quasi-Monte Carlo Simulation Research Articles

Decomposing high uncertainty in greenhouse gas mitigation pathways in emerging regions: An approach to evaluate risks toward carbon neutrality

In the context of ambitious greenhouse gas (GHG) mitigation strategies in emerging regions, addressing the prevailing uncertainty and its key influential factors is crucial. While uncertainty analysis of GHG mitigation pathways has been extensively explored, the systematic identification and quantification of pivotal factors has been notably absent. This study introduces a novel methodology that combines Quasi-Monte Carlo simulation with Sobol variance decomposition, which identifies key influential factors and traces the flow of uncertainties. Applied in Anhui Province, a rapidly developing region in China, our findings indicate an uncertain GHG emission proportion of 6.2 % by 2030, escalating to 68.6 % by 2060, with a 95 % confidence interval. This uncertainty results in a cumulative projection error of 2.1 billion tons of CO2e from 2020 to 2070. The per capita GDP factor emerges as the predominant influence, alongside the increasing impact of renewable energy factor and UHV import electricity factor. In our uncertainty flow analysis, the energy transformation sector is identified as the principal contributor to total uncertainty, driven significantly by economic and energy-related factors. These results underscore the critical need for policymakers in emerging regions to incorporate uncertainty decomposition analysis into their strategic planning to mitigate risks in GHG reduction efforts.

Thermal error prediction and reliability analysis of the main shaft bearing at wind turbines

During the operation of the gearless wind turbine, the phenomenon of heat generation in the main shaft bearing is inevitable and further affects the assembly preload. It is crucial to determine the effect of the thermal error on bearing assembly preload. In this paper, a reliability analysis method for main shaft bearing is proposed. Firstly, a finite element model for the thermal analysis of wind turbines is established based on heat transfer theory, and the thermal error of the preload is calculated. Subsequently, a reliability analysis of the main shaft bearing is conducted through Quasi-Monte Carlo simulation (QMCS) considering the influence of uncertainty factors. To further improve the computational efficiency, a surrogate model based on stochastic configuration network (SCN) is established to analyze the reliability and sensitivity of the bearing. Finally, the numerical example shows that the proposed model has high accuracy and applicability.

Research on uncertainties in fuel centrifugal pump based on prediction and reconstruction of internal flow field

Geometric machining errors in the blade profile and variable operating conditions in the extreme operating environment are primary factors leading to the uncertainties in pump performance. This paper presents an analysis of uncertainties of fuel centrifugal pumps by modeling the geometry uncertainty in blade machining based on the Karhunen–Loève (KL) expansion and using a polynomial chaos expansion (PCE) model. First, the geometric uncertainty in the blade machining is described by the KL expansion in three sections and a stochastic simulation of the blade geometry is performed. Then, a PCE surrogate model is trained based on the least angle regression method and validated by the bootstrap method to quantify the uncertainties of performance indices. Finally, the influence mechanism and relative importance of each input uncertainty parameter are investigated using a quasi-Monte Carlo simulation method. The results show that the KL expansion of the blade profile uses the random vector perturbation superposition of three stream surface, achieving the dimensional reduction in the blade machining error. The PCE surrogate model, trained with a dataset of 3 × 106 sample points, exhibits excellent fit, and the R-squared and adjusted R-squared for head coefficient and efficiency are both above 80%. The variance of parameter control points of the reconstructed flow field is less than 0.002. The uncertainties in both operating conditions and parameters have an influence on the distribution of the global flow field, while the influence of the uncertainty in machining error on the global flow field mainly concentrates on the power-generating positions of the blade.

Parameter subset reduction for imaging-based digital twin generation of patients with left ventricular mechanical discoordination

BackgroundIntegration of a patient’s non-invasive imaging data in a digital twin (DT) of the heart can provide valuable insight into the myocardial disease substrates underlying left ventricular (LV) mechanical discoordination. However, when generating a DT, model parameters should be identifiable to obtain robust parameter estimations. In this study, we used the CircAdapt model of the human heart and circulation to find a subset of parameters which were identifiable from LV cavity volume and regional strain measurements of patients with different substrates of left bundle branch block (LBBB) and myocardial infarction (MI). To this end, we included seven patients with heart failure with reduced ejection fraction (HFrEF) and LBBB (study ID: 2018-0863, registration date: 2019–10–07), of which four were non-ischemic (LBBB-only) and three had previous MI (LBBB-MI), and six narrow QRS patients with MI (MI-only) (study ID: NL45241.041.13, registration date: 2013–11–12). Morris screening method (MSM) was applied first to find parameters which were important for LV volume, regional strain, and strain rate indices. Second, this parameter subset was iteratively reduced based on parameter identifiability and reproducibility. Parameter identifiability was based on the diaphony calculated from quasi-Monte Carlo simulations and reproducibility was based on the intraclass correlation coefficient (ICC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${ICC}$$\\end{document}) obtained from repeated parameter estimation using dynamic multi-swarm particle swarm optimization. Goodness-of-fit was defined as the mean squared error (χ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{\\chi}}}^{{2}}$$\\end{document}) of LV myocardial strain, strain rate, and cavity volume.ResultsA subset of 270 parameters remained after MSM which produced high-quality DTs of all patients (χ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{\\chi}}}^{{2}}$$\\end{document} < 1.6), but minimum parameter reproducibility was poor (ICCmin\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{ICC}}_{{min}}$$\\end{document} = 0.01). Iterative reduction yielded a reproducible (ICCmin\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{ICC}}_{{min}}$$\\end{document} = 0.83) subset of 75 parameters, including cardiac output, global LV activation duration, regional mechanical activation delay, and regional LV myocardial constitutive properties. This reduced subset produced patient-resembling DTs (χ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{\\chi}}}^{{2}}$$\\end{document} < 2.2), while septal-to-lateral wall workload imbalance was higher for the LBBB-only DTs than for the MI-only DTs (p < 0.05).ConclusionsBy applying sensitivity and identifiability analysis, we successfully determined a parameter subset of the CircAdapt model which can be used to generate imaging-based DTs of patients with LV mechanical discoordination. Parameters were reproducibly estimated using particle swarm optimization, and derived LV myocardial work distribution was representative for the patient’s underlying disease substrate. This DT technology enables patient-specific substrate characterization and can potentially be used to support clinical decision making.

PRICING OF CALL OPTIONS USING THE QUASI MONTE CARLO METHOD

A call option is a type of option that grants the option holder the right to buy an asset at a specified price within a specified period of time. Determining the option price period of time within a certain period of time is the most important part of determining an investment strategy. Various methods can be employed to determine the prices of options, such as Quasi-Monte Carlo and Monte Carlo simulations. The purpose of this research is to determine the price of European-type call options using the Quasi-Monte Carlo method. The data used is daily stock closing price data on the Apple Inc. for the period October 1, 2021, to September 30, 2022. Apple Inc. stock options in this study were chosen because it is the largest technology company in the world in 2022. The steps taken in this study are to determine the parameters obtained from historical data such as the initial risk-free interest rate (r), stock price (, volatility , maturity time (T), and strike price (K). Next is to generate Halton’s quasi-randomized sequence and simulate the stock price by substituting the parameters by substituting the parameters. Then proceed to calculate the call option payoff and estimate the call option price by averaging the call option payoff values. The results showed that the call option price of the company Apple Inc. using the Quasi-Monte Carlo with Halton’s quasi-randomized sequence on the 1000000th simulation with a standard error of 0,045 is $90,163. The call option price obtained can be used as a reference for investors in purchasing options to minimize losses from call option investments in that period.

Calibration of a (6 + 3)-DOF kinematically redundant parallel mechanism based on hybrid identification algorithm and two-step compensation strategy

To improve both the absolute accuracy and pose repeatability of a (6 + 3)-DOF kinematically redundant parallel mechanism (KRPM), a novel kinematic calibration method based on hybrid identification algorithm and two-step compensation strategy is proposed. Firstly, the error model is established, and the joint clearances are treated equivalently. On this basis, the global sensitivity index (GSI) is introduced to analyze the influence of parameter errors. Secondly, the absolute accuracy and pose repeatability are evaluated with the quasi-Monte Carlo simulation method (QMCSM) and the three-dimensional ellipsoidal error probability method (ESEPM). Thirdly, a hybrid identification algorithm combining Levenberg-Marquardt (LM) algorithm and adaptive differential evolution (ADE) algorithm is proposed to ensure the identification speed and accuracy. Finally, the error compensation is carried out by two steps. The experimental results show that the absolute accuracy and pose repeatability of the KRPM are improved by at least 52.8 % and 12.5 % respectively based on the novel method.

Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing

AbstractIn financial engineering, prices of financial products are computed approximately many times each trading day with (slightly) different parameters in each calculation. In many financial models such prices can be approximated by means of Monte Carlo (MC) simulations. To obtain a good approximation the MC sample size usually needs to be considerably large resulting in a long computing time to obtain a single approximation. A natural deep learning approach to reduce the computation time when new prices need to be calculated as quickly as possible would be to train an artificial neural network (ANN) to learn the function which maps parameters of the model and of the financial product to the price of the financial product. However, empirically it turns out that this approach leads to approximations with unacceptably high errors, in particular when the error is measured in the ‐norm, and it seems that ANNs are not capable to closely approximate prices of financial products in dependence on the model and product parameters in real life applications. This is not entirely surprising given the high‐dimensional nature of the problem and the fact that it has recently been proved for a large class of algorithms, including the deep learning approach outlined above, that such methods are in general not capable to overcome the curse of dimensionality for such approximation problems in the ‐norm. In this article we introduce a new numerical approximation strategy for parametric approximation problems including the parametric financial pricing problems described above and we illustrate by means of several numerical experiments that the introduced approximation strategy achieves a very high accuracy for a variety of high‐dimensional parametric approximation problems, even in the ‐norm. A central aspect of the approximation strategy proposed in this article is to combine MC algorithms with machine learning techniques to, roughly speaking, learn the random variables (LRV) in MC simulations. In other words, we employ stochastic gradient descent (SGD) optimization methods not to train parameters of standard ANNs but instead to learn random variables appearing in MC approximations. In that sense, the proposed LRV strategy has strong links to Quasi‐Monte Carlo (QMC) methods as well as to the field of algorithm learning. Our numerical simulations strongly indicate that the LRV strategy might indeed be capable to overcome the curse of dimensionality in the ‐norm in several cases where the standard deep learning approach has been proven not to be able to do so. This is not a contradiction to the established lower bounds mentioned above because this new LRV strategy is outside of the class of algorithms for which lower bounds have been established in the scientific literature. The proposed LRV strategy is of general nature and not only restricted to the parametric financial pricing problems described above, but applicable to a large class of approximation problems. In this article we numerically test the LRV strategy in the case of the pricing of European call options in the Black‐Scholes model with one underlying asset, in the case of the pricing of European worst‐of basket put options in the Black‐Scholes model with three underlying assets, in the case of the pricing of European average put options in the Black‐Scholes model with three underlying assets and knock‐in barriers, as well as in the case of stochastic Lorentz equations. For these examples the LRV strategy produces highly convincing numerical results when compared with standard MC simulations, QMC simulations using Sobol sequences, SGD‐trained shallow ANNs, and SGD‐trained deep ANNs.

Weighted low-discrepancy samplings: A novel method for slope system reliability analysis

This paper presents an efficient method for system reliability analysis of geotechnical examples based on the low-discrepancy sequences (LDS) sampling technique in combination with probabilistic weighting of the samples, i.e., the weighted low-discrepancy samplings (WLDS). In the framework of quasi-Monte Carlo simulation (QMCS), the Sobol’s quasi-random LDS are employed to generate the basic random variable samples that can significantly improve the efficiency of probabilistic weighting for the probability of failure calculation. In particular, the most probable system failure points (MPPs) can be directly captured according to the probabilistic weighting calculation. Via Nataf transformation, the WLDS computational framework incorporating correlated random variables in soils is also illustrated. Focusing on geotechnical applications, the reliability analyses of example soil slopes are presented to verify the performance of the proposed method. It is found that the proposed WLDS method not only can accurately obtain the probability of system failure but also can capture the multi-MPPs defining different earth slope failure modes in a cost-effective manner. The results in this study demonstrate that the proposed WLDS method is a feasible and promising method for performing the system reliability analysis of complex geotechnical earth structures.

Error modelling and motion reliability analysis of a multi-DOF redundant parallel mechanism with hybrid uncertainties

Accuracy is one of the most important properties of mechanisms. However, the inherent uncertainties will cause an error between the actual motion and the desired motion, leading to a motion reliability problem. The error model and motion reliability of a type of multi-DOF (degrees of freedom) redundant parallel mechanism (MDRPM) with hybrid uncertainties are studied. Firstly, the error model of the mechanism is established and verified. Then, the motion inside the joints is analyzed, and the joint clearance is regarded as an interval variable, while the other variables are treated as random variables. On this basis, a hybrid motion reliability method based on the quasi-Monte Carlo simulation method (QMCSM) is developed for a mechanism with hybrid uncertainties. Compared with the traditional simulation method, the amount of computation required by the QMCSM is significantly reduced, and it is more suitable for solving the large-scale motion reliability problem of the MDRPM. A numerical simulation and experiments are performed. The simulation results show that the feasibility and adaptability of the motion reliability method in this paper are different. The experimental results demonstrate that the error model is effective, so the motion reliability analysis based on the error model is reliable.

A New Technique for Investigating Wind Power Prediction Error in the Multi-Objective Environmental Economics Problem

This paper develops a novel multi-objective evolutionary algorithm (MOEA), namely an enhanced multi-objective exchange market algorithm (EMEMA), to solve the multi-objective problems in the presence of uncertainty. In this paper, Quasi-Monte Carlo (QMC) simulation is implemented for the first time to address the overestimation and underestimation caused by the uncertain wind power generation. In QMC, instead of pseudorandom numbers, some samples are generated to reduce the computational burden using the Sobol sequences. In addition, a new procedure is introduced to increase the spread and maintain the diversity of solutions. The effectiveness of the EMEMA is tested on two well-known benchmark functions and Taiwan's well-known 40-unit test system in two different scenarios, with and without considering the uncertainty. Also, four different measurements are employed to scrutinize the algorithm performance. The results prove the superiority of the proposed method compared to the state-of-art algorithms. Finally, the robustness of the EMEMA is studied using parametric and non-parametric statistical tests. The statistical analysis corroborates the robustness of the proposed method.

Analysing Quantiles in Models of Forward Term Rates

The class of forward-LIBOR market models can, under certain volatility structures, produce unrealistically high long-dated forward rates, particularly for maturities and tenors beyond the liquid market calibration instruments. This paper presents a diagnostic tool for analysing the quantiles of distributions for forward term rates in a displaced lognormal forward-LIBOR model (DLFM). In particular, we provide a quantile approximation that can be used to assess whether the modelled term rates remain within realistic bounds with a high probability. Applying this diagnostic tool (verified using Quasi-Monte Carlo (QMC) simulations), we show that realised forward term rates for long time horizons may be kept within realistic limits by appropriately damping the tail of the DLFM volatility function.

Time-dependent non-linear buckling of 3D CFST arch structures with hybrid random interval uncertainties

Limit point and bifurcation buckling loads are critical concerns in structural stability design. With the inevitable viscoelastic effects of creep and shrinkage in concrete, such critical buckling points may vary due to the time-dependent change of equilibrium configuration. Furthermore, the intrinsic uncertainty and natural randomness in the geometry and material characteristics would affect the structural stability performance significantly. The present study provides a new robust method, called the generalized Chebyshev surrogate model-based sampling approach, in assessing the time-dependent nonlinear buckling behaviour of 3D concrete-filled steel tubular (CFST) arch structures when both random and interval uncertainties are involved. In the proposed approach, the relationships between the uncertain parameters and the critical nonlinear limit and bifurcation buckling loads are formulated using Chebyshev surrogate model strategy combined with finite element method. The extreme bounds of the statistical features, including means, standard deviations, of the critical nonlinear buckling loads are furnished by using Monte Carlo method and Quasi Monte Carlo simulation method. Finally, the applicability and the validity of the proposed approach are illustrated with a series of numerical investigations.

Quasi Monte Carlo for Periodic Review in Inventory Systems

Periodic Review as a method is widely used especially in inventory system. In this paper Quasi Monte Carlo is used for simulating Periodic Review. The problem: How to implement Quasi Monte Carlo simulation in Periodic Review for inventory system of MSMEs in order to achieve the expected minimum inventory cost? The solution offered: Implementation of Periodic Review with Quasi Monte Carlo in inventory system in MSMEs in order to achieve the expected minimum inventory cost. The method used in this article is a Literature Study on the use of Periodic Review optimization with Quasi Monte Carlo which is implemented in the inventory system. Result: Through the use of Quasi Monte Carlo in Periodic Review in Inventory System, the minimum inventory cost expected by MSMEs is achieved.

A probabilistic framework for the stability analysis of slopes considering the coupling influence of random ground motion and conditional random field

The uncertainty of the input parameters can have a significant influence on the evaluation of the slope stability. A simple probabilistic framework is presented in this paper to quantify the influence of the uncertainties of the stochastic ground motion coupled with the spatial variability of soil parameters. The robust quasi–Monte Carlo simulation in the proposed framework is employed to estimate the failure probability of slopes. The conditional random field (CRF) is manipulated to simulate the spatial variability of soil properties with the sample points where the soil parameters are determined, and the spectral representation of stochastic process is applied to generate the horizontal and vertical stochastic ground motions. In each run of the Quasi–Monte Carlo simulation, a modified Newmark method based on an infinite slope failure model is applied to determine the permanent displacement of slopes under horizontal and vertical ground motions. The originality and significance of this proposed probabilistic framework are validated by a real slope case. The results of the case show that the uncertainty affecting the stability of the slope can decrease when CRF is used in a random field simulation to incorporate the real values of soil parameters at sample points, and the failure probability of slopes can be underestimated without considering stochastic vertical ground motion. Moreover, the correlated distance and critical permanent displacement are significant in predicting the failure probability of slopes.

Reliability Sensitivity Analysis by the Axis Orthogonal Importance Sampling Method Based on the Box-Muller Transformation

The axis orthogonal importance sampling method proves to be one version of efficient importance sampling methods since the quasi-Monte Carlo simulation is its basic ingredient, in which it is now a common practice to transform low-discrepancy sequences from the uniform distribution to the normal distribution by the well-known inverse transformation. As a valid transformation method for low-discrepancy sequences, the Box-Muller transformation is introduced into the axis orthogonal importance sampling method and compared with the inverse transformation in this paper for structural reliability sensitivity analysis. Three representative quasi-random sequences with low discrepancy are presented to generate samples following the target distribution and explore the interaction with the transformation method, which is used as a sample plan along the tangent plane at the most probable failure point in the axial orthogonal importance sampling for structural reliability analysis and reliability sensitivity analysis. The numerical experiments show that the reliability sensitivity analysis method by means of the Box-Muller transformation is a good alternative to the inverse transformation to generate samples from low-discrepancy sequences to the normal distribution. In particular, the scheme of the Box-Muller transformation combined with the Sobol sequence needs fewer samples with more accuracy and is more applicable for solving reliability sensitivity analysis in various nonlinear problems.

Seismic reliability evaluation of spatially correlated pipeline networks by quasi-Monte Carlo simulation

The reliability assessment of pipeline networks under nature disasters helps to improve the disaster prevention capability and overall resilience of pipeline networks. For the earthquake hazard, the seismic damage of pipelines in the network is usually correlated due to the spatial correlation of critical parameters, including characteristics of pipeline, soil properties and ground motions. This paper proposes a framework to evaluate the seismic reliability of pipeline network under spatially correlated parameters based on the quasi-Monte Carlo (QMC) simulation. The influence of spatial correlated random variables on the seismic reliability of pipeline network is investigated with explicit consideration of the correlated random variables including the peak ground velocity, the wall thickness and the yield stress of pipe segment. The framework has been implemented in seismic reliability evaluation of two pipeline networks. The result shows that the QMC method has better simulation accuracy than the standard MC method under same sampling numbers. The spatial correlations have different influences on the seismic connectivity reliability of pipeline networks with different topological redundancy and parallelism of connective paths from sources to user nodes. For the pipeline network with greater redundancy and parallelism, the model without considering the spatial correlations overestimates the network connectivity reliability.

PENGGUNAAN METODE SIMULASI QUASI MONTE CARLO DENGAN BARISAN QUASI ACAK VAN DER COURPUT DALAM MENENTUKAN HARGA OPSI EROPA

An option is a right by a person or institution to sell or buy an investment instrument at a certain price for a certain period. The point of view for way an option is exercised, there are two types of options known today, namely European options and American options. European options are characterized by option contracts that can only be exercised at the expiration date of the option. American options are indicated by option contracts that can be exercised at any time within the validity of the option contract. There are several methods that can be used in determining option prices, including the Black Scholes method which can be used to calculate the standard price of options. Monte Carlo simulation is a method that gives all possible values of a variable using the average as an estimator of its exact value. Quasi Monte Carlo simulation is an alternative to Monte Carlo simulation which uses quasi-random sequences instead of random numbers. In this article, the quasi-random sequence used is Van der Courput. Calculate option prices with Monte Carlo simulation and Quasi Monte Carlo simulation using MATLAB.

A General Numerical Algorithm for CDO Pricing Based on Single Factor Copula Framework and Nonhomogeneous Assumptions

In view of the fact that different factor Copula models are only applicable to different practical problems in collateralized debt obligations (CDO) market and that there is no semianalytical solution under nonhomogeneous assumptions to CDO pricing model, we designed a general numerical algorithm which was based on the framework of single factor Copula model and randomized quasi-Monte Carlo (RQMC) simulation method. We took two single factor Copula models as examples to conduct empirical study, in which the simulation results of RQMC and Monte Carlo (MC) simulation method were compared and analyzed based on variance changes. The result showed that the algorithm in this paper was not only applicable to general single factor Copula model but also very stable. So, it was a general and efficient numerical method to solve the problem of CDO pricing under nonhomogeneous assumptions.

Robust topology optimization of structures under uncertain propagation of imprecise stochastic-based uncertain field

This study introduces a novel computational framework for Robust Topology Optimization (RTO) considering imprecise random field parameters. Unlike the worst-case approach, the present method provides upper and lower bounds for the mean and standard deviation of compliance as well as the optimized topological layouts of a structure for various scenarios. In the proposed approach, the imprecise random field variables are determined utilizing parameterized p-boxes with different confidence intervals. The Karhunen–Loève (K–L) expansion is extended to provide a spectral description of the imprecise random field. The linear superposition method in conjunction with a linear combination of orthogonal functions is employed to obtain explicit mathematical expressions for the first and second order statistical moments of the structural compliance. Then, an interval sensitivity analysis is carried out, applying the Orthogonal Similarity Transformation (OST) method with the boundaries of each of the intermediate variable searched efficiently at every iteration using a Combinatorial Approach (CA). Finally, the validity, accuracy, and applicability of the work are rigorously checked by comparing the outputs of the proposed approach with those obtained using the particle swarm optimization (PSO) and Quasi-Monte-Carlo Simulation (QMCS) methods. Three different numerical examples with imprecise random field loads are presented to show the effectiveness and feasibility of the study.

A spectral method for stochastic fractional PDEs using dynamically-orthogonal/bi-orthogonal decomposition

Modeling uncertainty propagation in anomalous transport applications leads to formulating stochastic fractional partial differential equations (SFPDEs), which require special algorithms for obtaining satisfactory accuracy at reasonable computational complexity. Here, we consider a stochastic fractional diffusion-reaction equation and combine a Galerkin spectral method based on poly-fractonomials with the modal decomposition of the stochastic fields to formulate effective numerical methods for SFPDEs. Specifically, we employ a generalized Karhunen-Loève (KL) expansion and proper dynamically-orthogonal/bi-orthogonal (DO/BO) constraints to derive new Galerkin formulations for the mean solution, the time-dependent spatial basis, and the stochastic time-dependent coefficients. In addition, we employ a hybrid approach to tackle the singular limit of DO and BO equations for the case of deterministic initial conditions. The DO and BO formulations are mathematically equivalent but they exhibit computationally complementary properties. In demonstration examples, we investigate the interplay between randomness and non-locality and quantify this interaction for first time. In particular, we apply generalized polynomial chaos (gPC), DO, BO, and hybrid methods (i.e., combining gPC with DO or BO) to linear problems and (combining Quasi Monte Carlo (QMC) method with DO or BO) to nonlinear problems, and we compare our results to reference solutions obtained by well-resolved QMC simulations. We find that the DO and BO methods are both accurate approaches suitable for SFPDEs, with the BO method seemingly more accurate overall. The fractional order α affects strongly the accuracy of the solution and hence the required number of modes for multi-dimensional problems with parametric uncertainty. Both the DO and BO methods converge fast with respect to the number of modes, and they are especially effective for nonlinear problems and long-time integration for a modest number of stochastic dimensions. With regards to temporal discretization, the mean of the solution using the backward differentiation formula (BDF3) exhibits very high accuracy.