We are concerned with the situation for which the discretization is required in multiple dimensions, for instance in space and time. In such contexts, it is known that the multi-index Monte Carlo (MIMC) method can improve upon i.i.d.~sampling from the most accurate approximation of the probability law. Indeed by a non-trivial modification of the multilevel Monte Carlo (MLMC) method and it can reduce the work to obtain a given level of error, relative to the afore mentioned i.i.d.~sampling and relative even to MLMC. In this article we consider the case when such probability laws are too complex to sampled independently. We develop a modification of the MIMC method which allows one to use standard Markov chain Monte Carlo (MCMC) algorithms to replace independent and coupled sampling, in certain contexts. We prove a variance theorem which shows that using our MIMCMC method is preferable, in the sense above, to i.i.d.~sampling from the most accurate approximation, under assumptions. The method is numerically illustrated on a problem associated to a stochastic partial differential equation (SPDE).