Enhanced-Precision LHSMC of Electrical Circuit Considering Low Discrepancy

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.

Increment variance reduction techniques are add-ons to Monte Carlo (MC) simulations. They make MC simulations converging faster by repeating the number of simulations with an incremental rate derived from mathematical functions. Besides speeding up MC simulations, the major advantage of increment techniques is their ability to handle large numbers of simulations avoiding memory saturation and overflow which occur when a plain MC simulation is involved in the pricing of multi-name credit derivatives. A trend among authors pricing financial securities with MC simulation has been to choose Quasi-Monte Carlo (QMC) methods using deterministic sequences instead of MC methods involving pseudorandom generators such as congruential generator and Mersenne twister. The Increment family models circumvent the constraint of identifying the optimal QMC sequence to price a given security by using a common generator such as Matlab-LCG-Xor RNG and determining the optimal mathematical function of incrementation of MC simulations that will make the pricing of the security adequate. Market participants in need of selecting a reliable numerical method for pricing complex financial securities such as multi-name credit derivatives will find our paper appealing.

<abstract> <bold><sc>Abstract. </sc></bold>Performing a comprehensive sensitivity/uncertainty analysis is a valuable step in understanding and using a predictive hydrologic/water quality (H/WQ) model. This article applies one-factor-at-a-time (OAT) sensitivity analysis (SA) and first-order error analysis (FOEA)/Monte Carlo simulation with Latin hypercube sampling (LHS) uncertainty analysis techniques for evaluation of a complex, process-based water erosion prediction tool, the USDA Water Erosion Prediction Project (WEPP) model (version 2010.1). Assessment of the WEPP hillslope profile model on a Midwestern U.S. Miami silt loam soil for three cropping/management scenarios and three erosion process cases (as defined by topography) is described. WEPP model runoff, soil loss, and corn (Zea mays L.) yield output responses in the form of expected values and error variances were determined to illustrate model prediction uncertainty. The OAT SA showed that WEPP runoff and soil loss output responses were most sensitive to changes in the baseline effective hydraulic conductivity (K_b) and sand content. WEPP model corn yield output response was most sensitive to crop input parameters affecting the simulation of biomass development. The FOEA showed that the largest contributions to runoff, soil loss, and corn yield total error variance came from K_b and sand/clay content, K_b and baseline soil erodibility factors, and the biomass energy ratio of a crop and harvest index, respectively. The FOEA total variances presented in this study for runoff and soil loss were considerably larger than the corresponding Monte Carlo LHS simulation total variances. The Monte Carlo LHS total variance results were reasonable, making Monte Carlo LHS appear to be a better alternative for quantifying WEPP output response error variance. The Monte Carlo LHS soil loss output responses were also compared to Universal Soil Loss Equation (USLE) soil loss predictions. The USLE soil loss estimates were within the Monte Carlo LHS 90% prediction intervals for six of the nine cropping/management and erosion process cases. Results of this study illustrate the usefulness of combining SA and Monte Carlo LHS for providing detailed uncertainty analysis information for complex, physically based models such as WEPP.

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

We propose an efficient stratified sampling (SS) algorithm for estimating the bit error rate (BER) of a digital communication system. Our algorithm efficiently exploits the observations of an approximate, but usually fast, model of the system under investigation to drive a clever Monte Carlo (MC) estimation based on SS. The proposed method is faster than standard MC even at BER in the range 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> to 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5</sup> . Moreover, it is possible to evaluate the estimated standard deviation of the measured BER, such as in an MC simulation, so that it is possible to associate a confidence to the results. We test the algorithm both in a simple optical system distorted by group velocity dispersion (GVD) and in more complex differential quadrature phase shift keying (DQPSK) systems. In the last case, we measured computational savings up to 70% compared with standard MC.

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

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.

One of the main challenges facing designers of high-speed integrated circuits and interconnects is predicting the effect of the variability of geometrical and physical parameters on the circuit performance. Monte Carlo (MC) simulation has been traditionally used in commercial circuit and electromagnetic (EM) simulators for predicting the statistical distribution of circuit and system performance metrics. However, the high computational cost associated with the MC simulation is a significant limiting factor especially for large circuits.

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.

Analog behavioral equivalence boundary computation under the effect of process variations

A new methodology is proposed to mitigate the high computation cost required to derive accurate Monte Carlo (MC) based system matrix for tomographic image reconstruction. The strategy consists of taking advantage of the symmetries between the lines of response to increase the statistics of data collected for the determination of the system matrix coefficients. By using the rotation and axial symmetries of a cylindrical camera, the number of MC generated events can be reduced substantially without affecting the matrix coefficient accuracy. Moreover, using the GATE simulator list-mode saving capabilities for storing coincidence events, single events and/or single hits with all their relevant information, the MC simulation can be performed only once and system matrices for different system configurations be derived from the same simulation. Using for example Positron emission tomography (PET), the processing of the collected MC data can be fine tuned to the imaging system characteristics by setting accordingly the time and energy blurring, the detector efficiencies, the coincidence time window width and the energy thresholds. The system matrix can also include or exclude PET events like scatters and randoms. Furthermore, the system matrix can be computed for different image grids and basis functions without requiring a new MC simulation to be performed.

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.

In this chapter we deal with numerical computation of integrals over the domain ℝ s (s >1) with respect to a positive weight function. For one-dimensional integrals Gauss Hermite formulas compute integrals with respect to a Gaussian weight with quite high accuracy but for high-dimensional integrals the effort increases exponentially. For integrals over the s-dimensional unit cube probabilistic methods like Monte Carlo (MC) are not affected by this so-called “curse of dimensions,” but the convergence rate is rather poor. If the integration nodes are not pure random points but special deterministic point sequences (the method is called Quasi Monte Carlo (QMC) due to this fact) this rate can be significantly improved. These lowdiscrepancy sequences appear also in the computation of integrals over ℝ s .

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.

A fast Monte-Carlo method to predict failure probability of offshore wind turbine caused by stochastic variations in soil