An FEM-MLMC algorithm for a moving shutter diffraction in time stochastic model

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 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.

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.

Multilevel sequential Monte Carlo samplers

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.

We show that Lasso and Bayesian Lasso are very close when the sparsity is large and the noise is small. We propose to solve Bayesian Lasso using multivalued stochastic differential equation. We derive four discretization algorithms, and present highly efficient multilevel Monte Carlo (MLMC) simulations. Additionally, we perform a numerical comparison of the Monte Carlo (MC), MLMC and proximal Markov chain Monte Carlo algorithm (PMALA).

The \(hp\)-FEM is an adaptive version of the Finite Element Method (FEM) that is capable of achieving fast exponential convergence rates by combining optimally spatial refinements (\(h\)-adaptivity) with varying element polynomial degrees (\(p\)-adaptivity). We present a novel space-time adaptive \(hp\)-FEM algorithm that is capable of capturing accurately traveling sharp fronts in nonlinear time-dependent problems. The algorithm tries to assess the true approximation error directly (not via residuals or related techniques), and it does not involve any tuning parameters exposed to the user. The algorithm is compared to non-adaptive FEM as well as to adaptive FEM with low-order elements in the context of a model problem with moving sharp front that consists of two coupled nonlinear parabolic equations. Also the treatment of nonlinear terms is discussed and it is shown that replacing the Newton’s method with simpler linearizations can lead to considerable errors.

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).

A hybrid method for modeling the creeping flow is proposed. In our method the so‐called marker‐in‐cell (MIC) and the Finite Element Method (FEM) algorithm are combined together to simulate the thermal creeping flow concerning heterogeneous medium deformation. In particular, the unknown parameters at the Euler mesh‐nodes are calculated using the FEM. The cell‐markers in each element carry the material composition and history variables during the flowing process. The momentum and continuity equation are solved in terms of the pressure‐stabilizing Petrov‐Galerkin method (PSPG) with the equal‐order interpolation of the velocity and pressure, and the energy equation is solved using the streamline upwind Petrov‐Galerkin method (SUPG). In the MIC algorithm, the bilinear interpolation corresponds to the interpolation function in the finite elements. The FEM and MIC algorithm are independent from each other. The data in these two processes communicate through the nodal points. In addition, the triangular element algorithm makes possible to solve the problems with irregular mesh‐grid in complex structures. Our computation program has been verified with the classical Rayleigh‐Benard convection problem. As an example, the descent of an irregular geometry block is calculated. The stability of numerical solution is also investigated.

This paper proposes a Petri Net based methodology for task scheduling on multiproces sor architectures. The aim is to build automati cally a stochastic model (markovian model or stochastic simulation model) for both perfor mance evaluation and task scheduling. The modelling process that we use consists in the consecutive elaboration of two models: the knowledge model and the action model. The knowledge model is a formalization of the structure and the working of the system, using Petri Nets. This knowledge model is made of three parts: a logical subsystem (precedence constraints between tasks), a physical subsystem (multiprocessor architecture) and a decision subsystem (scheduling method). The action model is a translation of the previous knowledge model in a programming language or in a mathematical formalism. We propose in this paper two kinds of action models: a stochastic simulation model and a markovian model. These models are dedicated to dynamic task allocation between identical processors without communi cation cost between tasks. It is to be remarked that these action models are automatically deduced from the corresponding knowledge models. Finally, we report some results obtained from these two action models. In particular, we highlight that these two independent models return identical results and so validate them selves.

Nonstationary oscillation (NSO) processes are observed in a number of hydroclimatic data series. Stochastic simulation models are useful to study the impacts of the climatic variations induced by NSO processes into hydroclimatic regimes. Reproducing NSO processes in a stochastic time series model is, however, a difficult task because of the complexity of the nonstationary behaviors. In the current study, a novel stochastic simulation technique that reproduces the NSO processes embedded in hydroclimatic data series is presented. The proposed model reproduces NSO processes by utilizing empirical mode decomposition (EMD) and nonparametric simulation techniques (i.e., k‐nearest‐neighbor resampling and block bootstrapping). The model was first tested with synthetic data sets from trigonometric functions and the Rössler system. The North Atlantic Oscillation (NAO) index was then examined as a real case study. This NAO index was then employed as an exogenous variable for the stochastic simulation of streamflows at the Romaine River in the province of Quebec, Canada. The results of the application to the synthetic data sets and the real‐world case studies indicate that the proposed model preserves well the NSO processes along with the key statistical characteristics of the observations. It was concluded that the proposed model possesses a reasonable simulation capacity and a high potential as a stochastic model, especially for hydroclimatic data sets that embed NSO processes.

Approaches to estimation of light-weight foam physico-mechanical characteristics are discussed; stochastic models are suggested followed by their computer study. Calculations result in macrostress diagrams in an ensemble of foam system gas structural elements. Model versions of varying density foam systems are considered and their programme implementation examined.

Civil engineering applications are often characterized by a large uncertainty on the material parameters. Discretization of the underlying equations is typically done by means of the Galerkin Finite Element method. The uncertain material parameter can be expressed as a random field represented by, for example, a Karhunen–Loève expansion. Computation of the stochastic responses, i.e., the expected value and variance of a chosen quantity of interest, remains very costly, even when state-of-the-art Multilevel Monte Carlo (MLMC) is used. A significant cost reduction can be achieved by using a recently developed multilevel method: p-refined Multilevel Quasi-Monte Carlo (p-MLQMC). This method is based on the idea of variance reduction by employing a hierarchical discretization of the problem based on a p-refinement scheme. It is combined with a rank-1 Quasi-Monte Carlo (QMC) lattice rule, which yields faster convergence compared to the use of random Monte Carlo points. In this work, we developed algorithms for the p-MLQMC method for two dimensional problems. The p-MLQMC method is first benchmarked on an academic beam problem. Finally, we use our algorithm for the assessment of the stability of slopes, a problem that arises in geotechnical engineering, and typically suffers from large parameter uncertainty. For both considered problems, we observe a very significant reduction in the amount of computational work with respect to MLMC.

Abstract. Pork is an important source of human Salmonellosis. Therefore efficient Salmonella surveillance programmes are recommended for minimising Salmonella entry into food production. With a stochastic simulation model, a Salmonella control programme (based on meat juice samples) was analysed depending on sample size (30 to 200 animals per finishing farm and year), prevalence in the population (8,13 and 24 %), farm size (400, 1500 and 3000 places per farm), threshold level classifying a farm as positive (20 and 40 %) and sensitivity of the test. The simulation model generates an integrated pig production chain with linkages between the stages farrowing, rearing, fattening and slaughter starting with the purchase of gilts. Salmonella may enter the system via purchase of animals, feedstuffs and other vectors. Crowding effects and pen-to-pen infection were also taken into account. Within each herd a dynamic pattern of Salmonella infection was simulated. The simulation covered a time interval of 24 months. Sensitivity, specificity and the predictive values were used for the evaluation of the different sampling strategies. At the base situation (prevalence 8%, 1500 places per finishing farm, 60 animals sampled per year and a threshold of 40 %) sensitivity was 65.9 and specificity 96.6 %. Increasing the sample size mainly improved the sensitivity (74.2 to 77.0 %). If the prevalence in the population is enhanced, more animals have to be sampled in order to realise the same detection rate. The effect of the farm size was negligible. The pattern of the Salmonella infection within a farm over 24 months demonstrated that a classification of farms supposes a handling time of 12 months.

A channel simulator that models accurately the physical channel statistics determined by the frequency separation between two carriers in frequency hopping (FH) systems is called a FH channel simulator. In this paper, a novel stochastic FH simulation model based on Rice's sum of sinusoids is proposed for wideband mobile fading channels. It is shown that a frequency hop is the physical channel model that corresponds to phase hops in the simulation model, while maintaining all other parameters unchanged. Furthermore, this simulator is capable of simulating both the fast FH and slow FH fading channel by only adjusting the phase hop rate. Due to the fact that closed-form expressions are given for all model parameters, we can study the correlation properties of the proposed simulation model from both the analytical and simulation point of view. A pretty good agreement between the correlation properties of the simulation model and those of the underlying reference model has been observed in all cases.