Monte Carlo Simulation Scheme Research Articles

A deep learning-based Monte Carlo simulation scheme for stochastic differential equations driven by fractional Brownian motion

Stochastic differential equations (SDEs) are widely used models to describe the evolution of stochastic processes. Among them, SDEs driven by fractional Brownian motion (fBm) have been shown to be capable of describing systems with temporal dependencies. In this paper, we develop a neural network based Monte Carlo methodology in which we can efficiently simulate SDEs that are governed by fBm. Particularly, we focus on large time step simulations. A property of fBm that complicates the development of such Monte Carlo schemes is the long-range temporal correlation. To this end, we build the network based on the encoder–decoder framework and employ the attention mechanism to learn the temporal relationships in the historical paths of such SDEs. In addition, a loss function based on the quantile loss is used, where the quantile levels to be predicted are determined by means of the stochastic collocation method. Experimental results show that this kind of loss function is superior to conventional loss functions in terms of solution accuracy, and the resulting scheme can learn and simulate SDEs driven by fBm accurately and highly efficiently.

Existence and approximation of densities of chord length- and cross section area distributions

In various stereological problems an $n$-dimensional convex body is intersected with an $(n-1)$-dimensional Isotropic Uniformly Random (IUR) hyperplane. In this paper the cumulative distribution function associated with the $(n-1)$-dimensional volume of such a random section is studied. This distribution is also known as chord length distribution and cross section area distribution in the planar and spatial case respectively. For various classes of convex bodies it is shown that these distribution functions are absolutely continuous with respect to Lebesgue measure. A Monte Carlo simulation scheme is proposed for approximating the corresponding probability density functions.

A study on burst failure mechanism analysis and quantitative risk assessment of corroded pipelines with random pitting clusters

Corrosion defect is the primary inducement for pipeline burst failures. Natural-occurred defects appear in the form of pitting clusters, exhibiting randomness and irregularity in local geometrics, which influences the failure behaviors. Considering this effect, a finite element analysis model was firstly derived to simulate the isolated and interacting random pitting clusters and estimate the burst pressure pb. Reasonable consistence with the existing results was achieved with the mean absolute relative error of 2.32%. Subsequently, through a Monte-Carlo Simulation scheme, the failure mechanisms and the probabilistic characteristics of pbs were investigated. Three failure modes were detected to link the failure development with the local geometric morphologies; Weak correlation between material loss and pb was found and explained; The pb for pitting clusters exhibited pronounced variabilities and the mean value was higher than the residual strength from traditional methods, which was aggravated when interacting effects were involved. Finally, a model uncertainty factor-based quantitative risk assessment framework was developed to estimate the life-cycle failure probabilities Pf. The Pf in traditional methods was overestimated by 70%–150% maximumly; Influential factors on failure risk were evaluated and their evolution tendencies were discussed; Safety margins were provided to determine operation pressures and inspection intervals.

The balanced discrete Burr–Hatke model and mixing INAR(1) process: properties, estimation, forecasting and COVID-19 applications

The main concern of this paper is providing a flexible discrete model that captures every kind of dispersion (equi-, over- and under-dispersion). Based on the balanced discretization method, a new discrete version of Burr–Hatke distribution is introduced with the partial moment-preserving property. Some statistical properties of the new distribution are introduced, and the applicability of proposed model is evaluated by considering counting series. A new integer-valued autoregressive (INAR) process based on the mixing Pegram and binomial thinning operators with discrete Burr–Hatke innovations is introduced, which can model contagious data properly. The different estimation approaches of parameters of the new process are provided and compared through the Monte Carlo simulation scheme. The performance of the proposed process is evaluated by four data sets of the daily death counts of the COVID-19 in Austria, Switzerland, Nigeria and Slovenia in comparison with some competitor INAR(1) models, along with the Pearson residual analysis of the assessing model. The goodness of fit measures affirm the adequacy of the proposed process in modeling all COVID-19 data sets. The fundamental prediction procedures are considered for new process by classic, modified Sieve bootstrap and Bayesian forecasting methods for all COVID-19 data sets, which is concluded that the Bayesian forecasting approach provides more reliable results.

A physics‐informed Bayesian framework for characterizing ground motion process in the presence of missing data

AbstractA Bayesian framework to characterize ground motions even in the presence of missing data is developed. This approach features the combination of seismological knowledge (a priori knowledge) with empirical observations (even incomplete) via Bayesian inference. At its core is a Bayesian neural network model that probabilistically learns temporal patterns from ground motion data. Uncertainties are accounted for throughout the framework. Performance of the approach has been quantitatively demonstrated via various missing data scenarios. This framework provides a general solution to dealing with missing data in ground motion records by providing various forms of representation of ground motions in a probabilistic manner, allowing it to be adopted for numerous engineering and seismological applications. Notably, it is compatible with the versatile Monte Carlo simulation scheme, such that stochastic dynamic analyses are still achievable even with missing data. Furthermore, it serves as a complementary approach to current stochastic ground‐motion models in data‐scarce regions under the growing interests of PBEE (performance‐based earthquake engineering), mitigating the data‐model dependence dilemma due to the paucity of data, and ultimately, as a fundamental solution to the limited data problem in data scarce regions.

Uncertainty quantification of multimodal surface wave inversion using artificial neural networks

An inversion of surface wave dispersion curves (DCs) is a nonunique and ill-conditioned problem. The inversion result has a probabilistic nature, which becomes apparent when simultaneously restoring the S-wave velocity and layer thickness. Therefore, the problem of uncertainty quantification is relevant. Existing methods through deterministic or global optimization approaches of uncertainty quantification via posterior probability density (PPD) of the model parameters are not computationally efficient because they demand multiple solutions to the inverse problem. We have developed an alternative method based on a multilayer fully connected artificial neural network (ANN). We improve the current unimodal approach, which is known from publications, to multimodal inversion. We use Cox’s and Teague’s algorithm to determine optimal parameterization (number of layers) and the ranges of possible model parameters. We uniformly draw training data sets within estimated ranges and train the ANN. Saved ANN weights map the phase velocity DCs to values of the S-wave velocity and layers thickness. To estimate the uncertainties, we adapt the Monte Carlo simulation strategy, according to which the frequency dependent data noise and the inverse operator errors are projected onto the resulting velocity model. The inverse operator errors are evaluated by the prediction of the training data set. The proposed combination of surface wave data processing methods, configured with each other, provides a novel surface wave multimodal dispersion data inversion and uncertainty quantification approach. We first test our approach on synthetic experiments for various velocity models: a positive gradient velocity, a low-velocity layer, and a high-velocity layer. This is done considering unimodal inversion at first and then compared with the multimodal inversion. Afterward, we apply our approach to field data and compare the resulting models with the body S-wave processing by the generalized reciprocal method. The experiments indicate high-potential results — using ANN yields the possibility to accurately estimate the PPD of restored model parameters without a significant computational effort. The PPD-based comparison demonstrates the advantages of a multimodal inversion over a unimodal inversion. The trained ANN provides reasonable model parameters predictions and related uncertainties in real time.

A New Viterbi-Based Decoding Strategy for Market Risk Tracking: an Application to the Tunisian Foreign Debt Portfolio During 2010–2012

In this paper, a novel market risk tracking and prediction strategy is introduced. Our approach takes volatility clustering into account and allows for the possibility of regime shifts in the intra-portfolio's latent correlation structure. The proposed specification combines hidden Markov models (HMM) with latent factor models that takes into account the presence of both the conditional skewness and leverage effects in stock returns. A computationally efficient expectation-maximization (EM) algorithm based on the Viterbi decoder is developed to estimate the model parameters. Using daily exchange rate data of the Tunisian dinar versus the currencies of the main Tunisian government's creditors, during the 2011 revolution period, the model parameters are estimated. Then, the suitable model is used in conjunction with a Monte Carlo simulation strategy to predict the Value-at-Risk (VaR) of the Tunisian government's foreign debt portfolio. The backtesting results indicate that the new approach appears to give a good fit to the data and can improve the VaR predictions, particularly during financial instability periods.

Influence of Target-Substrate Distance on the Transport Process of Sputtered Atoms: MC-MD Multiscale Coupling Simulation.

A Monte Carlo (MC) and molecular dynamics (MD) coupling simulation scheme for sputtered particle transport was first proposed in this work. In this scheme, the MC method was utilized to model the free-flight process of sputtered atoms, while the MD model was adopted to simulate the collision between the sputtered atom and background gas atom so as to self-consistently calculate the post-collision velocity of the sputtered atom. The reliability of the MD collision model has been verified by comparing the computation results of the MD model and of an analytical model. This MC-MD coupling simulation scheme was used to investigate the influence of target-substrate distance on the transport characteristic parameters of sputtered Cu atoms during magnetron sputtering discharge. As the target-substrate distance increased from 30 to 150 mm, the peak energy of the incident energy distribution of deposited Cu atoms decreased from 2 to 1 eV due to the gradual thermalization of sputtered atoms. The distribution of differential deposition rate in unit solid angle firstly became more forward-peaked and then reversely approached the cosine distribution, which was agreed with the existing experimental observations. This work is expected to provide a more realistic simulation scheme for sputtered particle transport, which can be further combined with the MD simulation of sputtered film growth to explore the influence mechanism of process parameters on the properties of sputtered film.

Global reliability evaluation of a high-pier long-span continuous RC rigid frame bridge subjected to multi-point and multi-component stochastic ground motions

High-pier long-span continuous RC rigid frame bridges would be more vulnerable to seismic spatial variability and multi-component nature due to their large geometric dimension. However, research on seismic performance of these bridges subjected to multi-point and multi-component ground motions is sparse. In this study, the unified expression of spectral decomposition integrating spectral representation method and proper orthogonal decomposition based on coherency function matrix is derived. Then the dimension-reduction representation is developed for sample realization of multi-point and multi-component non-stationary stochastic ground motions with merely three elementary random variables, which bypasses the challenges of high-dimensional randomness degree and huge computation expense faced by the Monte Carlo simulation scheme. Thereafter the accurate seismic response analysis of a high-pier long-span bridge is numerically scrutinized combining with the probability density evolution method to further expound the effectiveness of the dimension-reduction model. Finally, a heuristic global reliability evaluation strategy employing relative displacement angle of bridge pier segments as criterion is proposed inspired by the idea of equivalent extreme-value event. Benefiting from this, both the refined reliability evaluation of bridge members and the structural global reliability can be accomplished, providing the possibilities of quantitative seismic performance assessment and earthquake resistance optimization design from an overall perspective.

A Computational Monte Carlo Simulation Strategy to Determine the Temporal Ordering of Abnormal Age Onset Among Biomarkers of Alzheimer's Disease.

To quantitatively determining the temporal ordering of abnormal age onsets (AAO) among various biomarkers for Alzheimer's disease (AD), we introduced a computational Monte-Carlo simulation (CMCS) to statistically examine such ordering of an AAO pair or over all AAOs. The CMCS 1) simulates longitudinal data, estimates AAO for each iteration, and finally assesses the type-I error of an AAO pair or all AAO ordering. Using hippocampus volume (VHC), cerebral glucose hypometabolic convergence index (HCI), plasma neurofilament light (NfL), mini-mental state exam (MMSE), the auditory verbal learning test-long term memory (AVLT-LTM), short term memory (AVLT-STM) and clinical-dementia rating sum of box scale (CDR-SOB) from 382 mild cognitive impairment converters and non-converters, the CMCS estimated type-I error for the earlier AAO of VHC, AVLT_STM and AVLT_LTM each than MMSE was significant (p<0.002). The type-I error for the overall AAO temporal ordering of VHC ≤ AVLT_STM ≤ AVLT_LTM < HCI ≤ MMSE ≤ CDR-SOB ≤ NfL was p = 0.012. These findings showed that our CMCS is capable of providing statistical inferences for quantifying AAO ordering which has important implications in advancing our understanding of AD.

Allocation of geometric tolerances in one-dimensional stackup problems

Many tolerancing problems on mechanical assemblies involve a functional requirement depending on a chain of parallel dimensions on individual parts. In these one-dimensional cases, simple methods are available for the analysis and the allocation of dimensional tolerances. However, they are difficult to extend to geometric tolerances, which must be translated into equivalent dimensional tolerances; this allows the analysis but makes the allocation generally impossible without Monte Carlo simulation and complex search strategies. To overcome this difficulty, the paper proposes a way of dealing directly with geometric tolerances in the allocation problem. This consists in expressing the functional requirement as a linear model of geometric tolerances rather than equivalent dimensional tolerances; the coefficients of the model (sensitivities) are calculated considering both the dimension chain and the standard definition of the geometric tolerances. The approach can be combined with any constrained optimization method based on sensitivities. The optimal scaling method, previously proposed for dimensional tolerances, is extended to geometric tolerances and used in two examples to demonstrate the simplicity of the overall workflow and the quality of the optimal solution.

Signature-Based Analysis of the Weighted-r-within-Consecutive-k-out-of-n: F Systems

In the present paper, we deliver a reliability study of the weighted-r-consecutive-k-out-of-n: F reliability systems consisting of independent and identically distributed components. The signature vector of the structures is arithmetically determined by the aid of an appropriate Monte-Carlo simulation scheme. Several conclusions for the corresponding reliability polynomials are also drawn. An extensive numerical investigation is carried out to evaluate the performance of the weighted-r-consecutive-k-out-of-n: F reliability systems. Some stochastic signature-based comparisons among several members of the underlying family of reliability structures are also discussed.

Multi-centric analysis of linear energy transfer distribution from clinical proton beam based on TOPAS

Linear energy transfer (LET), a key parameter of relative biological effectiveness (RBE) in proton therapy, is not easy to measure or simulate in a standardized protocol. A multi-centric analysis of dose-averaged LET (LETd) and track-averaged LET (LETt) including individual particle contributions, scoring slab thickness, inhomogeneous medium and clinical beam configurations are investigated using the Monte Carlo code TOPAS. Primary and secondary protons are scored separately within the simulation. It is found that the primary protons are the major contribution to LET at the dose plateau and Bragg Peak (BP) regions, whilst the secondary protons have complete contribution to the tail-LET beyond the BP. The calculated LETt is always lower than LETd, nearly a factor of two lower in the region distal to the BP. Varying the scoring slab thickness shows a negligible influence on LETt, whereas, there is up to 26% discrepancy in LETd at 5-cm depth with thicknesses varying from 0.1 mm to 2 mm for 153.3 MeV proton beam. Moreover, smaller scoring thickness of 0.1 mm results in large fluctuations for tail LETd. LETd distributions with 3-cm thick bone- and lung-slab at the entrance of water phantom against primary and secondary protons' contribution are derived. A small trough appears at the position of inserted inhomogeneous material and large spikes which could account for the changes of cross section in nuclear interactions by secondary protons display on the tail of LETd curve. Regarding the modulated beams, substantial-high LET values are obtained at both-sides along the beam path which should gain more attention when calculating LET in clinical applications. In conclusion, the selection of Monte Carlo simulation schemes indeed affects the calculated LET values, more procedures should be considered when building bridges between LET and radiological effectiveness.

Asymptotic Bayesian Optimization: A Markov sampling-based framework for design optimization

This paper presents a Markov sampling-based framework, called Asymptotic Bayesian Optimization, for solving a class of constrained design optimization problems. The optimization problem is converted into a unified two-phase sample generation problem which is solved by an effective Markov chain Monte Carlo simulation scheme. First, an exploration phase generates designs distributed over the feasible design space. Based on this information, an exploitation phase obtains a set of designs lying in the vicinity of the optimal solution set. The proposed formulation can handle continuous, discrete, or mixed discrete-continuous design variables. Appropriate adaptive proposal distributions for the continuous and discrete design variables are suggested. The set of optimal solutions provides valuable sensitivity information of the different quantities involved in the problem with respect to the design variables. Representative examples including an analytical problem involving nonlinear benchmark functions, a classical engineering design problem, and a performance-based design optimization problem of a structural system under stochastic excitation are presented to show the effectiveness and potentiality of the proposed optimization scheme. Validation calculations show that the scheme is a flexible, efficient and competitive choice for solving a wide range of classical and complex engineering design problems.

Fundamental Aspects of Semiconductor Device Modeling Associated With Discrete Impurities: Monte Carlo Simulation Scheme

We discuss the theoretical basis of the Boltzmann transport equation (BTE) and the Monte Carlo (MC) simulation scheme to take into account the discrete nature of impurities in semiconductor substrates. This is achieved by explicitly considering the self-consistent coupling between the BTE and Poisson’s equation and by separating the impurity potential into long- and short-range parts in those equations. We show that the Hartree potential is represented by the long-range part of the potential, whereas the scattering potential for the BTE is naturally derived within the framework, rather than an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ad hoc</i> assumption. The proposed scheme is validated by showing that MC simulations with discrete impurities in bulk can reproduce the conventional simulation results with the “jellium” impurity. Also, by including both static and dynamic potential fluctuations, the MC simulation can predict the correct mobility even under discrete impurities.

A hybrid residual neural network–Monte Carlo approach to invert surface wave dispersion data

ABSTRACTSurface‐wave inversion is a non‐linear and ill‐conditioned problem usually solved through deterministic or global optimization approaches. Here, we present an alternative method based on machine learning. Under the assumption of a local one‐dimensional model, we train a residual neural network to predict the non‐linear mapping between the full dispersion image and the model space, parameterized in terms of shear wave velocity and layer thicknesses. On the one hand, compared to standard convolutional neural networks, the residual network prevents the vanishing gradient problem when training a deep network. On the other hand, the use of the full dispersion image avoids the time‐consuming and often ambiguous picking procedure and allows considering higher modes in the inversion framework. One key aspect of any machine learning inversion strategy is the definition of an appropriate training set. In this case, the models forming the training and validation examples are uniformly drawn from previously defined ranges that cover a wide range of possible near‐surface layered Vs models. The reflectivity method constitutes the forward modelling operator that converts the model parameters into the observed shot gathers. The inversion also includes a Monte Carlo simulation strategy that propagates onto the model space the uncertainties related to noise in the data and the modelling error introduced by the network approximation. We first discuss synthetic inversions to assess the applicability of the proposed method and to analyse the effect of erroneous model parameterizations. The inversion results are also benchmarked with those provided by a more standard approach in which the particle swarm optimization algorithm inverts the fundamental mode only. Then, we discuss a field data application. Our tests confirm that the residual neural network inversion provides accurate model estimations and reliable uncertainty appraisals. One of the main benefits of the proposed approach is that once the network is trained it provides the near‐surface shear wave velocity profile in near real‐time.

Structural synthesis considering mixed discrete–continuous design variables: A Bayesian framework

In this work attention is directed to general structural optimization problems considering discrete–continuous design variables. The optimization problem is formulated as the minimization of an objective function subject to multiple design requirements. The mathematical programming statement is set into the framework of a Bayesian model updating problem. Constraints are handled directly within the proposed scheme, generating designs distributed over the feasible design space. Based on these samples, a set of designs lying in the vicinity of the optimal solution set is obtained. The Bayesian model updating problem is solved by an effective Markov chain Monte Carlo simulation scheme, where appropriate proposal distributions are introduced for the continuous and discrete design variables. The approach can efficiently estimate the sensitivity of the final design and constraints with respect to the design variables. In addition, the numerical implementation of the optimization algorithm depends on few control parameters. For illustration purposes, the general formulation is applied to an important class of problems, specifically, reliability-based design optimization of structural systems under stochastic excitation. Three numerical examples showing the effectiveness and potentiality of the approach reported herein are presented.

Magnitude Estimation of Overpressure Generation Mechanisms Using Quantitative Stochastic 2D Basin Models: A Case Study from the Danube-Tisza Interfluve Area in Hungary

The overpressure formation in the Pannonian basin, Hungary, was investigated but has not been properly understood for the last 40 years because at least two different explanations were delineated. The first explanation considers the hydrocarbon generation as the main overpressure generation mechanism with some undercompaction contribution. On the contrary, another explanation assumes tectonic stress as the main trigger of abnormal pressure. The following research delivers a suitable workflow to understand which generation mechanisms were active in the study area and estimate the quantitative contribution of the mechanisms. The developed workflow relies on the basin modeling principles that were designed to simulate subsurface processes on a geological timeframe. Moreover, the uncertainty of input parameters was considered, and the joint application of a heuristic Monte Carlo simulation scheme and improved basin modeling resulted in stochastic pore pressure models. The most frequent value (MFV) method was applied on the simulated values to test a robust statistical method in pore pressure prediction. The study has identified not only the four main overpressure generation mechanisms, but it could calculate the individual contribution to the subsurface pressure. Finally, two independent and stochastic pore pressure prediction methods have been developed that could be used in the pre-drill well planning phase and the real-time prediction during drilling.

On the application of Wishart process to the pricing of equity derivatives: the multi-asset case

Given the inherent complexity of financial markets, a wide area of research in the field of mathematical finance is devoted to develop accurate models for the pricing of contingent claims. Focusing on the stochastic volatility approach (i.e. we assume to describe asset volatility as an additional stochastic process), it appears desirable to introduce reliable dynamics in order to take into account the presence of several assets involved in the definition of multi-asset payoffs. In this article we deal with the multi asset Wishart Affine Stochastic Correlation model, that makes use of Wishart process to describe the stochastic variance covariance matrix of assets return. The resulting parametrization turns out to be a genuine multi-asset extension of the Heston model: each asset is exactly described by a single instance of the Heston dynamics while the joint behaviour is enriched by cross-assets and cross-variances stochastic correlation, all wrapped in an affine modeling. In this framework, we propose a fast and accurate calibration procedure, and two Monte Carlo simulation schemes.

Markovian Approximation of the Rough Bergomi Model for Monte Carlo Option Pricing

The recently developed rough Bergomi (rBergomi) model is a rough fractional stochastic volatility (RFSV) model which can generate a more realistic term structure of at-the-money volatility skews compared with other RFSV models. However, its non-Markovianity brings mathematical and computational challenges for model calibration and simulation. To overcome these difficulties, we show that the rBergomi model can be well-approximated by the forward-variance Bergomi model with wisely chosen weights and mean-reversion speed parameters (aBergomi), which has the Markovian property. We establish an explicit bound on the L2-error between the respective kernels of these two models, which is explicitly controlled by the number of terms in the aBergomi model. We establish and describe the affine structure of the rBergomi model, and show the convergence of the affine structure of the aBergomi model to the one of the rBergomi model. We demonstrate the efficiency and accuracy of our method by implementing a classical Markovian Monte Carlo simulation scheme for the aBergomi model, which we compare to the hybrid scheme of the rBergomi model.