Monte Carlo Algorithm Research Articles

Covariate-Dependent Clustering of Undirected Networks with Brain-Imaging Data

This article focuses on model-based clustering of subjects based on the shared relationships of subject-specific networks and covariates in scenarios when there are differences in the relationship between networks and covariates for different groups of subjects. It is also of interest to identify the network nodes significantly associated with each covariate in each cluster of subjects. To address these methodological questions, we propose a novel nonparametric Bayesian mixture modeling framework with an undirected network response and scalar predictors. The symmetric matrix coefficients corresponding to the scalar predictors of interest in each mixture component involve low-rankness and group sparsity within the low-rank structure. While the low-rank structure in the network coefficients adds parsimony and computational efficiency, the group sparsity within the low-rank structure enables drawing inference on network nodes and cells significantly associated with each scalar predictor. Being a principled Bayesian mixture modeling framework, our approach allows model-based identification of the number of clusters, offers clustering uncertainty in terms of the co-clustering matrix and presents precise characterization of uncertainty in identifying network nodes significantly related to a predictor in each cluster. Empirical results in various simulation scenarios illustrate substantial inferential gains of the proposed framework in comparison with competitors. Analysis of a real brain connectome dataset using the proposed method provides interesting insights into the brain regions of interest (ROIs) significantly related to creative achievement in each cluster of subjects. Supplementary material shows the convergence rate for the posterior predictive density of the proposed model, additional simulation examples with model misspecification, full conditional distributions to run the Markov chain Monte Carlo (MCMC) algorithm and also presents traceplots for various model parameters to demonstrate convergence of the MCMC algorithm.

Low-mode deflation for twisted-mass and RHMC reweighting in lattice QCD

We propose improved estimators to compute the reweighting factors which are needed for lattice QCD calculations that rely on twisted-mass reweighting for the light quark contribution and the Rational Hybrid Monte Carlo (RHMC) algorithm for non-degenerate quark masses. This is the case for a number of modern large-scale simulations based on O(a) improved Wilson fermions. We find a significant reduction of uncertainties for the reweighting factors at similar computational cost compared to the conventional estimation. This leads to a significant increase in precision for phenomenologically relevant observables with high correlation to the low eigenmodes of the Wilson-Dirac operator in the presence of exceptionally small eigenvalues. Supplementary details regarding the spectral gap of the light quark Dirac operator on the 2+1 flavor large-volume ensembles explored in this study can be found in an accompanying appendix.

Uncertainty assessment of aquifer hydraulic parameters from pumping test data

Data from pumping tests is a noisy process, and therefore, performing the pumping test numerous times will not get the same drawdown values. As a consequence, various pumping experiments lead to different values for aquifer parameter estimates. The data of pumping tests are usually analyzed using traditional methods (aquifer tests and AQtesolv software), which depend on trial and error technique. During these methods, non-unique values of hydraulic parameters are selected, which usually have a high level of uncertainty. Uncertainty must be taken into account in determining aquifer parameters, especially when using groundwater models for decision makers. The main goal of this study is to build a comprehensive tool for quantifying uncertainty associated with hydraulic parameter estimation from different pumping test conditions for fully penetrating wells in confined and semi-confined aquifers. To achieve the previous objective, a FORTRAN code was developed to apply the Markov Chain Monte Carlo (MCMC) algorithm using different likelihood functions (exponential, inverse, and log). This developed tool can be used to detect the most probable range of aquifer parameters that are consistent with pumping test data with a high degree of confidence. The tool was successfully used to several hypothetical cases to demonstrate the uncertainty in the quantification of aquifer parameters and compare the findings to the standard method's results. Also, the concept was verified numerically (using Modflow program) with satisfactory results using a hypothetical case with well-known aquifer parameters. Finally, the tool was applied for actual pumping test data with good results.

Optimising CH4 simulations from the LPJ-GUESS model v4.1 using an adaptive Markov chain Monte Carlo algorithm

Abstract. The processes responsible for methane (CH4) emissions from boreal wetlands are complex; hence, their model representation is complicated by a large number of parameters and parameter uncertainties. The arctic-enabled dynamic global vegetation model LPJ-GUESS (Lund–Potsdam–Jena General Ecosystem Simulator) is one such model that allows quantification and understanding of the natural wetland CH4 fluxes at various scales, ranging from local to regional and global, but with several uncertainties. The model contains detailed descriptions of the CH4 production, oxidation, and transport controlled by several process parameters. Complexities in the underlying environmental processes, warming-driven alternative paths of meteorological phenomena, and changes in hydrological and vegetation conditions highlight the need for a calibrated and optimised version of LPJ-GUESS. In this study, we formulated the parameter calibration as a Bayesian problem, using knowledge of reasonable parameters values as priors. We then used an adaptive Metropolis–Hastings (MH)-based Markov chain Monte Carlo (MCMC) algorithm to improve predictions of CH4 emission by LPJ-GUESS and to quantify uncertainties. Application of this method on uncertain parameters allows for a greater search of their posterior distribution, leading to a more complete characterisation of the posterior distribution with a reduced risk of the sample impoverishment that can occur when using other optimisation methods. For assimilation, the analysis used flux measurement data gathered during the period from 2005 to 2014 from the Siikaneva wetlands in Southern Finland with an estimation of measurement uncertainties. The data are used to constrain the processes behind the CH4 dynamics, and the posterior covariance structures are used to explain how the parameters and the processes are related. To further support the conclusions, the CH4 flux and the other component fluxes associated with the flux are examined. The results demonstrate the robustness of MCMC methods to quantitatively assess the interrelationship between objective function choices, parameter identifiability, and data support. The experiment using real observations from Siikaneva resulted in a reduction in the root-mean-square error (RMSE), from 0.044 to 0.023 gC m−2 d−1, and a 93.89 % reduction in the cost function value. As a part of this work, knowledge about how CH4 data can constrain the parameters and processes is derived. Although the optimisation is performed based on a single site's flux data from Siikaneva, the algorithm is useful for larger-scale multi-site studies for a more robust calibration of LPJ-GUESS and similar models, and the results can highlight where model improvements are needed.

Asteroid reflectance spectra from Gaia DR3: Near-UV in primitive asteroids

In the context of charge-coupled devices (CCDs), the ultraviolet (UV) region has mostly remained unexplored after the 1990s. Gaia DR3 offers the community a unique opportunity to explore tens of thousands of asteroids in the near-UV as a proxy of the UV absorption. This absorption has been proposed in previous works as a diagnostic of hydration, organics, and space weathering. In this work, we aim to explore the potential of the NUV as a diagnostic region for primitive asteroids using Gaia DR3. We used a corrective factor over the blue part of $Gaia$ spectra to erase the solar analog selection effect. We identified an artificial relation between the band noise and slope and applied a signal-to-noise ratio (S/N) threshold for $Gaia$ bands. Meeting the quality standards, we employed a Markov chain Monte Carlo (MCMC) algorithm to compute the albedo threshold, maximizing primitive asteroid inclusion. Utilizing one- and two-dimensional (1D and 2D) projections, along with dimensionality-reduction methods (such as PCA and UMAP), we identified primitive asteroid populations. We uncovered: (a) the first observational evidence linking UV absorption to the 0.7 mu m band, tied to hydrated iron-rich phyllosilicates; and (b) a 2D space revealing a split in C-type asteroids based on spectral features, including UV absorption. The computed average depth (3.5 pm 1.0 <!PCT!>) and center (0.70 pm 0.03 mu m) of the 0.7 mu m absorption band for primitive asteroids observed with $Gaia$ is in agreement with the literature values. In this paper, we shed light on the importance of the UV absorption feature to discriminate among different mineralogies (i.e., iron-rich phyllosilicates vs. iron-poor) or to identify taxonomies that are conflated in the visible (i.e., F-types vs. B-types). We have shown that this is a promising region for diagnostic studies of the composition of primitive asteroids.

Bayesian mixed model inference for genetic association under related samples with brain network phenotype.

Genetic association studies for brain connectivity phenotypes have gained prominence due to advances in noninvasive imaging techniques and quantitative genetics. Brain connectivity traits, characterized by network configurations and unique biological structures, present distinct challenges compared to other quantitative phenotypes. Furthermore, the presence of sample relatedness in the most imaging genetics studies limits the feasibility of adopting existing network-response modeling. In this article, we fill this gap by proposing a Bayesian network-response mixed-effect model that considers a network-variate phenotype and incorporates population structures including pedigrees and unknown sample relatedness. To accommodate the inherent topological architecture associated with the genetic contributions to the phenotype, we model the effect components via a set of effect network configurations and impose an inter-network sparsity and intra-network shrinkage to dissect the phenotypic network configurations affected by the risk genetic variant. A Markov chain Monte Carlo (MCMC) algorithm is further developed to facilitate uncertainty quantification. We evaluate the performance of our model through extensive simulations. By further applying the method to study, the genetic bases for brain structural connectivity using data from the Human Connectome Project with excessive family structures, we obtain plausible and interpretable results. Beyond brain connectivity genetic studies, our proposed model also provides a general linear mixed-effect regression framework for network-variate outcomes.

On the power of threshold-based algorithms for detecting cycles in the CONGEST model

It is known that, for every k≥2, C2k-freeness can be decided by a generic Monte-Carlo algorithm running in n1−1/Θ(k2) rounds in the congest model. For 2≤k≤5, faster Monte-Carlo algorithms do exist, running in O(n1−1/k) rounds, based on upper bounding the number of messages to be forwarded, and aborting search sub-routines for which this number exceeds certain thresholds. We investigate the possible extension of these threshold-based algorithms, for the detection of larger cycles. We first show that, for every k≥6, there exists an infinite family of graphs containing a 2k-cycle for which any threshold-based algorithm fails to detect that cycle. Hence, in particular, neither C12-freeness nor C14-freeness can be decided by threshold-based algorithms. Nevertheless, we show that {C12,C14}-freeness can still be decided by a threshold-based algorithm, running in O(n1−1/7)=O(n0.857…) rounds, which is faster than using the generic algorithm, which would run in O(n1−1/22)≃O(n0.954…) rounds. Moreover, we exhibit an infinite collection of families of cycles such that threshold-based algorithms can decide F-freeness for every F in this collection.

Non-Hermitian Strongly Interacting Dirac Fermions.

Exotic quantum phases and phase transition in the strongly interacting Dirac systems have attracted tremendous interests. On the other hand, non-Hermitian physics, usually associated with dissipation arising from the coupling to environment, emerges as a frontier of modern physics in recent years. In this Letter, we investigate the interplay between non-Hermitian physics and strong correlation in Dirac-fermion systems. We generalize the projector quantum Monte-Carlo (PQMC) algorithm to the non-Hermitian interacting fermionic systems. Employing PQMC simulation, we decipher the ground-state phase diagram of the honeycomb Hubbard model with spin resolved non-Hermitian asymmetric hopping processes. The antiferromagnetic (AFM) ordering induced by Hubbard interaction is enhanced by the non-Hermitian asymmetric hopping. Combining PQMC simulation and renormalization group analysis, we reveal that the quantum phase transition between Dirac semi-metal and AFM phases belongs to Hermitian chiral XY universality class, implying that a Hermitian Gross-Neveu transition is emergent at the quantum critical point although the model is non-Hermitian.

Quantum Monte Carlo algorithm for arbitrary spin-1/2 Hamiltonians

Quantum Monte Carlo algorithm for arbitrary spin-1/2 Hamiltonians

Modeling high-frequency financial data using R and Stan: A bayesian autoregressive conditional duration approach

In econometrics, Autoregressive Conditional Duration (ACD) models use high-frequency economic or financial duration data, which mostly exhibit irregular time intervals. The ACD model is widely used to examine the duration of transaction volume and duration of price variations in stock markets. In this work, our goal is to devise testing that will aid in the identification of the best potential duration model among a set of four models using Bayesian approach. We test three models that rely on conditional mean duration (Weibull ACD, Log Weibull ACD, Generalized Gamma ACD) and one conditional median duration model (Birnbaum-Saunders ACD), and are being compared each other. The study was done in Rstan, a programming language for statistical inference, and the simulation uses the Hamiltonian Monte Carlo (HMC) algorithm of Markov Chain Monte Carlo (MCMC) to sample from the posterior distribution. Our findings show that Log Weibull ACD (second-generation model) as best among the four models followed by Birnbaum-Saunders ACD (third-generation model). The result offers methodological implications for algorithmic trading (algo-trading), high-frequency trading and risk management.

PDMP Monte Carlo methods for piecewise smooth densities

Abstract There has been substantial interest in developing Markov chain Monte Carlo algorithms based on piecewise deterministic Markov processes. However, existing algorithms can only be used if the target distribution of interest is differentiable everywhere. The key to adapting these algorithms so that they can sample from densities with discontinuities is to define appropriate dynamics for the process when it hits a discontinuity. We present a simple condition for the transition of the process at a discontinuity which can be used to extend any existing sampler for smooth densities, and give specific choices for this transition which work with popular algorithms such as the bouncy particle sampler, the coordinate sampler, and the zigzag process. Our theoretical results extend and make rigorous arguments that have been presented previously, for instance constructing samplers for continuous densities restricted to a bounded domain, and we present a version of the zigzag process that can work in such a scenario. Our novel approach to deriving the invariant distribution of a piecewise deterministic Markov process with boundaries may be of independent interest.

Parameter estimation for Gipps’ car following model in a Bayesian framework

Car following model is an important part in traffic modelling and has attracted a lot of attentions in the literature. As the proposed car following models become more complex with more components, reliably estimating their parameters becomes crucial to enhance model predictive performance. While most studies adopt an optimisation-based approach for parameters estimation, we present a statistically rigorous method that quantifies uncertainty of the estimates. We present a Bayesian approach to estimate parameters using the popular Gipps’ car following model as demonstration, which allows proper uncertainty quantification and propagation. Since the parameters of the car following model enter the statistical model through the solution of a delay-differential equation, the posterior is analytically intractable so we implemented an adaptive Markov Chain Monte Carlo algorithm to sample from it. Our results show that predictive uncertainty using a point estimator versus a full Bayesian approach are similar with sufficient data. In the absence of adequate data, the former can make over-confident predictions while such uncertainty is more appropriately incorporated in a Bayesian framework. Furthermore, we found that the congested flow parameters in the Gipps’ car following model are strongly correlated in the posterior, which not only causes issues for sampling efficiency but more so suggests the potential ineffectiveness of a point estimator in an optimisation-based approach. Lastly, an application of the Bayesian approach to a car following episode in the NGISM dataset is presented.

Adaptive tempered reversible jump algorithm for Bayesian curve fitting

Bayesian curve fitting plays an important role in inverse problems, and is often addressed using the reversible jump Markov chain Monte Carlo (RJMCMC) algorithm. However, this algorithm can be computationally inefficient without appropriately tuned proposals. As a remedy, we present an adaptive RJMCMC algorithm for the curve fitting problems by extending the adaptive Metropolis sampler from a fixed-dimensional to a trans-dimensional case. In this presented algorithm, both the size and orientation of the proposal function can be automatically adjusted in the sampling process. Specifically, the curve fitting setting allows for the approximation of the posterior covariance of the a priori unknown function on a representative grid of points. This approximation facilitates the definition of efficient proposals. In addition, we introduce an auxiliary-tempered version of this algorithm via non-reversible parallel tempering. To evaluate the algorithms, we conduct numerical tests involving a series of controlled experiments. The results demonstrate that the adaptive algorithms exhibit significantly higher efficiency compared to the conventional ones. Even in cases where the posterior distribution is highly complex, leading to ineffective convergence in the auxiliary-tempered conventional RJMCMC, the proposed auxiliary-tempered adaptive RJMCMC performs satisfactorily. Furthermore, we present a realistic inverse example to test the algorithms. The successful application of the adaptive algorithm distinguishes it again from the conventional one that fails to converge effectively even after millions of iterations.

Cosmic insights from galaxy clusters: Exploring magnification bias on sub-millimetre galaxies

Magnification bias, an observational effect of gravitational lensing in the weak regime, allows the cosmological model to be tested through angular correlations of sources at different redshifts. This effect has been observed in various contexts, particularly with sub-millimetre galaxies (SMGs), offering valuable astrophysical and cosmological insights. The study aims to investigate the magnification bias effect exerted by galaxy clusters on SMGs and its implications for astrophysical and cosmological parameters within the Lambda cold dark matter model. Magnification bias was explored by quantifying the cross-correlation function,which we then utilised to derive constraints on cosmological and astrophysical parameters with a Markov chain Monte Carlo algorithm. Two distinct galaxy cluster samples were used to assess result robustness and understand the influence of sample characteristics. Cluster samples show higher cross-correlation values than galaxies, with an excess at larger scales suggesting contributions from additional large-scale structures. The parameters obtained, while consistent with those of galaxies, are less constrained due to broader redshift distributions and limited cluster statistics. Results align with weak lensing studies, hinting at slightly lower $ and $ values than Planck 's cosmic microwave background data, emphasising the need for enhanced precision and alternative low-redshift universe tests. While this method yields constraints that are compatible with the Lambda cold dark matter model, its limitations include broader redshift distributions and a limited number of lenses, resulting in less constrained parameters compared to previous galaxy studies. Nonetheless, our study underscores the potential of using galaxy clusters as lenses for magnification bias studies, capitalising on their elevated masses and thus providing a promising avenue to test current cosmology theories. Further progress can be made by expanding the lens sample size.

Bayesian variable selection for matrix autoregressive models

A Bayesian method is proposed for variable selection in high-dimensional matrix autoregressive models which reflects and exploits the original matrix structure of data to (a) reduce dimensionality and (b) foster interpretability of multidimensional relationship structures. A compact form of the model is derived which facilitates the estimation procedure and two computational methods for the estimation are proposed: a Markov chain Monte Carlo algorithm and a scalable Bayesian EM algorithm. Being based on the spike-and-slab framework for fast posterior mode identification, the latter enables Bayesian data analysis of matrix-valued time series at large scales. The theoretical properties, comparative performance, and computational efficiency of the proposed model is investigated through simulated examples and an application to a panel of country economic indicators.

From linear to nonlinear Breit-Wheeler pair production in laser-solid interactions.

During the ultraintense laser interaction with solids (overdense plasmas), the competition between two possible quantum electrodynamics (QED) mechanisms responsible for e^{±} pair production, i.e., linear and nonlinear Breit-Wheeler (BW) processes, remains to be studied. Here, we have implemented the linear BW process via a Monte Carlo algorithm into the QED particle-in-cell (PIC) code yunic, enabling us to self-consistently investigate both pair production mechanisms in the plasma environment. By a series of two-dimensional QED-PIC simulations, the transition from the linear to the nonlinear BW process is observed with the increase of laser intensities in the typical configuration of a linearly polarized laser interaction with solid targets. A critical normalized laser amplitude about a_{0}∼400-500 is found under a large range of preplasma scale lengths, below which the linear BW process dominates over the nonlinear BW process. This work provides a practicable technique to model linear QED processes via integrated QED-PIC simulations. Moreover, it calls for more attention to be paid to linear BW pair production in near future 10-PW-class laser-solid interactions.

Self-supervised Bipartite Graph Representation Learning: A Dirichlet Max-margin Matrix Factorization Approach

Bipartite graph representation learning aims to obtain node embeddings by compressing sparse vectorized representations of interactions between two types of nodes, e.g., users and items. Incorporating structural attributes among homogeneous nodes, such as user communities, improves the identification of similar interaction preferences, namely, user/item embeddings, for downstream tasks. However, existing methods often fail to proactively discover and fully utilize these latent structural attributes. Moreover, the manual collection and labeling of structural attributes is always costly. In this paper, we propose a novel approach called Dirichlet Max-margin Matrix Factorization (DM3F), which adopts a self-supervised strategy to discover latent structural attributes and model discriminative node representations. Specifically, in self-supervised learning, our approach generates pseudo group labels (i.e., structural attributes) as a supervised signal using the Dirichlet process without relying on manual collection and labeling, and employs them in a max-margin classification. Additionally, we introduce a Variational Markov Chain Monte Carlo algorithm (Variational MCMC) to effectively update the parameters. The experimental results on six real datasets demonstrate that, in the majority of cases, the proposed method outperforms existing approaches based on matrix factorization and neural networks. Furthermore, the modularity analysis confirms the effectiveness of our model in capturing structural attributes to produce high-quality user embeddings.

Black-Box Boundary Attack Based on Gradient Optimization

Deep neural networks have gained extensive applications in computer vision, demonstrating significant success in fundamental research tasks such as image classification. However, the robustness of these networks faces severe challenges in the presence of adversarial attacks. In real-world scenarios, addressing hard-label attacks often requires the execution of tens of thousands of queries. To combat these challenges, the Black-Box Boundary Attack leveraging Gradient Optimization (GOBA) has been introduced. This method employs a binary search strategy to acquire an initial adversarial example with significant perturbation. The Monte Carlo algorithm is utilized to estimate the gradient of the sample, facilitating iterative movement along the estimated gradient and the direction of the malicious label. Moreover, query vectors positively correlated with the gradient are extracted to construct a sampling space with an optimal scale, thereby enhancing the efficiency of the Monte Carlo algorithm. Experimental evaluations were conducted using the HSJA, QEBA, and NLBA attack methodologies on the ImageNet, CelebA, and MNIST datasets, respectively. The results indicate that, under the constraint of 3 k query times, the GOBA, compared to other methods, can, on average, reduce perturbation (L2 distance) by 55.74% and simultaneously increase the attack success rate by an average of 13.78%.

Impact of new direct-acting antiviral therapy on the prevalence and undiagnosed proportion of chronic hepatitis C infection.

Patients with chronic hepatitis C (CHC) can be cured with the new highly effective interferon-free combination treatments (DAA) that were approved in 2014. However, CHC is a largely silent disease, and many individuals are unaware of their infections until the late stages of the disease. The impact of wider access to effective treatments and improved awareness of the disease on the number of infections and the number of patients who remain undiagnosed is not known in Canada. Such evidence can guide the development of strategies and interventions to reduce the burden of CHC and meet World Health Organization's (WHO) 2030 elimination targets. The purpose of this study is to use a back-calculation framework informed by provincial population-level health administrative data to estimate the prevalence of CHC and the proportion of cases that remain undiagnosed in the three most populated provinces in Canada: British Columbia (BC), Ontario and Quebec. We have conducted a population-based retrospective analysis of health administrative data for the three provinces to generate the annual incidence of newly diagnosed CHC cases, decompensated cirrhosis (DC), hepatocellular carcinoma (HCC) and HCV treatment initiations. For each province, the data were stratified in three birth cohorts: individuals born prior to 1945, individuals born between 1945 and 1965 and individuals born after 1965. We used a back-calculation modelling approach to estimate prevalence and the undiagnosed proportion of CHC. The historical prevalence of CHC was inferred through a calibration process based on a Bayesian Markov chain Monte Carlo (MCMC) algorithm. The algorithm constructs the historical prevalence of CHC for each cohort by comparing the model-generated outcomes of the annual incidence of the CHC-related health events against the data set of observed diagnosed cases generated in the retrospective analysis. The results show a decreasing trend in both CHC prevalence and undiagnosed proportion in BC, Ontario and Quebec. In 2018, CHC prevalence was estimated to be 1.23% (95% CI: .96%-1.62%), .91% (95% CI: .82%-1.04%) and .57% (95% CI: .51%-.64%) in BC, Ontario and Quebec respectively. The CHC undiagnosed proportion was assessed to be 35.44% (95% CI: 27.07%-45.83%), 34.28% (95% CI: 26.74%-41.62%) and 46.32% (95% CI: 37.85%-52.80%) in BC, Ontario and Quebec, respectively, in 2018. Also, since the introduction of new DAA treatment in 2014, CHC prevalence decreased from 1.39% to 1.23%, .97% to .91% and .65% to .57% in BC, Ontario and Quebec respectively. Similarly, the CHC undiagnosed proportion decreased from 38.78% to 35.44%, 38.70% to 34.28% and 47.54% to 46.32% in BC, Ontario and Quebec, respectively, from 2014 to 2018. We estimated that the CHC prevalence and undiagnosed proportion have declined for all three provinces since the new DAA treatment has been approved in 2014. Yet, our findings show that a significant proportion of HCV cases remain undiagnosed across all provinces highlighting the need to increase investment in screening. Our findings provide essential evidence to guide decisions about current and future HCV strategies and help achieve the WHO goal of eliminating hepatitis C in Canada by 2030.

Wigner transport in linear electromagnetic fields

Applying a Weyl–Stratonovich transform to the evolution equation of the Wigner function in an electromagnetic field yields a multidimensional gauge-invariant equation which is numerically very challenging to solve. In this work, we apply simplifying assumptions for linear electromagnetic fields and the evolution of an electron in a plane (two-dimensional transport), which reduces the complexity and enables to gain first experiences with a gauge-invariant Wigner equation. We present an equation analysis and show that a finite difference approach for solving the high-order derivatives allows for reformulation into a Fredholm integral equation. The resolvent expansion of the latter contains consecutive integrals, which is favorable for Monte Carlo solution approaches. To that end, we present two stochastic (Monte Carlo) algorithms that evaluate averages of generic physical quantities or directly the Wigner function. The algorithms give rise to a quantum particle model, which interprets quantum transport in heuristic terms.