Monte Carlo Variance Reduction Research Articles

Investigation of the anisotropic distribution of microdosimetric quantities in the vicinity of X-ray-irradiated gold nanoparticles

High-Z nanoparticles incorporated in tumors have great potential to enhance radiotherapy treatments by increasing the absorbed radiation dose in the vicinity of the nanoparticle. However, current Monte Carlo simulations estimate dose enhancement metrics in 1D, neglecting the anisotropy of the energy deposition around the nanoparticle. This could have radiobiological consequences and affect treatment accuracy. Accurate knowledge of the spatial distribution of the dose or other microdosimetric quantities is necessary for predicting radiobiological outcomes using biophysical models. In this study, we utilized a new combination of two Monte Carlo Variance Reduction Techniques (VRTs) to estimate microdosimetric quantities at the nanoscale, taking into account the anisotropy of the microdosimetric distributions. Our results show significant anisotropy in multi-events frequency-mean specific energy, as well as in microdosimetric distributions. These findings highlight the importance of considering anisotropy in microdosimetric distributions when predicting radiobiological outcomes and suggest that current 1D simulations may not accurately represent the real microdosimetric effects. Our work contributes to the development of multiscale dosimetry approaches for radiotherapy treatments with high-Z nanoparticles.

Batching Adaptive Variance Reduction

Adaptive Monte Carlo variance reduction is an effective framework for running a Monte Carlo simulation along with a parameter search algorithm for variance reduction, whereas an initialization step is required for preparing problem parameters in some instances. In spite of the effectiveness of adaptive variance reduction in various fields of application, the length of the preliminary phase has often been left unspecified for the user to determine on a case-by-case basis, much like in typical sequential frameworks. This uncertain element may possibly be even fatal in realistic finite-budget situations, since the pilot run may take most of the budget, or possibly use up all of it. To unnecessitate such an ad hoc initialization step, we develop a batching procedure in adaptive variance reduction, and provide an implementable formula of the learning rate in the parameter search which minimizes an upper bound of the theoretical variance of the empirical batch mean. We analyze decay rates of the minimized upper bound towards the minimal estimator variance with respect to the predetermined computing budget, and provide convergence results as the computing budget increases progressively when the batch size is fixed. Numerical examples are provided to support theoretical findings and illustrate the effectiveness of the proposed batching procedure.

Reliability assessment methods to address fast transient of reactor core

In order to enhance the safety of new advanced reactors, reliability based approach to the design of thermal hydraulic system becomes necessary. In this work, “load exceeds capacity” based approach of structural reliability analysis is employed and probability of failure of the system was then assessed in terms of a limit state function while probabilistic measure of limit state function violation is performed through different methods of reliability assessment. Here, we have focused on TRIGA core subjected to reactivity initiated fast transient. Initially, response surface design method has been used for approximating true failure surface, and then FORM-SORM analysis has been carried out. But, due to non linearity involved with failure surface, there have been noticed instability in FORM-SORM implementation. Later, directional simulation approach of Monte Carlo variance reduction techniques has been employed to illustrate such fast transient. In the investigation, there have been several aspects considered and in each case directional simulation method has shown its ability to give valid results. Hence, the method could be recommended as a viable and efficient scheme to solve even fast transient problem in design and analysis of any reactor.

Accelerated radiation transport modeling techniques for pencil beam computed tomography using gamma rays

Monte Carlo radiation transport modeling studies were performed for a compact, and high-resolution gamma-ray computed tomography system designed for imaging irradiated nuclear fuel. The system comprises a 60Co source – chosen for its highly penetrating 1173 keV and 1332 keV gamma rays – a pair of high-aspect-ratio pencil beam collimators, and an inorganic scintillator detector. Two acceleration methods are proposed to rapidly model a transmission type gamma-ray tomography system. The first, a variance reduction technique, is based on performing Monte Carlo simulations with a monodirectionally-biased source, sampled from a characteristic sub-volume of the full source volume. The second acceleration method is based on the deterministic calculations using the Beer–Lambert law and detector response characteristics. Comparison of simulations using acceleration approaches with analog simulations of the fully isotropic, full-volume equivalent, show that the Monte Carlo variance reduction technique gives quantitatively accurate predictions for large collimator aspect ratios while the deterministic calculations are semi-quantitative but converge close to the correct result as the collimator aspect ratio increases. As such, these techniques can be used to reduce the computational cost in generating simulated radiographs and tomographs by several orders of magnitude. Experimental validation efforts are currently underway and will be demonstrated in future work.

Progress on neutronic analysis for CFETR

The neutron-induced irradiation field is a key problem in a fusion reactor related to nuclear responses, shielding design, nuclear safety and thermo-hydraulic analysis. To support the system design of the China Fusion Engineering Test Reactor (CFETR), a comprehensive analysis of the irradiation field has been conducted in support of many newly developed advanced tools. This paper first summarizes recent progress on the related effort to develop neutronics code including the geometry conversion tool cosVMPT, Monte Carlo variance reduction technology and ‘on-the-fly’ global variance reduction. Such developed tools have been fully validated and applied to the CFETR nuclear analysis. The neutron irradiation has been evaluated on the CFETR water-cooled ceramic breeder blanket, divertor, vacuum vessel (VV), superconductive coils and four kinds of heating systems: electron cyclotron resonance heating, ion cyclotron resonance heating, low hybrid wave and neutral beam injection. The nuclear responses of tritium breeding ratio, heating, irradiation damage and the hydrogen/helium production rate of material have been analyzed. In the case of neutron damage and overheating, deposition on the superconductive coils and VV, the interface and the shielding design among heating systems, and the blanket and other systems, has been initialized. The results show the shielding design can meet the requirement of coil and VV after several iterated neutronics calculations.

Monte Carlo Variance Reduction and American Option Exercise Strategies

Monte Carlo Variance Reduction and American Option Exercise Strategies

Comparison of a semi-analytic variance reduction technique to classical Monte Carlo variance reduction techniques for high aspect ratio pencil beam collimators for emission tomography applications

A semi-analytic variance reduction technique developed for collimated gamma emission tomography problems was compared to classic Monte Carlo variance reduction techniques within the Monte Carlo N Particle Transport (MCNP) code. In the semi-analytic technique, a computationally efficient, non-analog, monodirectional source biased Monte Carlo simulation is first performed. Analytical expressions or empirical values are then used to correct for solid angle and field-of-view effects introduced by the non-analog source definition. This variance reduction technique was compared with deterministic transport sphere (DXTRAN) and geometry splitting variance reduction schemes to determine the accuracy and computational savings of each technique relative to an analog pulse height tally (F8 tally) at 1, 3, 5, and 10 mm collimator aperture radii. For radii 0.5 mm and 0.3 mm a DXTRAN sphere was used in place of an analog F8 tally, due to large particle history demands, to analyze the accuracy of the semi-analytic variance reduction technique. The computational savings and accuracy were evaluated for six to seven photopeaks depending on the method used. The monodirectional source biasing technique overestimated the count rates by approximately 9%–19% when the radius is less than 3 mm, but the technique overestimated by a factor of 2 to 7, when the radius is greater than or equal to 3 mm. The monodirectionally source biased technique offered computational saving factors on the order of 108-1013 over 1-10 mm collimator radii studied. DXTRAN and geometry splitting methods yielded higher accuracy, but computational savings range from approximately 0.13 to 2.2 and 0.07 to 2.9, respectively indicating marginal improvement at best.

A hybrid Monte Carlo acceleration method of pricing basket options based on splitting

Pricing basket options has always been one of the key problems in financial engineering because of high dimensionality and low convergence rate. This paper proposes a hybrid Monte Carlo variance reduction method for pricing basket options. First, by splitting the payoff of the basket option into two parts, we can price basket options by value the two parts respectively. The first part has a closed-form expectation formula, the second part can be considered as a small probability event. To reduce variance for simulating the second part, the conditional Monte Carlo (CMC) method combined with the importance sampling(IS) method is adapted. Because these two methods are all effective to deal with small probability events. For IS method, it is a challenge to compute the optimal parameters with as little computational cost as possible. Therefore, an efficient prediction–correction(PC) iteration algorithm based on moment estimation is proposed to determine the optimal parameters in the importance sampling method. Some theoretical analyses for the existence and uniqueness of the optimal parameters in the IS method and the convergence of the PC method are also given. Numerical results show that the hybrid variance reduction method has great variance reduction effect and PC iteration algorithm can save a lot of computing costs comparing with the traditional Newton’s iteration method.

Multilevel rejection sampling for approximate Bayesian computation

Likelihood-free methods, such as approximate Bayesian computation, are powerful tools for practical inference problems with intractable likelihood functions. Markov chain Monte Carlo and sequential Monte Carlo variants of approximate Bayesian computation can be effective techniques for sampling posterior distributions in an approximate Bayesian computation setting. However, without careful consideration of convergence criteria and selection of proposal kernels, such methods can lead to very biased inference or computationally inefficient sampling. In contrast, rejection sampling for approximate Bayesian computation, despite being computationally intensive, results in independent, identically distributed samples from the approximated posterior. An alternative method is proposed for the acceleration of likelihood-free Bayesian inference that applies multilevel Monte Carlo variance reduction techniques directly to rejection sampling. The resulting method retains the accuracy advantages of rejection sampling while significantly improving the computational efficiency.

Analysis of inconsistent source sampling in monte carlo weight-window variance reduction methods

Abstract The application of Monte Carlo (MC) to large-scale fixed-source problems has recently become possible with new hybrid methods that automate generation of parameters for variance reduction techniques. Two common variance reduction techniques, weight windows and source biasing, have been automated and popularized by the consistent adjoint-driven importance sampling (CADIS) method. This method uses the adjoint solution from an inexpensive deterministic calculation to define a consistent set of weight windows and source particles for a subsequent MC calculation. One of the motivations for source consistency is to avoid the splitting or rouletting of particles at birth, which requires computational resources. However, it is not always possible or desirable to implement such consistency, which results in inconsistent source biasing. This paper develops an original framework that mathematically expresses the coupling of the weight window and source biasing techniques, allowing the authors to explore the impact of inconsistent source sampling on the variance of MC results. A numerical experiment supports this new framework and suggests that certain classes of problems may be relatively insensitive to inconsistent source sampling schemes with moderate levels of splitting and rouletting.

Improved algorithms and coupled neutron-photon transport for auto-importance sampling method**Supported by the subject of National Science and Technology Major Project of China (2013ZX06002001-007, 2011ZX06004-007) and National Natural Science Foundation of China (11275110, 11375103)

The Auto-Importance Sampling (AIS) method is a Monte Carlo variance reduction technique proposed for deep penetration problems, which can significantly improve computational efficiency without pre-calculations for importance distribution. However, the AIS method is only validated with several simple examples, and cannot be used for coupled neutron-photon transport. This paper presents improved algorithms for the AIS method, including particle transport, fictitious particle creation and adjustment, fictitious surface geometry, random number allocation and calculation of the estimated relative error. These improvements allow the AIS method to be applied to complicated deep penetration problems with complex geometry and multiple materials. A Completely coupled Neutron-Photon Auto-Importance Sampling (CNP-AIS) method is proposed to solve the deep penetration problems of coupled neutron-photon transport using the improved algorithms. The NUREG/CR-6115 PWR benchmark was calculated by using the methods of CNP-AIS, geometry splitting with Russian roulette and analog Monte Carlo, respectively. The calculation results of CNP-AIS are in good agreement with those of geometry splitting with Russian roulette and the benchmark solutions. The computational efficiency of CNP-AIS for both neutron and photon is much better than that of geometry splitting with Russian roulette in most cases, and increased by several orders of magnitude compared with that of the analog Monte Carlo.

Multilevel Ensemble Transform Particle Filtering

This paper extends the multilevel Monte Carlo variance reduction technique to nonlinear filtering. In particular, multilevel Monte Carlo is applied to a certain variant of the particle filter, the ensemble transform particle filter (EPTF). A key aspect is the use of optimal transport methods to re-establish correlation between coarse and fine ensembles after resampling; this controls the variance of the estimator. Numerical examples present a proof of concept of the effectiveness of the proposed method, demonstrating significant computational cost reductions (relative to the single-level ETPF counterpart) in the propagation of ensembles.

Randomisation and recursion methods for mixed-exponential Lévy models, with financial applications

We develop a new Monte Carlo variance reduction method to estimate the expectation of two commonly encountered path-dependent functionals: first-passage times and occupation times of sets. The method is based on a recursive approximation of the first-passage time probability and expected occupation time of sets of a Lévy bridge process that relies in part on a randomisation of the time parameter. We establish this recursion for general Lévy processes and derive its explicit form for mixed-exponential jump-diffusions, a dense subclass (in the sense of weak approximation) of Lévy processes, which includes Brownian motion with drift, Kou's double-exponential model, and hyperexponential jump-diffusion models. We present a highly accurate numerical realisation and derive error estimates. By way of illustration the method is applied to the valuation of range accruals and barrier options under exponential Lévy models and Bates-type stochastic volatility models with exponential jumps. Compared with standard Monte Carlo methods, we find that the method is significantly more efficient.

Randomisation and recursion methods for mixed-exponential Lévy models, with financial applications

We develop a new Monte Carlo variance reduction method to estimate the expectation of two commonly encountered path-dependent functionals: first-passage times and occupation times of sets. The method is based on a recursive approximation of the first-passage time probability and expected occupation time of sets of a Lévy bridge process that relies in part on a randomisation of the time parameter. We establish this recursion for general Lévy processes and derive its explicit form for mixed-exponential jump-diffusions, a dense subclass (in the sense of weak approximation) of Lévy processes, which includes Brownian motion with drift, Kou's double-exponential model, and hyperexponential jump-diffusion models. We present a highly accurate numerical realisation and derive error estimates. By way of illustration the method is applied to the valuation of range accruals and barrier options under exponential Lévy models and Bates-type stochastic volatility models with exponential jumps. Compared with standard Monte Carlo methods, we find that the method is significantly more efficient.

Monte Carlo simulation of prompt γ-ray emission in proton therapy using a specific track length estimator

A Monte Carlo (MC) variance reduction technique is developed for prompt-γ emitters calculations in proton therapy. Prompt-γ emitted through nuclear fragmentation reactions and exiting the patient during proton therapy could play an important role to help monitoring the treatment. However, the estimation of the number and the energy of emitted prompt-γ per primary proton with MC simulations is a slow process. In order to estimate the local distribution of prompt-γ emission in a volume of interest for a given proton beam of the treatment plan, a MC variance reduction technique based on a specific track length estimator (TLE) has been developed. First an elemental database of prompt-γ emission spectra is established in the clinical energy range of incident protons for all elements in the composition of human tissues. This database of the prompt-γ spectra is built offline with high statistics. Regarding the implementation of the prompt-γ TLE MC tally, each proton deposits along its track the expectation of the prompt-γ spectra from the database according to the proton kinetic energy and the local material composition. A detailed statistical study shows that the relative efficiency mainly depends on the geometrical distribution of the track length. Benchmarking of the proposed prompt-γ TLE MC technique with respect to an analogous MC technique is carried out. A large relative efficiency gain is reported, ca. 105.

Shutdown Dose Rate Analysis Using the Multi-Step CADIS Method

The Multi-Step Consistent Adjoint Driven Importance Sampling (MS-CADIS) hybrid Monte Carlo (MC)/deterministic radiation transport method was proposed to speed up the shutdown dose rate (SDDR) neutron MC calculation using an importance function that represents the neutron importance to the final SDDR. In this work, the MS-CADIS method was applied to the ITER SDDR benchmark problem. The MS-CADIS method was also used to calculate the SDDR uncertainty resulting from uncertainties in the MC neutron calculation and to determine the degree of undersampling in SDDR calculations because of the limited ability of the MC method to tally detailed spatial and energy distributions. The analysis that used the ITER benchmark problem compared the efficiency of the MS-CADIS method to the traditional approach of using global MC variance reduction techniques for speeding up SDDR neutron MC calculation. Compared to the standard Forward-Weighted-CADIS (FW-CADIS) method, the MS-CADIS method increased the efficiency of the SDDR neutron MC calculation by 69%. The MS-CADIS method also increased the fraction of nonzero scoring mesh tally elements in the space-energy regions of high importance to the final SDDR.

Improved Monte Carlo Variance Reduction for Space and Energy Self-Shielding

Continued demand for accurate and computationally efficient transport methods to solve optically thick, fixed-source transport problems has inspired research on variance-reduction (VR) techniques for Monte Carlo (MC). Methods that use deterministic results to create VR maps for MC constitute a dominant branch of this research, with Forward Weighted–Consistent Adjoint Driven Importance Sampling (FW-CADIS) being a particularly successful example. However, locations in which energy and spatial self-shielding are combined, such as thin plates embedded in concrete, challenge FW-CADIS. In these cases the deterministic flux cannot appropriately capture transport behavior, and the associated VR parameters result in high variance in and following the plate.This work presents a new method that improves performance in transport calculations that contain regions of combined space and energy self-shielding without significant impact on the solution quality in other parts of the problem. This method is based on FW-CADIS and applies a Resonance Factor correction to the adjoint source. The impact of the Resonance Factor method is investigated in this work through an example problem. It is clear that this new method dramatically improves performance in terms of lowering the maximum 95% confidence interval relative error and reducing the compute time. Based on this work, we recommend that the Resonance Factor method be used when the accuracy of the solution in the presence of combined space and energy self-shielding is important.

Using the Monte Carlo method to solve integral equations using a modified control variate

One of the numerical methods for solving definite integrals is Monte Carlo (MC) randomized simulation. In this paper we intend to apply a Monte Carlo variance reduction technique to solve the linear Fredholm integral equations, which is based on a modified control variate. Taylor expansion to each subinterval of the objective function is regarded as the modified control variate. And thus we solve the linear Fredholm integral equations of the second kind more accurately. One of the main advantage of the proposed method is reducing the problem caused by the linear system of equations.

Novel hybrid Monte Carlo/deterministic technique for shutdown dose rate analyses of fusion energy systems

The rigorous 2-step (R2S) computational system uses three-dimensional Monte Carlo transport simulations to calculate the shutdown dose rate (SDDR) in fusion reactors. Accurate full-scale R2S calculations are impractical in fusion reactors because they require calculating space- and energy-dependent neutron fluxes everywhere inside the reactor. The use of global Monte Carlo variance reduction techniques was suggested for accelerating the R2S neutron transport calculation. However, the prohibitive computational costs of these approaches, which increase with the problem size and amount of shielding materials, inhibit their ability to accurately predict the SDDR in fusion energy systems using full-scale modeling of an entire fusion plant. This paper describes a novel hybrid Monte Carlo/deterministic methodology that uses the Consistent Adjoint Driven Importance Sampling (CADIS) method but focuses on multi-step shielding calculations. The Multi-Step CADIS (MS-CADIS) methodology speeds up the R2S neutron Monte Carlo calculation using an importance function that represents the neutron importance to the final SDDR. Using a simplified example, preliminary results showed that the use of MS-CADIS enhanced the efficiency of the neutron Monte Carlo simulation of an SDDR calculation by a factor of 550 compared to standard global variance reduction techniques, and that the efficiency enhancement compared to analog Monte Carlo is higher than a factor of 10,000.

Variance reduction techniques for estimation of integrals over a set of branching trajectories

Monte Carlo variance reduction techniques within the supertrack approach are justified as applied to estimating non-Boltzmann tallies equal to the mean of a random variable defined on the set of all branching trajectories. For this purpose, a probability space is constructed on the set of all branching trajectories, and the unbiasedness of this method is proved by averaging over all trajectories. Variance reduction techniques, such as importance sampling, splitting, and Russian roulette, are discussed. A method is described for extending available codes based on the von Neumann-Ulam scheme in order to cover the supertrack approach.