Increment Variance Reduction Techniques with an Application to Multi-name Credit Derivatives

The way of measuring diameter by use of measuring bow height and chord length is commonly adopted for the large diameter work piece. In the process of computing the diameter of large work piece, measurement uncertainty is an important parameter and is always employed to evaluate the reliability of the measurement results. Therefore, it is essential to present reliable methods to evaluate the measurement uncertainty, especially in precise measurement. Because of the limitations of low convergence and unstable results of the Monte-Carlo (MC) method, the quasi-Monte-Carlo (QMC) method is used to estimate the measurement uncertainty. The QMC method is an improvement of the ordinary MC method which employs highly uniform quasi random numbers to replace MC's pseudo random numbers. In the process of evaluation, first, more homogeneous random numbers (quasi random numbers) are generated based on Halton's sequence. Then these random numbers are transformed into the desired distribution random numbers. The desired distribution random numbers are used to simulate the measurement errors. By computing the simulation results, measurement uncertainty can be obtained. An experiment of cylinder diameter measurement and its uncertainty evaluation are given. In the experiment, the guide to the expression of uncertainty in measurement method, MC method, and QMC method are validated. The result shows that the QMC method has a higher convergence rate and more stable evaluation results than that of the MC method. Therefore, the QMC method can be applied effectively to evaluate the measurement uncertainty.

A quasi-Monte Carlo solver for thermal radiation in participating media

Capturing stochastic variabilities in pedestrian flows: A dynamic continuum modeling approach

Thermal-pulse method is a powerful tool for measuring space charge distributions in polymer films. The data analysis for thermal-pulse method involves the Fredholm integral equation of the first kind, which requires an appropriate numerical procedure to obtain a solution. Various numerical techniques, including scale transformation and regulation method, are proposed. Of those numerical methods, the scale transformation (ST) is the simplest and the most widely used method. However, it presents a high spatial resolution only near the sample surface. Monte Carlo (MC) method is one of the recently proposed ways to solve the equation numerically and has been successfully applied to the analysis of laser intensity modulation method data, which also involves the Fredholm integral equation of the first kind. In this paper we attempt to analyze thermal-pulse data in frequency domain with the MC method and discuss its effectiveness based on some numerical simulations. The simulation results indicate that the electric field profiles can be effectively extracted by the MC method. The computed profiles by the MC method consist well with the supposed distributions in the entire thickness of the sample, while the profiles reconstructed by the ST method fit very well to the supposed one at the vicinity of the target surface and distort sharply along the direction of the thermal pulse propagation in the sample bulk. On the other hand, the oscillations in the computed results by the MC method could deteriorate its accuracy in this study. The influence of noise level on the analysis based on the MC method is also tested by the use of the simulated data. The results show that the computed profiles would become more fluctuant as the noise level increases. This problem can be solved by selecting a larger value of tolerance during the singular value decomposition procedure. Thus, the value of tolerance is considered to be one of the key parameters in this algorithm, which is actually hard to determine. Additionally, the experimental data obtained from a polypropylene film under applied electric field are analyzed to illustrate the feasibility of MC method to be applied to the thermal-pulse experimental data. The results also show that the spatial accuracy by the MC method in the entire sample thickness is higher than by the ST method, which verifies that the MC method is more suitable for detecting the electric field distribution in the deep bulk of the sample. Owing to noise and error, the accuracy of MC calculation depends on the chosen tolerance value, which is now considered to be an obstacle in applying this method to the practical thermal-pulse measurement.

Property variations in a structure strongly impact the macroscopic mechanical performance as regions with lower strength will be prone to damage initiation or acceleration. Consideration of the variability in material property is critical for high-resolution simulations of damage initiation and propagation. While the recent progressive damage analyses consider randomness in property fields, accurately quantifying the uncertainty in damage measures remains computationally expensive. Stochastic damage analyses require extensive sampling of random property fields and numerous replications of the underlying nonlinear deterministic simulations. This paper demonstrates that a Quasi-Monte Carlo (QMC) method, which uses a multidimensional low discrepancy sobol sequence, is a computationally economical way to obtain the mean and standard deviations in cracks evolving in composites. An extended finite element method (XFEM) method with spatially random strength fields simulates the damage initiation and evolution in a model composite. We compared the number of simulations required for Monte Carlo (MC) and QMC techniques to measure the influence of input variability on the mean crack-length in an open-hole angle-ply tensile test. We conclude that the low discrepancy sampling and QMC technique converges substantially faster than traditional MC methods.

Analog behavioral equivalence boundary computation under the effect of process variations

We are interested in Monte Carlo (MC) methods for solving the diffusion equation: in the case of a constant diffusion coefficient, the solution is approximated by using particles and in every time step, a constant stepsize is added to or subtracted from the coordinates of each particle with equal probability. For a spatially dependent diffusion coefficient, the naive extension of the previous method using a spatially variable stepsize introduces a systematic error: particles migrate in the directions of decreasing diffusivity. A correction of stepsizes and stepping probabilities has recently been proposed and the numerical tests have given satisfactory results. In this paper, we describe a quasi-Monte Carlo (QMC) method for solving the diffusion equation in a spatially nonhomogeneous medium: we replace the random samples in the corrected MC scheme by low-discrepancy point sets. In order to make a proper use of the better uniformity of these point sets, the particles are reordered according to their successive coordinates at each time step. We illustrate the method with numerical examples: in dimensions 1 and 2, we show that the QMC approach leads to improved accuracy when compared with the original MC method using the same number of particles.

Background Small megavoltage photon fields are increasingly used in modern radiotherapy techniques such as stereotactic radiotherapy. Therefore, it is important to study the reliability of dosimetry in the small-field conditions. The IBA Razor Nano Chamber (Nano chamber) ionization chamber is particularly intended for small-field measurements. In this work, properties of the Nano chamber were studied with both measurements and Monte Carlo (MC) simulations. Material and methods The measurements and MC simulations were performed with 6 MV, 6 MV FFF and 10 MV FFF photon beams from the Varian TrueBeam linear accelerator. The source-to-surface distance was fixed at 100 cm. The measurements and MC simulations included profiles, percentage depth doses (PDD), and output factors (OF) in square jaw-collimated fields. The MC simulations were performed with the EGSnrc software system in a large water phantom. Results The measured profiles and PDDs obtained with the Nano chamber were compared against IBA Razor Diode, PTW microDiamond and the PTW Semiflex ionization chamber. These results indicate that the Nano chamber is a high-resolution detector and thus suitable for small field profile measurements down to field sizes 2 × 2 cm2 and appropriate for the PDD measurements. The field output correction factors and field OFs were determined according to TRS-483 protocol In the 6 MV FF and FFF beams, the determined correction factors were within 1.2% for the field sizes of 1 × 1 cm2–3 × 3 cm2 and the experimental and MC defined field output factors showed good agreement. Conclusion The Nano chamber with its small cavity volume is a potential detector for the small-field dosimetry. In this study, the properties of this detector were characterized with measurements and MC simulations. The determined correction factors are novel results for the NC in the TrueBeam fields.

In these days, Monte Carlo (MC) simulation is a method that can calculate the radiation dose that occurs in an environment in the most accurate way. The correct measurement of the dose occurring on the patient’s surface is of great importance to estimate the reactions that may occur on the patient’s skin. This importance encouraged us to do this study. The aim of this study is to determine buildup region and surface doses using MC simulation and to compare them with results of the parallel plane ion chamber and Treatment Planning System (TPS) measurements for 6-MV photon beams. Surface doses normalized to the maximum dose for the parallel plane ion chamber, MC simulation, fast photon (FP) algorithm, and collapsed cone convolution superposition (CC) algorithm are 13.6%, 30.28%, 0%, and 27.33%, respectively. The CC algorithm and parallel plane ion chamber measurements are compatible with MC simulation but the FP algorithm has calculated the dose less to a depth of 0.8 cm. Measuring the surface dose and the doses in the buildup region is of great importance in terms of accurately predicting the complications that may occur in the patient’s skin and taking precautions early. Using some methods and correction factors, the surface dose and the doses that may occur in the buildup region can be accurately calculated. It is recommended not to use the FP algorithm for stereotactic body radiation therapy and intensity-modulated radiation therapy treatments, as it cannot calculate doses correctly in the buildup region and surface.

As energy demand increases and global warming progresses, expectations for fuel cells, which generate electricity through chemical reactions between hydrogen and oxygen, are rising. There are various types of fuel cells, which are classified according to the electrolyte. Polymer electrolyte fuel cells (PEFCs) are expected to be used in stationary power sources for home use and fuel cell vehicles because of their low operating temperature, short start-up time, and ease of miniaturization. The cost has been an issue in the widespread use of fuel cells. The amount of platinum used in the catalyst layer (CL) must be reduced to reduce costs. However, it increases the current density per unit area of platinum. There are three factors that contribute to voltage drop: ohmic polarization, activation polarization, and diffusion polarization. At high current densities, the voltage drop due to diffusion polarization is dominant. The main cause of diffusion polarization is the transport resistance of oxygen molecules in the CL, and improving the oxygen transport properties in the CL will lead to lower cost PEFCs. In this study, we analyzed the transport properties of oxygen molecules in the CLs of PEFCs using the Monte Carlo (MC) method. The effect of oxygen scattering on the overall oxygen transport properties was investigated by considering the behavior on the surface of the ionomer film (surface diffusion) into the MC method, which simulates the overall transport in the PEFC CL. The objective of this study is to reproduce accurate oxygen transport by comparing the results of MC and molecular dynamics (MD) methods. First, we used the MC method to analyze oxygen scattering phenomena. In this study, we used the 3D structure data of CL in the MC simulations. The three-dimensional structure of the sample was reconstructed from sequential slice images. Molecules were assumed to reflect specularly on the boundaries of the simulation domain. Oxygen molecules were randomly placed in the simulation system. The time step was set at 5 ps and the simulation time was set at 4 μs. In order to examine the effect of surface diffusion on the transport properties of oxygen molecules, the simulation was performed using various conditions of surface diffusion. We calculated the effective diffusion coefficient of oxygen in the CL as the transport property of oxygen molecules in the MC method. As the surface diffusion coefficient increases, the effective diffusion coefficient of oxygen increases. When the surface diffusion coefficient is small, the effective diffusion coefficient of oxygen increases due to a decrease in the time constant, which determines the probability distribution of the surface residence time. However, when the surface diffusion coefficient is large, a different trend emerges. As the time constant increases, the effective diffusion coefficient has a peak value and then decreases. Next, to verify the model of surface diffusion in the MC method, we analyzed the scattering phenomenon of oxygen molecules in a simulation system that simulates the pores in the CL using the MD method. We modeled the nano pores in the CL as a slit between the ionomer walls in the MD method. Nafion with the equivalent weight (EW) of 1146 was used as polymer model. The number of Nafion chains is set at 4, and the water content λ is set at 7. Three-dimensional periodic boundary conditions were applied to the simulation domain. After the system was equilibrated, 1 ns of NVT simulation was performed for production. The temperature was set at 297 K. The behavior of oxygen molecules impacting on the ionomer wall was analyzed. We analyzed the behavior of oxygen molecules colliding with the ionomer membrane using the MD method. The average residence time of surface diffusion on the ionomer thin film was of the order of 10-10s. When the calculation results were compared with the MC results, it was found that the average residence time obtained by the MD method was the value where the surface diffusion coefficient affects much on the effective diffusion coefficient. Comparing the results of the MD method with the surface diffusion model used in the MC method, there are some discrepancies between the results and the model. For example, the MC method used a model in which oxygen molecules reflect diffusively on the ionomer surface. However, in the MD method, it was found that oxygen molecules tend to reflect in the direction of travel (Fig. 1). Therefore, a more accurate model of the MC method needs to be developed in the future. Figure 1

High-dimensional multivariate integration is a difficult problem for which the Monte Carlo method is often the only viable approach. This method provides an unbiased estimator of the integral, together with a probabilistic error estimate (e.g., in the form of a confidence interval). The aim of randomized quasi-Monte Carlo (QMC) methods is to provide lower-variance unbiased estimators, also with error estimates. This talk will concentrate on one class of randomized QMC methods: randomized lattice rules.We will explain how these methods fit into QMC methods in general and why they are interesting, how to choose their parameters, and how they can be used for medium and large-scale simulations. Numerical examples will be given to illustrate their effectiveness.

For the approximation of multidimensional integrals, two types of methods are widely used. Monte Carlo (MC) methods are the best known and require the use of a pseudorandom generator. Quasi-Monte Carlo (QMC) methods use low discrepancy point sets and are deterministic. The idea is to use points that are more regularly distributed over the integration space than random points. The best known methods to achieve this are the lattice rules and (t,s) sequences (or (t,m,s) nets) (A.B. Owen, 1998; H. Niederreiter; 1992; I.H. Sloan and S. Joe, 1994). The paper compares Monte Carlo methods, lattice rules, and other low discrepancy point sets on the problem of evaluating Asian options. The combination of these methods with variance reduction techniques is also explored.

Purpose: To create a Monte Carlo (MC) model of an ocular proton therapy device, or ocular nozzle, that can predict relative doses to within 3% or 0.5 mm, D/MU values, and be used to develop and commission an analytical algorithm to predict doses and D/MU values to within similar criteria. Methods and Materials: Depth dose profiles and D/MU values were measured in eight ocular proton therapy fields including four unmodulated and four modulated beams using a parallel‐plane ionization chamber in water. Crossfield profiles were measured at three or more depths for each of the modulated beams using film. A MC model of the ocular nozzle was constructed using the MCNPX radiation transport code. After benchmark tests against the measurements, the MC model was used to provide dosimetric data needed to develop a broad‐beam algorithm capable of predicting relative and absolute doses and to test the algorithm's predictions more comprehensively than could be afforded with measured data. Specifically, MC and analytical predictions were calculated and compared in two‐dimensions in a water phantom and in an anthropomorphic model of the eye. Results: Relative depth dose profiles between MC and measurements agreed to within 3% or 0.5 mm. MC predicted D/MU values agreed to within 1% of the measured values. Similarly, the analytical algorithm accurately predicted dose distributions where the predictions were generally within 3% or 0.5 mm of measured and simulated values at water depth > 6 mm. At shallower depths, the analytical model underestimated the dose by approximately 3% to 6%, which MC simulations revealed was due to edge‐scattered protons from the nozzle's collimators. Conclusion: The MC model accurately predicted dose distributions and D/MU values in ocular proton therapy beams. Furthermore, the present work demonstrates the value of the MC method for developing and testing contemporary analytical dose algorithms.

We combine molecular dynamics (MD) and Monte Carlo (MC) simulations to study the properties of gas molecules confined between two hard walls of a microchannel or nanochannel. The coupling between MD and MC simulations is introduced by performing MD near the boundaries for accuracy and MC in the bulk because of the low computational cost. We characterize the influence of different densities and molecule sizes on the equilibrium properties of the gas in the microchannel. The effect of the particle size on the simulation results is very small in the case of a dilute gas and increases with the density. The hybrid MD-MC simulation method is validated by comparing the results for density and temperature profiles with those of pure MD and pure MC simulations. These results compare well for pure MD and pure MC, as well as hybrid MD-MC, both in the bulk and near the boundaries, when hard-sphere interactions are used. When Lennard-Jones potentials are used to accurately model the interactions between the gas and wall molecules instead, the results of pure MD simulations differ significantly from the pure MC simulations near the boundaries, but the results of the hybrid method compare well with the pure MD results near the wall, and with the pure MC and pure MD results in the middle of the channel. The hybrid method also very accurately simulates the interface between the MD and MC simulation domains. Comparisons between MD, MC, and hybrid MD-MC computational costs are outlined. The speedup when using 50% of the domain for MD simulations and 50% for MC simulations is very small compared to pure MD simulations times, but this speedup increases drastically for more realistic situations where the region near the wall is small compared to the bulk region.

Purpose: For microdosimetric calculation event-by-event Monte Carlo (MC) simulation is considered the most accurate, but it is very time-consuming. In this work we present an event-by-event MC simulation of low energy electron and proton for accelerated microdosimetric MC simulations on a graphic processing unit (GPU). Methods: The MC simulation of particle was implemented in C and executed on a multi-core CPU, and a commercially available general purpose GPU using the compute unified device architecture (CUDA). Additionally, a hybrid implementation scheme was realized by employing OpenMP and CUDA so that both GPU and multi-core CPU were utilized simultaneously. The two implementation schemes have been tested and compared with the sequential single threaded MC simulation on the CPU. A calculation time comparison was established on the speed-up for a set of benchmarking cases of electron and proton. Results: Dosimetric results were obtained with both the parallel and serial MC codes. A GPU over CPU speed-up of 67.2 and 19.2 times was achieved for 300 eV and 2 keV electron tracks respectively. For proton tracks, the GPU-based code was approximately 5 times faster than the CPU-single-thread code. By incorporating a multi-core CPU and running the MC code simultaneously on the GPU and CPU, an increase of 2%-7% and 20% in the speedup was noticed for electrons and protons tracks respectively. A good dosimetric agreement between the CPU- and GPU-based MC methods was observed for both electrons and proton. Conclusions: A GPU-based MC method for microdosimetric calculations of low energy electron and proton has been presented. The results indicate the capability of our GPU-based implementation for accelerated MC simulations of both electron and proton without loss of accuracy. Lastly, the potential of a hybrid approach by utilizing simultaneously a GPU and a multi-core CPU for further acceleration of MC microdosimetric calculations has been demonstrated.