Symmetries and exponential error reduction in Yang-Mills theories on the lattice

The partition function of a quantum field theory with an exact symmetry can be decomposed into a sum of functional integrals each giving the contribution from states with definite symmetry properties. The composition rules of the corresponding transfer matrix elements can be exploited to devise a multi-level Monte Carlo integration scheme for computing correlation functions whose numerical cost, at a fixed precision and at asymptotically large times, increases power-like with the time extent of the lattice. As a result the numerical effort is exponentially reduced with respect to the standard Monte Carlo procedure. We test this strategy in the SU(3) Yang-Mills theory by evaluating the relative contribution to the partition function of the parity odd states.     

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.     

The stationary phase auxiliary field Monte Carlo method: a new strategy for reducing the sign problem of auxiliary field methodologies

In this paper we focus on a new computational procedure, which permits an efficient calculation within the classical auxiliary field methodology. As has been previously reported, the method suffers from a sign problem, typically encountered in methodologies based on a field-theoretical approach. To ameliorate its statistical convergence, the efforts have so far exclusively been concentrated on the development of efficient analytical integral transformation techniques, such as the method of Gaussian equivalent representation of Efimov et al. In the present work we reformulate the classical auxiliary field methodology according to the concepts of the stationary phase Monte Carlo method of Doll et al., a numerical strategy originally developed for the simulation with real-time path integrals. The procedure, which is here employed for the first time for auxiliary field computation, utilizes an importance sampling strategy, to identify the regions of configuration space that contribute most strongly to the functional integral averages. Its efficiency is here compared to the method of Gaussian equivalent representation.     

A transmission matrix model for ion beam energy deposition in stack detectors

The accurate reconstruction of the energy deposition of finite-divergence wide energy spectrum ion beams in multi-material stack detectors calls for substantial computing resources. A transmission matrix model has been developed that uses statistical information extracted from a series of Monte Carlo simulations to build a fast, accurate and flexible algorithm for computing the energy loss of ions in complex heterogeneous detectors. In particular, the ion range and the broadening of the Bragg peak with increasing ion energy were correctly reproduced by the model.     

马尔可夫系统瞬态不可用度计算的高效率蒙特卡罗方法

当马尔可夫系统规模较大时,需要采用蒙特卡罗方法计算其瞬态不可用度,如果系统的 不可用度很小,则需要采用高效率的蒙特卡罗方法.本文在马尔可夫系统寿命过程的积分方程的 基础上,给出了系统瞬态不可用度计算的蒙特卡罗方法的统一描述,由此设计了马尔可夫系统瞬 态不可用度计算的直接统计估计方法和加权统计估计方法.用直接仿真方法、拟仿真方法、基于 直接仿真的统计估计方法、基于拟方仿真的统计估计方法和加权统计估计方法计算了-可修 Con/3/30:F系统的瞬态不可用度.结果表明,由于同时采用了偏倚的抽样空间和逐次事件估计 量,加权统计估计方法的方差最小,当系统不可用度很小时,该方法效率最高.     

Quantum-dot cellular automata based reversible low power parity generator and parity checker design for nanocommunication

Quantum-dot cellular automata (QCA) is an emerging area of research in reversible computing. It can be used to design nanoscale circuits. In nanocommunication, the detection and correction of errors in a received message is a major factor. Besides, device density and power dissipation are the key issues in the nanocommunication architecture. For the first time, QCA-based designs of the reversible low-power odd parity generator and odd parity checker using the Feynman gate have been achieved in this study. Using the proposed parity generator and parity checker circuit, a nanocommunication architecture is proposed. The detection of errors in the received message during transmission is also explored. The proposed QCA Feynman gate outshines the existing ones in terms of area, cell count, and delay. The quantum costs of the proposed conventional reversible circuits and their QCA layouts are calculated and compared, which establishes that the proposed QCA circuits have very low quantum cost compared to conventional designs. The energy dissipation by the layouts is estimated, which ensures the possibility of QCA nano-device serving as an alternative platform for the implementation of reversible circuits. The stability of the proposed circuits under thermal randomness is analyzed, showing the operational efficiency of the circuits. The simulation results of the proposed design are tested with theoretical values, showing the accuracy of the circuits. The proposed circuits can be used to design more complex low-power nanoscale lossless nanocommunication architecture such as nano-transmitters and nano-receivers.     

Monte Carlo algorithms for evaluating Sobol’ sensitivity indices

Sensitivity analysis is a powerful technique used to determine robustness, reliability and efficiency of a model. The main problem in this procedure is the evaluating total sensitivity indices that measure a parameter’s main effect and all the interactions involving that parameter. From a mathematical point of view this problem is presented by a set of multidimensional integrals. In this work a simple adaptive Monte Carlo technique for evaluating Sobol’ sensitivity indices is developed. A comparison of accuracy and complexity of plain Monte Carlo and adaptive Monte Carlo algorithms is presented. Numerical experiments for evaluating integrals of different dimensions are performed.     

Monte Carlo methods for matrix computations on the grid

Many scientific and engineering applications involve inverting large matrices or solving systems of linear algebraic equations. Solving these problems with proven algorithms for direct methods can take very long to compute, as they depend on the size of the matrix. The computational complexity of the stochastic Monte Carlo methods depends only on the number of chains and the length of those chains. The computing power needed by inherently parallel Monte Carlo methods can be satisfied very efficiently by distributed computing technologies such as Grid computing. In this paper we show how a load balanced Monte Carlo method for computing the inverse of a dense matrix can be constructed, show how the method can be implemented on the Grid, and demonstrate how efficiently the method scales on multiple processors.     

Compressive estimation for signal integration in rendering

In rendering applications, we are often faced with the problem of computing the integral of an unknown function. Typical approaches used to estimate these integrals are often based on Monte Carlo methods that slowly converge to the correct answer after many point samples have been taken. In this work, we study this problem under the framework of compressed sensing and reach the conclusion that if the signal is sparse in a transform domain, we can evaluate the integral accurately using a small set of point samples without requiring the lengthy iterations of Monte Carlo approaches. We demonstrate the usefulness of our framework by proposing novel algorithms to address two problems in computer graphics: image antialiasing and motion blur. We show that we can use our framework to generate good results with fewer samples than is possible with traditional approaches.     

Uncertainty quantification through the Monte Carlo method in a cloud computing setting

The Monte Carlo (MC) method is the most common technique used for uncertainty quantification, due to its simplicity and good statistical results. However, its computational cost is extremely high, and, in many cases, prohibitive. Fortunately, the MC algorithm is easily parallelizable, which allows its use in simulations where the computation of a single realization is very costly. This work presents a methodology for the parallelization of the MC method, in the context of cloud computing. This strategy is based on the MapReduce paradigm, and allows an efficient distribution of tasks in the cloud. This methodology is illustrated on a problem of structural dynamics that is subject to uncertainties. The results show that the technique is capable of producing good results concerning statistical moments of low order. It is shown that even a simple problem may require many realizations for convergence of histograms, which makes the cloud computing strategy very attractive (due to its high scalability capacity and low-cost). Additionally, the results regarding the time of processing and storage space usage allow one to qualify this new methodology as a solution for simulations that require a number of MC realizations beyond the standard.     

Optimal digital simulations for random linear systems with integration constraints

A generalized approach involving concepts from optimization theory is developed for realizing optimal digital simulations for linear, time-varying, continuous dynamical systems having random inputs by modifying discrete input signal variances. The minimization of a cost functional based on the state covariance matrices of the continuous system and its discrete model leads to a two-point boundary value problem which can be solved by known numerical techniques. The result is a systematic procedure for determining optimal digital simulations under the constraints that the numerical integration formula and integration step size have been specified in advance. An example is presented to illustrate the procedure, including a verification using Monte Carlo simulation runs.     

Probabilistic Marching Cubes

In this paper we revisit the computation and visualization of equivalents to isocontours in uncertain scalar fields. We model uncertainty by discrete random fields and, in contrast to previous methods, also take arbitrary spatial correlations into account. Starting with joint distributions of the random variables associated to the sample locations, we compute level crossing probabilities for cells of the sample grid. This corresponds to computing the probabilities that the well‐known symmetry‐reduced marching cubes cases occur in random field realizations. For Gaussian random fields, only marginal density functions that correspond to the vertices of the considered cell need to be integrated. We compute the integrals for each cell in the sample grid using a Monte Carlo method. The probabilistic ansatz does not suffer from degenerate cases that usually require case distinctions and solutions of ill‐conditioned problems. Applications in 2D and 3D, both to synthetic and real data from ensemble simulations in climate research, illustrate the influence of spatial correlations on the spatial distribution of uncertain isocontours.     

The Path Integral Approach to Financial Modeling and Options Pricing

In this paper we review some applications of the path integral methodology of quantum mechanics to financial modeling and options pricing. A path integral is defined as a limit of the sequence of finite-dimensional integrals, in a much the same way as the Riemannian integral is defined as a limit of the sequence of finite sums. The risk-neutral valuation formula for path-dependent options contingent upon multiple underlying assets admits an elegant representation in terms of path integrals (Feynman–Kac formula). The path integral representation of transition probability density (Green's function) explicitly satisfies the diffusion PDE. Gaussian path integrals admit a closed-form solution given by the Van Vleck formula. Analytical approximations are obtained by means of the semiclassical (moments) expansion. Difficult path integrals are computed by numerical procedures, such as Monte Carlo simulation or deterministic discretization schemes. Several examples of path-dependent options are treated to illustrate the theory (weighted Asian options, floating barrier options, and barrier options with ladder-like barriers).     

Determination of exact upper estimates of time taken to perform complex sets of problems in control,parallel computing systems

The actual problem for prediction of the reliable completion of the user-prescribed complex of interconnected jobs (CIJs)—the collection of interdependent program modules with random times of their realization—is solved in control, parallel computing systems by determining the exact upper estimate of the distribution functions of the time taken to complete the CIJs on condition that the number of computing devices of the system is no less than the coefficient of parallelism of the CIJs. Usually, a rather cumbersome obtaining of similar estimates is replaced by the construction of an aggregate of multiple integrals evaluated by the Monte Carlo method.     

Usage of a reconfigurable computer to simulate multiparticle systems

In this article we focus on the implementation of a Lattice Monte Carlo simulation for a generic pair potential within a reconfigurable computing platform. The approach presented was used to simulate a specific soft matter system.We found the performed simulations to be in excellent accordance with previous theoretical and simulation studies. By taking advantage of the shortened processing time, we were also able to find new micro- and macroscopic properties of this system. Furthermore we analyzed analytically the effects of the spatial discretization introduced by the Lattice Monte Carlo algorithm.     

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.     

Neumann dynamic stochastic finite element method of vibration for structures with stochastic parameters to random excitation

This paper presents a method for the dynamic analysis of structures with stochastic parameters to random excitation. A procedure to derive the statistical characteristics of the dynamic response for structure is proposed by using dynamic Neumann stochastic finite element method presented herein. Random equation of motion for structure is transformed into a quasi-static equilibrium equation for the solution of displacement in time domain. Neumann expansion method is developed and applied to the equation for deriving the statistical solution of the dynamic response of such a random structure system, within the framework of Monte Carlo simulation. Then, the results from Neumann dynamic stochastic finite element method are compared with those from the first- and second-order perturbation stochastic finite element methods and the direct Monte Carlo simulation with respect to accuracy, convergence and computational efficiency. Numerical examples are examined to show that the approach proposed in this paper has a very high accuracy and efficiency in the analysis of compound random vibration.     

Four-index integral transformation exploiting symmetry

We present a Fortran implementation of four-index integral transformation in the LCAO-MO (linear combination of atomic orbitals-molecular orbitals) framework that exploits symmetry. Electron correlation calculations, such as configuration interaction (CI) calculations, usually require electron repulsion integrals to be transformed to a molecular orbital basis from a basis using atomic orbitals. In large molecular systems it is vital to exploit the sparsity of integrals in making this transformation. By exploiting symmetry, the sparsity of integrals is fully utilized, the size of intermediate file is minimized, and the computational cost is reduced. The present algorithm is simple and can readily be added to existing quantum chemistry program packages.

Program summary

Title of program: SYM4TR (symmetry adapted 4-index integral transformation)Catalogue identifier: ADUWProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUWProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandComputers: IBM/AIX, HP Alpha server/Tru64, PC's/LinuxProgram language used: Fortran 95Number of lines in distributed program, including test data, etc.: 4519No. of bytes in distributed program, including test data, etc.: 32 095Distributed format: tar gzip fileNature of physical problem: Molecular orbital calculations including electron correlation effects usually require electron repulsion integrals to be transformed from an atomic orbital (AO) basis to a molecular orbital (MO) basis. By exploiting the sparsity of molecular integrals, the computational cost and memory needed for the transformation are minimized.Method of solution: The sparsity of molecular integrals is exploited. The program treats only nonzero integrals. The length of running indices in DO loops is reduced using the block-diagonal form of the MO coefficient matrix. In the present program, the point group is limited to D_2h and its subgroups.     

Investigating the performance of an adiabatic quantum optimization processor

Adiabatic quantum optimization offers a new method for solving hard optimization problems. In this paper we calculate median adiabatic times (in seconds) determined by the minimum gap during the adiabatic quantum optimization for an NP-hard Ising spin glass instance class with up to 128 binary variables. Using parameters obtained from a realistic superconducting adiabatic quantum processor, we extract the minimum gap and matrix elements using high performance Quantum Monte Carlo simulations on a large-scale Internet-based computing platform. We compare the median adiabatic times with the median running times of two classical solvers and find that, for the considered problem sizes, the adiabatic times for the simulated processor architecture are about 4 and 6 orders of magnitude shorter than the two classical solvers’ times. This shows that if the adiabatic time scale were to determine the computation time, adiabatic quantum optimization would be significantly superior to those classical solvers for median spin glass problems of at least up to 128 qubits. We also discuss important additional constraints that affect the performance of a realistic system.     

Improved algorithm for tolerance allocation based on Monte Carlo simulation and discrete optimization

The allocation of design and manufacturing tolerances has a significant effect on both manufacturing cost and quality. This paper considers nonlinearly constrained tolerance allocation problems. The purpose is to minimize the ratio between the sum of the manufacturing costs (tolerances costs) and the risk (probability of the respect of geometrical requirements). The techniques of Monte Carlo simulation and genetic algorithm are adopted to solve these problems. As the simplest and the popular method for non-linear statistical tolerance analysis, the Monte Carlo simulation is introduced into the frame. Moreover, in order to make the frame efficient, the genetic algorithm is improved according to the features of the Monte Carlo simulation. An illustrative example (hyperstatic mechanism) is given to demonstrate the efficiency of the proposed approach.