Direct Monte Carlo Sampling Research Articles

Effectiveness of Random Field Approach in Serviceability Limit State Analysis of Strip Foundation

This work conducts a probabilistic inquiry on how the variability of the parameter defining soil deformability affects the settlement of the foundation located on the soil. The analysis addresses the random foundation model to relevantly estimate the probability of allowable deflection exceedance. The constitutive model parameter is based either on a single random variable or a random field. The computations incorporate direct Monte Carlo sampling and the variance reduction techniques, i.e. Stratified Sampling and Latin Hypercube Sampling. The work focuses on soil parameter modelling by random fields defined by various correlation functions. One of the analytical means is the application of a non-homogeneous function capable of reflecting the soil strata. The probabilistic methods proposed in the paper are tested on a standard example of a foundation strip featuring load eccentricity. It is proved that the modelling mode of the soil—a single variable or a random field—substantially affects the results, i.e. foundation settlements. It was detected that incorporating random fields in soil analysis allows for a valid reliability assessment of a foundation in respect to Serviceability Limit State. The relevantly adjusted correlation functions of random fields allow for a realistic subsoil analysis even in the case of a limited in situ measurement database.

Fast Posterior Estimation of Cardiac Electrophysiological Model Parameters via Bayesian Active Learning.

Probabilistic estimation of cardiac electrophysiological model parameters serves an important step toward model personalization and uncertain quantification. The expensive computation associated with these model simulations, however, makes direct Markov Chain Monte Carlo (MCMC) sampling of the posterior probability density function (pdf) of model parameters computationally intensive. Approximated posterior pdfs resulting from replacing the simulation model with a computationally efficient surrogate, on the other hand, have seen limited accuracy. In this study, we present a Bayesian active learning method to directly approximate the posterior pdf function of cardiac model parameters, in which we intelligently select training points to query the simulation model in order to learn the posterior pdf using a small number of samples. We integrate a generative model into Bayesian active learning to allow approximating posterior pdf of high-dimensional model parameters at the resolution of the cardiac mesh. We further introduce new acquisition functions to focus the selection of training points on better approximating the shape rather than the modes of the posterior pdf of interest. We evaluated the presented method in estimating tissue excitability in a 3D cardiac electrophysiological model in a range of synthetic and real-data experiments. We demonstrated its improved accuracy in approximating the posterior pdf compared to Bayesian active learning using regular acquisition functions, and substantially reduced computational cost in comparison to existing standard or accelerated MCMC sampling.

Modeling neutronic transients with Galerkin projection onto a greedy-sampled, POD subspace

Presented here is an application of POD-Galerkin projection to develop a reduced-order-model (ROM) that approximates the solution of the multi-group, time-dependent, diffusion equation with a lower computational cost. Such reduced models are often sought in the context of parametric studies and uncertainty quantification, which require successive evaluation of the full-order model (FOM). To construct the ROM efficiently, greedy sampling was used to generate a reduced subspace that represents the entire parameter (here, cross-sections) domain. By using this subspace, the ROM is expected to yield accurate predictions when used in place of the FOM for uncertainty quantification conducted, e.g., with direct Monte-Carlo sampling. The TWIGL benchmark was used here for illustration. A ROM constructed with a POD basis of rank 10 was able to reproduce the core power with a maximum RMS error on the order of 10–6, and for the precursors concentration it was 10–9. Moreover, the statistical moments obtained from the ROM were in good agreement with those of the full-order-model. The mean relative error of the power sample mean prediction was about 4 ×10–6 for both the ramp and the step perturbations. In terms of the computational efficiency, the ROM time, including the projection, is approximately four-times faster than the full-order-model.

Non-intrusive uncertainty quantification in the simulation of turbulent spray combustion using Polynomial Chaos Expansion: A case study

A major source of input uncertainties in the simulation of turbulent spray combustion is introduced by the need to specify the state of the liquid spray after primary breakup, i.e. a spray boundary condition for the lagrangian transport equations. To further enhance the credibility and predictive capabilities of such simulations, output uncertainties should be reported in addition to the quantities of interest. Therefore, this paper presents results from a comprehensive quantification of uncertainties from the specification of a spray boundary condition and numerical approximation errors. A well characterized lab-scale spray flame is studied by means of an Euler-Lagrange simulation framework using detailed finite rate chemistry. As direct Monte Carlo sampling of the simulation model is prohibitive, non-intrusive Polynomial Chaos expansion (PCE) is used for forward propagation of the uncertainties. Uncertain input parameters are prioritized in a screening study, which allows for a reduction of the parameter space. The computation of probabilistic bounds reveals an extensive uncertainty region around the deterministic reference simulation. In an a posteriori sensitivity analysis, the majority of this variance is traced back to the uncertain spray cone angle of the atomizer. The explicit computation of input uncertainties finally allows for an evaluation of total predictive uncertainty in the case considered.

Sampling-based stochastic analysis of the PKN model for hydraulic fracturing

Hydraulic fracturing processes are surrounded by uncertainty, as available data is typically scant. In this work, we present a sampling-based stochastic analysis of the hydraulic fracturing process by considering various system parameters to be random. Our analysis is based on the Perkins-Kern-Nordgren (PKN) model for hydraulic fracturing. This baseline model enables computation of high fidelity solutions, which avoids pollution of our stochastic results by inaccuracies in the deterministic solution procedure. In order to obtain the desired degree of accuracy of the computed solution, we supplement the employed time-dependent moving-mesh finite element method with two new enhancements: (i) global conservation of volume is enforced through a Lagrange multiplier; (ii) the weakly singular behavior of the solution at the fracture tip is resolved by supplementing the solution space with a tip enrichment function. This tip enrichment function enables the computation of the tip speed directly from its associated solution coefficient. A novel incremental-iterative solution procedure based on a backward-Euler time-integrator with sub-iterations is employed to solve the PKN model. Direct Monte-Carlo sampling is performed based on random variable and random field input parameters. The presented stochastic results quantify the dependence of the fracture evolution process—in particular the fracture length and fracture opening—on variations in the elastic properties and leak-off coefficient of the formation, and the height of the fracture.

Optimal Reliability Allocation of ±800 kV Ultra HVDC Transmission Systems

In this paper, an optimal reliability allocation (ORA) model of ±800 kV ultra-high voltage direct current (HVDC) transmission system is proposed. The model intends to provide the means to determine the availability requirements of components in the bidding process of ±800 kV ultra HVDC transmission projects. In this model, the component availability is optimized based on the prespecified system reliability. The objective function is composed of the component investment cost and the operation and maintenance cost. The ORA is formulated as a constrained nonlinear programming problem, and the genetic algorithm (GA) is used to find its solution. An improved dagger sampling (IDS) method is proposed to evaluate the reliability of ultra HVDC systems, and a bisection method is used to generate the initial population of the GA. An actual ultra HVDC project is used to conduct the case study. Results show that the proposed ORA model can minimize the component investment, operation, and maintenance costs, while achieving optimal allocation of the system reliability index to obtain the availability of components. In addition, the impact of the component spare on the ORA result is investigated. Performance comparison on the sampling efficiency of the IDS method and the direct Monte Carlo sampling method is also presented.

Quantifying the uncertainty in model parameters using Gaussian process-based Markov chain Monte Carlo in cardiac electrophysiology.

Model personalization requires the estimation of patient-specific tissue properties in the form of model parameters from indirect and sparse measurement data. Moreover, a low-dimensional representation of the parameter space is needed, which often has a limited ability to reveal the underlying tissue heterogeneity. As a result, significant uncertainty can be associated with the estimated values of the model parameters which, if left unquantified, will lead to unknown variability in model outputs that will hinder their reliable clinical adoption. Probabilistic estimation of model parameters, however, remains an unresolved challenge. Direct Markov Chain Monte Carlo (MCMC) sampling of the posterior distribution function (pdf) of the parameters is infeasible because it involves repeated evaluations of the computationally expensive simulation model. To accelerate this inference, one popular approach is to construct a computationally efficient surrogate and sample from this approximation. However, by sampling from an approximation, efficiency is gained at the expense of sampling accuracy. In this paper, we address this issue by integrating surrogate modeling of the posterior pdf into accelerating the Metropolis-Hastings (MH) sampling of the exact posterior pdf. It is achieved by two main components: (1) construction of a Gaussian process (GP) surrogate of the exact posterior pdf by actively selecting training points that allow for a good global approximation accuracy with a focus on the regions of high posterior probability; and (2) use of the GP surrogate to improve the proposal distribution in MH sampling, in order to improve the acceptance rate. The presented framework is evaluated in its estimation of the local tissue excitability of a cardiac electrophysiological model in both synthetic data experiments and real data experiments. In addition, the obtained posterior distributions of model parameters are interpreted in relation to the factors contributing to parameter uncertainty, including different low-dimensional representations of the parameter space, parameter non-identifiability, and parameter correlations.

Hybrid benchmarking of arbitrary quantum gates

We present a protocol for Interleaved Randomized Benchmarking of arbitrary quantum gates using Monte Carlo sampling of quantum states. It is generally applicable, including non-Clifford gates while preserving key advantages of Randomized Benchmarking such as error amplification as well as independence from state preparation and measurement errors. This property is crucial for implementations in many contemporary systems. Although the protocol scales exponentially in the number of qubits, it is superior to direct Monte Carlo sampling of the average gate fidelity in both the total number of experiments by orders of magnitude and savings in classical preprocessing, that are exponential.

Learning thermodynamics with Boltzmann machines

A Boltzmann machine is a stochastic neural network that has been extensively used in the layers of deep architectures for modern machine learning applications. In this paper, we develop a Boltzmann machine that is capable of modelling thermodynamic observables for physical systems in thermal equilibrium. Through unsupervised learning, we train the Boltzmann machine on data sets constructed with spin configurations importance-sampled from the partition function of an Ising Hamiltonian at different temperatures using Monte Carlo (MC) methods. The trained Boltzmann machine is then used to generate spin states, for which we compare thermodynamic observables to those computed by direct MC sampling. We demonstrate that the Boltzmann machine can faithfully reproduce the observables of the physical system. Further, we observe that the number of neurons required to obtain accurate results increases as the system is brought close to criticality.

Alternative environmental contours for structural reliability analysis

This paper presents alternative methods for constructing environmental contours for probabilistic structural reliability analysis of structures exposed to environmental forces such as wind and waves. For such structures, it is important to determine the environmental loads to apply in structural reliability calculations and structural design. The environmental contour concept is an effective, risk-based approach in establishing such design conditions. Traditionally, such contours are established by way of a Rosenblatt transformation from the environmental parameter space to a standard normal space, which introduces uncertainties and may lead to biased results. The proposed alternative approach, however, eliminates the need for such transformations and established environmental contours based on direct Monte Carlo sampling from the joint distribution of the relevant environmental parameters. In this paper, three alternative implementations of the proposed generic approach will be outlined.

Uncertainty estimation of core safety parameters using cross-correlations of covariance matrix

An uncertainty reduction method for core safety parameters, for which measurement values are not obtained, is proposed. We empirically recognize that there exist some correlations among the prediction errors of core safety parameters, e.g., a correlation between the control rod worth and the assembly relative power at corresponding position. Correlations of errors among core safety parameters are theoretically estimated using the covariance of cross sections and sensitivity coefficients of core parameters. The estimated correlations of errors among core safety parameters are verified through the direct Monte Carlo sampling method. Once the correlation of errors among core safety parameters is known, we can estimate the uncertainty of a safety parameter for which measurement value is not obtained.

Modeling the commodity fluctuations of OPEX terms

A feasibility study of a new plant or even of a revamped one bases the forecast of incomes and outcomes on a discounting back approach. This means that both prices and costs of commodities (i.e. raw materials and products) are assumed constant for long time-horizons. Commodities together with utilities play a major role in the economic assessment of OPEXs (operative expenditures). The paper tackles the “discounting back” problem that sees a coming apart between the dynamics of real market prices/costs (subject to fluctuations, volatility, and the “supply and demand” law) and the constant prices/costs assumed in conventional feasibility studies. The manuscript presents and discusses a methodology to model the time evolution of prices and costs of commodities for the feasibility-study framework of dynamic conceptual design. It also provides an improved methodology respect to direct Monte Carlo sampling of quotations over historical ranges, which is effective for repeated design optimization.

Modeling the Sequence Dependent Rigidity of DNA with a Molecular Dynamics Parameterized Rigid Base Model

The sequence of nucleotides in a DNA molecule varies its mechanical properties. This provides a mechanism for sequence specificity in the binding of proteins that distort the shape of DNA, such as in the nucleosome. Additionally, the formation of DNA loops, which are used in gene regulation, can be effected by the sequence dependent mechanical properties. To better understand and predict this sequence specificity necessitates an accurate mechanical model of DNA.Previous work in our lab has shown that a rigid base model performs significantly better than a rigid base pair model. We determine all parameters for this coarse grained model from molecular dynamics simulations. Using a direct Monte Carlo sampling method the sequence averaged persistence length of the coarse grained model with these parameters was determined and found to be in excellent agreement with experimental results. To evaluate whether the sequence dependent mechanical properties are accurately captured experimental cyclization data was reproduced.

A statistical method for verifying mesh convergence in Monte Carlo simulations with application to fragmentation

AbstractA novel method is presented for assessing the convergence of a sequence of statistical distributions generated by direct Monte Carlo sampling. The primary application is to assess the mesh or grid convergence, and possibly divergence, of stochastic outputs from non‐linear continuum systems. Example systems include those from fluid or solid mechanics, particularly those with instabilities and sensitive dependence on initial conditions or system parameters. The convergence assessment is based on demonstrating empirically that a sequence of cumulative distribution functions converges in the Linfty norm. The effect of finite sample sizes is quantified using confidence levels from the Kolmogorov–Smirnov statistic. The statistical method is independent of the underlying distributions.The statistical method is demonstrated using two examples: (1) the logistic map in the chaotic regime, and (2) a fragmenting ductile ring modeled with an explicit‐dynamics finite element code. In the fragmenting ring example the convergence of the distribution describing neck spacing is investigated. The initial yield strength is treated as a random field. Two different random fields are considered, one with spatial correlation and the other without. Both cases converged, albeit to different distributions. The case with spatial correlation exhibited a significantly higher convergence rate compared with the one without spatial correlation. Copyright © 2011 John Wiley & Sons, Ltd.

Reliability of deterministic non‐linear systems subjected to stochastic dynamic excitation

AbstractEngineering structures react to exceptionally high forces caused by, for example, extreme winds, sea waves, earthquakes, avalanches, etc. in a non‐linear way, before they finally collapse. Mostly these environmental loadings cause dynamic excitations which are adequately modeled by the so‐called stochastic processes.To identify subsets of the excitation, which may trigger failure, methods based on power inputs of the stochastic excitation will be exploited. This procedure is based on the simple consideration that any excitation that maximizes the energy input into the system has the potential to adversely affect the integrity of the structure. This method considers the velocity of the displacement field of the structure and the energy dissipation induced by viscous damping, friction and hysteresis.For an efficient reliability estimation, the n‐dimensional standard normal space S[0]∈ℜ︁n, in which the stochastic excitation is modeled, is split into two disjunct subspaces S[1]∈ℜ︁m and S[2]∈ℜ︁n−m. The subset S[1]∈ℜ︁m represents the space of important directions, which is identified by a procedure based on an approximation of the gradient of the energy input. Directional sampling in the subspace S[1] and direct Monte Carlo sampling in the subspace S[2] are combined to established an efficient estimator for the structural reliability. The proposed methodology is generally applicable to finite element models with strong non‐conservative non‐linearities. Copyright © 2010 John Wiley & Sons, Ltd.

Hierarchical Bayesian analysis of the seemingly unrelated regression and simultaneous equations models using a combination of direct Monte Carlo and importance sampling techniques

Computationally efficient simulation methods for hierarchical Bayesian analysis of the seemingly unrelated regression (SUR) and simultaneous equations models (SEM) are proposed and applied. These methods combine a direct Monte Carlo (DMC) approach and an importance sampling procedure to calculate Bayesian estimation and prediction results, namely, Bayesian posterior densities for parameters, predictive densities for future values of variables and associated moments, intervals and other quantities. The results obtained by our approach are compared to those yielded by use of MCMC techniques. Finally, we show that our algorithm can be applied to the Bayesian analysis of state space models.

Inferring phenotypic evolution in the fossil record by Bayesian inversion

This paper takes an alternative approach to the problem of inferring patterns of phenotypic evolution in the fossil record. Reconstructing temporal biological signal from noisy stratophenetic data is an inverse problem analogous to subsurface reconstructions in geophysics, and similar methods apply. To increase the information content of stratophenetic series, available geological data on sample ages and environments are included as prior knowledge, and all inferences are conditioned on the uncertainty in these geological variables. This uncertainty, as well as data error and the stochasticity of fossil preservation and evolution, prevents any unique solution to the stratophenetic inverse problem. Instead, the solution is defined as a distribution of model parameter values that explain the data to varying degrees. This distribution is obtained by direct Monte Carlo sampling of the parameter space, and evaluated with Bayesian integrals. The Bayesian inversion is illustrated with Miocene stratigraphic data from the ODP Leg 174AX Bethany Beach borehole. A sample of the benthic foraminiferPseudononion pizarrensisis used to obtain a phenotypic covariance matrix for outline shape, which constrains a model of multivariate shape evolution. The forward model combines this evolutionary model and stochastic models of fossil occurrence with the empirical sedimentary record to generate predicted stratophenetic series. A synthetic data set is inverted, using the Neighbourhood Algorithm to sample the parameter space and characterize the posterior probability distribution. Despite small sample sizes and noisy shape data, most of the generating parameter values are well resolved, and the underlying pattern of phenotypic evolution can be reconstructed, with quantitative measures of uncertainty. Inversion of a stratigraphic series into a time series can significantly improve our perception and interpretation of an evolutionary pattern.

Direct Monte Carlo sampling of the short-range screening potentials for classical Coulomb liquids.

A Monte Carlo (MC) simulation scheme to sample directly the screening potentials for Coulomb liquids at short interparticle distances is formulated on the basis of the importance sampling techniques. It is applied to the one-component plasmas at various degrees of the Coulomb coupling to obtain the screening potentials with high accuracy; a fitting formula for the MC values of the potentials is presented. The results for the screening potentials at zero separation are compared with those obtained in various approximation methods, and are utilized for the analyses of the excess free energy for binary-ionic-mixture plasmas through a thermodynamic self-consistency equation with respect to the screening potentials. \textcopyright{} 1996 The American Physical Society.

Critical surface free energies and universal finite-size scaling amplitudes of three-dimensional XY models by direct Monte Carlo sampling.

Direct Monte Carlo sampling is employed to obtain estimates of excess surface free energies of three-dimensional XY models at criticality. Results for simple-cubic and body-centered-cubic lattices are consistent with universality of finite-size scaling amplitudes. The results are used to estimate the magnitude of the thinning effect, mediated by the incipient long-ranged correlations, on liquid-helium films at the \ensuremath{\lambda} point.

Interfacial tension and correlation lengths for the three-state Potts model in two dimensions.

Estimates for the critical interfacial tension amplitudes of the three-state Potts model in two dimensions via direct Monte Carlo sampling are presented for the first time. Both the square and triangular lattices are analyzed with use of finite-size scaling considerations. Also presented for the first time are estimates for the correlation-length amplitudes as derived from high-temperature series. These numerical results allow investigation of the ratio U=\ensuremath{\tau}${\ensuremath{\xi}}^{d\mathrm{\ensuremath{-}}1}$/${k}_{B}$T as T\ensuremath{\rightarrow}${T}_{c}$. The results confirm the universality of U and are in agreement with exact results for hard hexagons. The finite-size scaling amplitude for the interfacial free energy at criticality is also shown to be universal within the available precision.