Exploiting symmetries for exponential error reduction in path integral Monte Carlo

The path integral of a quantum system with an exact symmetry can be written as a sum of functional integrals each giving the contribution from quantum states with definite symmetry properties. We propose a strategy to compute each of them, normalized to the one with vacuum quantum numbers, by a Monte Carlo procedure whose cost increases power-like with the time extent of the lattice. This is achieved thanks to a multi-level integration scheme, inspired by the transfer matrix formalism, which exploits the symmetry and the locality in time of the underlying statistical system. As a result the cost of computing the lowest energy level in a given channel, its multiplicity and its matrix elements is exponentially reduced with respect to the standard path-integral Monte Carlo. We test the strategy with a one-dimensional harmonic oscillator, by computing the ratio of the parity odd over the parity even functional integrals and the two-point correlation function. The cost of the simulations scales as expected. In particular the effort for computing the lowest energy eigenvalue in the parity odd sector grows linearly with the time extent. At a fixed CPU time, the statistical error on the two-point correlation function is exponentially reduced with respect to the standard Monte Carlo procedure, and at large time distances it is lowered by many orders of magnitude.     

Numerical integration using Wang-Landau sampling

We report a new application of Wang-Landau sampling to numerical integration that is straightforward to implement. It is applicable to a wide variety of integrals without restrictions and is readily generalized to higher-dimensional problems. The feasibility of the method results from a reinterpretation of the density of states in statistical physics to an appropriate measure for numerical integration. The properties of this algorithm as a new kind of Monte Carlo integration scheme are investigated with some simple integrals, and a potential application of the method is illustrated by the evaluation of integrals arising in perturbation theory of quantum many-body systems.     

A polynomial approximation-based approach for chance-constrained optimization

We proposed a polynomial approximation-based approach to solve a specific type of chance-constrained optimization problem that can be equivalently transformed into a convex programme. This type of chance-constrained optimization is in great needs of many applications and most solution techniques are problem-specific. Our key contribution is to provide an all-purpose solution approach through Monte Carlo and establish the linkage between our obtained optimal solution with the true optimal solution. Our approach performs well because: First, our method controls approximation errors for both the function value and its gradient (or subgradient) at the same time. This is the primary advantage of our method in comparison to the commonly used finite difference method. Second, the approximation error is well bounded in our method and, with a properly chosen algorithm, the total computational complexity will be polynomial. We also address issues associated with Monte Carlo, such as discontinuity and nondifferentiability of the function. Thanks to fast-advancing computer hardware, our method would be increasingly appealing to businesses, including small businesses. We present the numerical results to show that our method with Monte Carlo will yield high-quality, timely, and stable solutions.     

Parallel Monte Carlo Driver (PMCD)—a software package for Monte Carlo simulations in parallel

Thanks to the dramatic decrease of computer costs and the no less dramatic increase in those same computer's capabilities and also thanks to the availability of specific free software and libraries that allow the set up of small parallel computation installations the scientific community is now in a position where parallel computation is within easy reach even to moderately budgeted research groups. The software package PMCD (Parallel Monte Carlo Driver) was developed to drive the Monte Carlo simulation of a wide range of user supplied models in parallel computation environments. The typical Monte Carlo simulation involves using a software implementation of a function to repeatedly generate function values. Typically these software implementations were developed for sequential runs. Our driver was developed to enable the run in parallel of the Monte Carlo simulation, with minimum changes to the original code that implements the function of interest to the researcher. In this communication we present the main goals and characteristics of our software, together with a simple study its expected performance. Monte Carlo simulations are informally classified as “embarrassingly parallel”, meaning that the gains in parallelizing a Monte Carlo run should be close to ideal, i.e. with speed ups close to linear. In this paper our simple study shows that without compromising the easiness of use and implementation, one can get performances very close to the ideal.     

Re‐Weighting Firefly Samples for Improved Finite‐Sample Monte Carlo Estimates

Samples with high contribution but low probability density, often called fireflies, occur in all practical Monte Carlo estimators and are part of computing unbiased estimates. For finite‐sample estimates, however, they can lead to excessive variance. Rejecting all samples classified as outliers, as suggested in previous work, leads to estimates that are too low and can cause undesirable artefacts. In this paper, we show how samples can be re‐weighted depending on their contribution and sampling frequency such that the finite‐sample estimate gets closer to the correct expected value and the variance can be controlled. For this, we first derive a theory for how samples should ideally be re‐weighted and that this would require the probability density function of the optimal sampling strategy. As this probability density function is generally unknown, we show how the discrepancy between the optimal and the actual sampling strategy can be estimated and used for re‐weighting in practice. We describe an efficient algorithm that allows for the necessary analysis of per‐pixel sample distributions in the context of Monte Carlo rendering without storing any individual samples, with only minimal changes to the rendering algorithm. It causes negligible runtime overhead, works in constant memory and is well suited for parallel and progressive rendering. The re‐weighting runs as a fast post‐process, can be controlled interactively and our approach is non‐destructive in that the unbiased result can be reconstructed at any time.     

毁伤概率评估的加权蒙特卡洛方法

由于经典蒙特卡洛方法的仿真效率不高,文中利用概率论和数理统计的基本原理,可以推导得到加权蒙特卡洛方法.加权蒙特卡罗方法不但能有效的缩小了样本方差,还能提高目标事件出现概率,相当于提高了样本的抽样效率,从而提高了仿真效率.与此同时,文中分别利用经典蒙特卡洛和加权蒙特卡洛这两种方法对典型目标进行了仿真计算,仿真计算最终证明了加权蒙特卡洛方法的仿真效率明显优于经典蒙特卡洛方法,能使仿真工作量成数量级的衰减.     

Cluster Monte Carlo and numerical mean field analysis for the water liquid-liquid phase transition

Using Wolff's cluster Monte Carlo simulations and numerical minimization within a mean field approach, we study the low temperature phase diagram of water, adopting a cell model that reproduces the known properties of water in its fluid phases. Both methods allow us to study the thermodynamic behavior of water at temperatures, where other numerical approaches - both Monte Carlo and molecular dynamics - are seriously hampered by the large increase of the correlation times. The cluster algorithm also allows us to emphasize that the liquid-liquid phase transition corresponds to the percolation transition of tetrahedrally ordered water molecules.     

蒙特卡洛方法在仿真中的应用

本文介绍了蒙特卡洛方法,一种利用随机数(或伪随机数)来解决许多类型计算问题的通用算法。首先描述了蒙特卡洛方法的基本原理,并且通过两个典型应用案例,讨论了蒙特卡洛方法的适用范围和使用条件,展示了该方法的优点,体现了该方法在解决高自由度问题方面的优势。     

Exploiting Visibility Correlation in Direct Illumination

The visibility function in direct illumination describes the binary visibility over a light source, e.g., an environment map. Intuitively, the visibility is often strongly correlated between nearby locations in time and space, but exploiting this correlation without introducing noticeable errors is a hard problem. In this paper, we first study the statistical characteristics of the visibility function. Then, we propose a robust and unbiased method for using estimated visibility information to improve the quality of Monte Carlo evaluation of direct illumination. Our method is based on the theory of control variates, and it can be used on top of existing state‐of‐the‐art schemes for importance sampling. The visibility estimation is obtained by sparsely sampling and caching the 4D visibility field in a compact bitwise representation. In addition to Monte Carlo rendering, the stored visibility information can be used in a number of other applications, for example, ambient occlusion and lighting design.     

Fast recall of state-history in kinetic Monte Carlo simulations utilizing the Zobrist key

We present a novel application of the Zobrist hashing method, known in the computer chess literature, to simulation of diffusional phase transformations in metal alloys. A history of previously visited states can be easily maintained, allowing very fast lookup of energies and transition rates calculated earlier in the simulation. The method has been applied to the simulation of a Fe-1at.%Cu system, with simple potentials and a transition rate for diffusional events approximated from the difference in internal energy between trial states. In this simple model at temperatures of 1073 K we find that 61.2% of the states considered during the simulation have been seen previously, and that this proportion rises to 85.1% at 773 K and even to 99.9% at 373 K. Rapid recall of these states reduces the computational time taken for the same sequence of atom-vacancy exchange moves by a factor of 6.3 at 773 K rising to over 100 at 373 K. We suggest that a similar speedup factor will be found using more sophisticated models of diffusion and that the method can, with small modifications, be applied to a wide range of kinetic Monte Carlo simulations of atomistic diffusion processes.     

Feed-back neural networks with discrete weights

We use the Monte Carlo Adaptation learning algorithm to design feed-back neural networks with discrete weights. The dynamic properties of these types of neural networks are investigated as a function of the states of weights. The numerical results of these networks show three phases: the “chaos phase,” the “pure memory phase” and the “mixture phase” in the parameter space. The maximum storage ratio for the “pure memory phase” increases with the increasing of the states of the weights, which is favorable for practical applications.     

基于蒙特卡罗方法的目标跟踪

为了更鲁棒和快速地进行目标跟踪,在基于粒子滤波的目标跟踪方法的启发下,提出了一种新的基于蒙特卡罗方法的目标跟踪方法。该方法首先运用蒙特卡罗技术对下一帧目标可能出现的位置和尺度进行抽样;然后计算各抽样与参考目标的相似度;最后通过估计目标状态来获得跟踪目标。实验表明,该方法无需目标运动信息,特别适用于目标灵活运动时的跟踪,与现有的算法相比,不仅算法实现简单,同时有较好的鲁棒性和通用性。     

Error in Monte Carlo, quasi-error in Quasi-Monte Carlo

While the Quasi-Monte Carlo method of numerical integration achieves smaller integration error than standard Monte Carlo, its use in particle physics phenomenology has been hindered by the absence of a reliable way to estimate that error. The standard Monte Carlo error estimator relies on the assumption that the points are generated independently of each other and, therefore, fails to account for the error improvement advertised by the Quasi-Monte Carlo method. We advocate the construction of an estimator of stochastic nature, based on the ensemble of pointsets with a particular discrepancy value. We investigate the consequences of this choice and give some first empirical results on the suggested estimators.     

Monte Carlo integration with subtraction

This paper investigates a class of algorithms for numerical integration of a function in d$d$ dimensions over a compact domain by Monte Carlo methods. We construct a histogram approximation to the function using a partition of the integration domain into a set of bins specified by some parameters. We then consider two adaptations: the first is to subtract the histogram approximation, whose integral we may easily evaluate explicitly, from the function and integrate the difference using Monte Carlo; the second is to modify the bin parameters in order to make the variance of the Monte Carlo estimate of the integral the same for all bins. This allows us to use Student’s t$t$-test as a trigger for rebinning, which we claim is more stable than the χ²${\chi }^2$ test that is commonly used for this purpose. We provide a program that we have used to study the algorithm for the case where the histogram is represented as a product of one-dimensional histograms. We discuss the assumptions and approximations made, as well as giving a pedagogical discussion of the myriad ways in which the results of any such Monte Carlo integration program can be misleading.     

Sewing algorithm

We present a procedure that in many cases enables the Monte Carlo sampling of states of a large system from the sampling of states of a smaller system. We illustrate this procedure, which we call the sewing algorithm, for sampling states from the transfer matrix of the two-dimensional Ising model.     

Connections between score matching, contrastive divergence, and pseudolikelihood for continuous-valued variables.

Score matching (SM) and contrastive divergence (CD) are two recently proposed methods for estimation of nonnormalized statistical methods without computation of the normalization constant (partition function). Although they are based on very different approaches, we show in this letter that they are equivalent in a special case: in the limit of infinitesimal noise in a specific Monte Carlo method. Further, we show how these methods can be interpreted as approximations of pseudolikelihood.     

Multi-electron systems in strong magnetic fields II: A fixed-phase diffusion quantum Monte Carlo application based on trial functions from a Hartree–Fock–Roothaan method

We present a quantum Monte Carlo application for the computation of energy eigenvalues for atoms and ions in strong magnetic fields. The required guiding wave functions are obtained with the Hartree–Fock–Roothaan code described in the accompanying publication (Schimeczek and Wunner, 2014). Our method yields highly accurate results for the binding energies of symmetry subspace ground states and at the same time provides a means for quantifying the quality of the results obtained with the above-mentioned Hartree–Fock–Roothaan method.     

Robust Monte Carlo Method for R&D Real Options Valuation

This paper is devoted to developing a robust numerical analysis of least squares Monte Carlo (LSM) in valuing R&D investment opportunities. As it is well known, R&D projects are characterized by sequential investments and therefore they can be considered as compound options involving a set of interacting American-type options. The basic Monte Carlo simulation takes a long time and it is computationally intensive and inefficient. In this context, LSM method is a powerful and flexible tool for capital budgeting decisions and for valuing R&D investments. In particular way, numerical tests are performed to examine the optimal choice of basis function and polynomial degree in terms of reduction of the execution time, accuracy and improvement in the simulation.     

量子退火算法研究进展

在数学和应用领域,量子退火算法是一类新的量子优化算法.不同于经典模拟退火算法利用热波动来搜寻问题的最优解,量子退火算法利用量子波动产生的量子隧穿效应来使算法摆脱局部最优,而实现全局优化.在已有的研究中,量子退火算法在某些问题上展现出良好的优化效果.系统地综述了量子退火算法的基本原理和近年来的主要研究进展,较为详细地介绍了几个主要的量子退火算法,对量子退火算法的优点和可能的不足进行了分析评述,并对今后的研究方向进行了展望.     

A fast approximate method for parameter sensitivity estimation in Monte Carlo structural reliability

In estimating the effect of a change in a random variable parameter on the (time-invariant) probability of structural failure estimated through Monte Carlo methods the usual approach is to carry out a duplicate simulation run for each parameter being varied. The associated computational cost may become prohibitive when many random variables are involved. Herein a procedure is proposed in which the numerical results from a Monte Carlo reliability estimation procedure are converted to a form that will allow the basic ideas of the first order reliability method to be employed. Using these allows sensitivity estimates of low computational cost to be made. Illustrative examples with sensitivities computed both by conventional Monte Carlo and the proposed procedure show good agreement over a range of probability distributions for the input random variables and for various complexities of the limit state function.