Importance Sampling Monte Carlo Methods Research Articles

H-T magnetic phase diagram of CuCrO2: A Monte Carlo study based on a realistic 3D classical Heisenberg model

In the multiferroic geometrically frustrated triangular antiferromagnet CuCrO2, we investigate the magnetic phases as function of temperature under a magnetic field 70 T ≤ B ≤ 120 T along the [001] axis using the importance sampling Monte Carlo method. We consider a realistic three-dimensional classical Heisenberg model, which takes into consideration the distortion and exchange interactions up to the third-nearest neighbors in the ab planes and an interplane interaction, in addition to hard and easy anisotropy axes along the [110] and the [001] directions, respectively. For 70 T ≤ B < 106 T, our calculations show that two phase transitions occur when decreasing the temperature. The first one, which takes place around 30 K, is a “paramagnetic ↔ up-up-down (UUD)” transition and the second one, at lower temperatures, is an “UUD ↔ commensurate Y (CY)” transition. Thus, the colinear UUD phase is stable in an intermediate temperature range. For B = 90 T, our estimates of the critical exponents indicate that the two transitions belong to the universality class of the two-dimensional Ising model. For 106 T ≤ B < 110 T, since the CY phase is identical to the UUD one, only a “paramagnetic ↔ up-up-down (UUD)” transition occurs. For 110 T ≤ B ≤ 120 T, only a “paramagnetic ↔ V” transition is observed at ∼27 K. Our results are in good agreement with the existing experimental data on the stable phases in this magnetic field range. Moreover, we have determined the temperature range where the CY and UUD phases are stable which was not done previously.

Estimating Tail Probabilities of Random Sums of Phase-Type Scale Mixture Random Variables

We consider the problem of estimating tail probabilities of random sums of scale mixture of phase-type distributions—a class of distributions corresponding to random variables which can be represented as a product of a non-negative but otherwise arbitrary random variable with a phase-type random variable. Our motivation arises from applications in risk, queueing problems for estimating ruin probabilities, and waiting time distributions, respectively. Mixtures of distributions are flexible models and can be exploited in modelling non-life insurance loss amounts. Classical rare-event simulation algorithms cannot be implemented in this setting because these methods typically rely on the availability of the cumulative distribution function or the moment generating function, but these are difficult to compute or are not even available for the class of scale mixture of phase-type distributions. The contributions of this paper are that we address these issues by proposing alternative simulation methods for estimating tail probabilities of random sums of scale mixture of phase-type distributions which combine importance sampling and conditional Monte Carlo methods, showing the efficiency of the proposed estimators for a wide class of scaling distributions, and validating the empirical performance of the suggested methods via numerical experimentation.

Computational optimization of the SARS-CoV-2 receptor-binding-motif affinity for human ACE2

The coronavirus severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), which is responsible for the coronavirus disease 2019 pandemic, and the closely related SARS-CoV coronavirus enter cells by binding at the human angiotensin converting enzyme 2 (hACE2). The stronger hACE2 affinity of SARS-CoV-2 has been connected with its higher infectivity. In this work, we study hACE2 complexes with the receptor-binding domains (RBDs) of the human SARS-CoV-2 and human SARS-CoV viruses, using all-atom molecular dynamics simulations and computational protein design with a physics-based energy function. The molecular dynamics simulations identify charge-modifying substitutions between the CoV-2 and CoV RBDs, which either increase or decrease the hACE2 affinity of the SARS-CoV-2 RBD. The combined effect of these mutations is small, and the relative affinity is mainly determined by substitutions at residues in contact with hACE2. Many of these findings are in line and interpret recent experiments. Our computational protein design calculations redesign positions 455, 493, 494, and 501 of the SARS-CoV-2 receptor binding motif, which contact hACE2 in the complex and are important for ACE2 recognition. Sampling is enhanced by an adaptive importance sampling Monte Carlo method. Sequences with increased affinity replace CoV-2 glutamine by a negative residue at position 493; serine by a nonpolar or aromatic residue or an asparagine at position 494; and asparagine by valine or threonine at position 501. Substitutions at positions 455 and 501 have a smaller effect on affinity. Substitutions suggested by our design are seen in viral sequences encountered in other species, including bat and pangolin. Our results might be used to identify potential virus strains with higher human infectivity and assist in the design of peptide-based or peptidomimetic compounds with the potential to inhibit SARS-CoV-2 binding at hACE2.

RETRACTED: Fatigue reliability assessment for hangers of a special-shaped CFST arch bridge

Abstract The main bridge of the Yingzhou Bridge is a special -shaped concrete-filled steel tubular (CFST) arch bridge. As a key load-carrying member, the safety of a hanger affects the service life of arch bridges. This paper presents a comprehensive frame work for fatigue reliability analysis of hangers in CFST arch bridges. A simplified finite element (FE) model is established to satisfy the fatigue analysis accuracy. The mathematical model for traffic flow of the Yingzhou Bridge is simulated using the Monte Carlo method. Then the stress time history of the hanger is calculated. The rain flow counting method is used to obtain the stress range values and the number of cycles. Based on the Miner cumulative damage rule, considering the effect of mean stress, the fatigue reliability for hangers with and without corrosion is calculated using the Monte Carlo Importance Sampling method. It is found that the reliability index of inclined hangers is lower than that of vertical hangers. Inclined hangers have a higher probability of failure earlier. The reliability index of the inclined hanger X3 in the position where the stiffness changes is the lowest, which should be paid greater attention to in daily maintenance. The reliability index of hangers with corrosion is significantly lower than that of hangers without corrosion, and the decline rate of the reliability index is increased with time.

Practical Product Sampling by Fitting and Composing Warps

AbstractWe introduce a Monte Carlo importance sampling method for integrands composed of products and show its application to rendering where direct sampling of the product is often difficult. Our method is based on warp functions that operate on the primary samples in [0,1)n, where each warp approximates sampling a single factor of the product distribution. Our key insight is that individual factors are often well‐behaved and inexpensive to fit and sample in primary sample space, which leads to a practical, efficient sampling algorithm. Our sampling approach is unbiased, easy to implement, and compatible with multiple importance sampling. We show the results of applying our warps to projected solid angle sampling of spherical triangles, to sampling bilinear patch light sources, and to sampling glossy BSDFs and area light sources, with efficiency improvements of over 1.6× on real‐world scenes.

Current fluctuations in noninteracting run-and-tumble particles in one dimension

We present a general framework to study the distribution of the flux through the origin up to time t, in a noninteracting one-dimensional system of particles with a step initial condition with a fixed density ρ of particles to the left of the origin. We focus principally on two cases: (i) particles undergoing diffusive dynamics (passive case) and (ii) run-and-tumble dynamics for each particle (active case). In analogy with disordered systems, we consider the flux distribution for both the annealed and the quenched initial conditions, for passive and active particles. In the annealed case, we show that, for arbitrary particle dynamics, the flux distribution is a Poissonian with a mean μ(t) that we compute exactly in terms of the Green's function of the single-particle dynamics. For the quenched case, we show that, for the run-and-tumble dynamics, the quenched flux distribution takes an anomalous large-deviation form at large times, P_{qu}(Q,t)∼exp[-ρv_{0}γt^{2}ψ_{RTP}(Q/ρv_{0}t)], where γ is the rate of tumbling and v_{0} is the ballistic speed between two successive tumblings. In this paper, we compute the rate function ψ_{RTP}(q) and show that it is nontrivial. Our method also gives access to the probability of the rare event that, at time t, there is no particle to the right of the origin. For diffusive and run-and-tumble dynamics, we find that this probability decays with time as a stretched exponential, ∼exp(-csqrt[t]), where the constant c can be computed exactly. We verify our results for these large deviations by using an importance sampling Monte Carlo method.

Analytic Sensor Rules for Optimal Distributed Decision Given $K$-Out-of-$L$ Fusion Rule Under Monte Carlo Approximation

With the development of large-scale sensor networks, there is an increasing interest for high dimensional distributed decision fusion problem in statistical decision, machine learning, control theory, etc. In this article, we investigate the optimal sensor quantized rules of the distributed decision under a given fusion rule for conditionally dependent and high dimensional sensor observations. We provide a Monte Carlo importance sampling method to reduce the computational complexity of the distributed decision fusion. For the K-out-of-L rule, we derive a group of analytic sensor rules based on the necessary and sufficient condition of the optimal sensor rules. In term of efficiency, it is significantly better than the traditional iterative search algorithms, such as the Riemann sum approximation iteration algorithm. The analytic solution is also a general result since it does not rely on the specific K value of the K-out-of-L rule, the specific conditional probability density function and the dimension of sensor observations. Thus, it can be applied to multisensor decision fusion problems with high dimensional dependent sensor observations. For the general fusion rules rather than the K-out-of-L rule, the Monte Carlo Gauss-Seidel algorithm is also provided. The numerical examples demonstrate the effectiveness of the analytic solution and the Monte Carlo Gauss-Seidel algorithm.

Distributed track‐to‐track fusion for non‐linear systems with Gaussian mixture noise

The authors propose a distributed state estimation algorithm based on optimal track-to-track fusion for local posteriors in terms of Gaussian mixtures. The track-to-track fusion system is implemented with both parallel and sequential structure based on generalised covariance intersection rule. They obtain the optimal fusion coefficients in a computationally efficient manner via Monte Carlo importance sampling method. The Dirac mixture approximation method is proposed for the computation of arbitrary power of a Gaussian mixture density. The resulting Gaussian mixture fusion rule is analytical and applicable to the multi-sensor case. Numerical examples are presented to demonstrate the performance advantages of the proposed method in comparison with existing track-to-track fusion algorithms.

A weighted discrepancy bound of quasi-Monte Carlo importance sampling

Importance sampling Monte-Carlo methods are widely used for the approximation of expectations with respect to partially known probability measures. In this paper we study a deterministic version of such an estimator based on quasi-Monte Carlo. We obtain an explicit error bound in terms of the star-discrepancy for this method.

Hybrid importance sampling Monte Carlo approach for yield estimation in circuit design

The dimension of transistors shrinks with each new technology developed in the semiconductor industry. The extreme scaling of transistors introduces important statistical variations in their process parameters. A large digital integrated circuit consists of a very large number (in millions or billions) of transistors, and therefore the number of statistical parameters may become very large if mismatch variations are modeled. The parametric variations often cause to the circuit performance degradation. Such degradation can lead to a circuit failure that directly affects the yield of the producing company and its fame for reliable products. As a consequence, the failure probability of a circuit must be estimated accurately enough. In this paper, we consider the Importance Sampling Monte Carlo method as a reference probability estimator for estimating tail probabilities. We propose a Hybrid ISMC approach for dealing with circuits having a large number of input parameters and provide a fast estimation of the probability. In the Hybrid approach, we replace the expensive to use circuit model by its cheap surrogate for most of the simulations. The expensive circuit model is used only for getting the training sets (to fit the surrogates) and near to the failure threshold for reducing the bias introduced by the replacement.

Adaptive stratified importance sampling: hybridization of extrapolation and importance sampling Monte Carlo methods for estimation of wind turbine extreme loads

Abstract. Wind turbine extreme load estimation is especially difficult because turbulent inflow drives nonlinear turbine physics and control strategies; thus there can be huge differences in turbine response to essentially equivalent environmental conditions. The two main current approaches, extrapolation and Monte Carlo sampling, are both unsatisfying: extrapolation-based methods are dangerous because by definition they make predictions outside the range of available data, but Monte Carlo methods converge too slowly to routinely reach the desired 50-year return period estimates. Thus a search for a better method is warranted. Here we introduce an adaptive stratified importance sampling approach that allows for treating the choice of environmental conditions at which to run simulations as a stochastic optimization problem that minimizes the variance of unbiased estimates of extreme loads. Furthermore, the framework, built on the traditional bin-based approach used in extrapolation methods, provides a close connection between sampling and extrapolation, and thus allows the solution of the stochastic optimization (i.e., the optimal distribution of simulations in different wind speed bins) to guide and recalibrate the extrapolation. Results show that indeed this is a promising approach, as the variance of both the Monte Carlo and extrapolation estimates are reduced quickly by the adaptive procedure. We conclude, however, that due to the extreme response variability in turbine loads to the same environmental conditions, our method and any similar method quickly reaches its fundamental limits, and that therefore our efforts going forward are best spent elucidating the underlying causes of the response variability.

A note on random walks with absorbing barriers and sequential Monte Carlo methods

ABSTRACTIn this article, we consider importance sampling (IS) and sequential Monte Carlo (SMC) methods in the context of one-dimensional random walks with absorbing barriers. In particular, we develop a very precise variance analysis for several IS and SMC procedures. We take advantage of some explicit spectral formulae available for these models to derive sharp and explicit estimates; this provides stability properties of the associated normalized Feynman–Kac semigroups. Our analysis allows one to compare the variance of SMC and IS techniques for these models. The work in this article is one of the few to consider an in-depth analysis of an SMC method for a particular model-type as well as variance comparison of SMC algorithms.

Passive source localization using importance sampling based on TOA and FOA measurements

Passive source localization via a maximum likelihood (ML) estimator can achieve a high accuracy but involves high calculation burdens, especially when based on time-of-arrival and frequency-of-arrival measurements for its internal nonlinearity and nonconvex nature. In this paper, we use the Pincus theorem and Monte Carlo importance sampling (MCIS) to achieve an approximate global solution to the ML problem in a computationally efficient manner. The main contribution is that we construct a probability density function (PDF) of Gaussian distribution, which is called an important function for efficient sampling, to approximate the ML estimation related to complicated distributions. The improved performance of the proposed method is attributed to the optimal selection of the important function and also the guaranteed convergence to a global maximum. This process greatly reduces the amount of calculation, but an initial solution estimation is required resulting from Taylor series expansion. However, the MCIS method is robust to this prior knowledge for point sampling and correction of importance weights. Simulation results show that the proposed method can achieve the Cramer-Rao lower bound at a moderate Gaussian noise level and outperforms the existing methods.

Rare event simulation via importance sampling for linear SPDE’s

The goal of this paper is to develop provably efficient importance sampling Monte Carlo methods for the estimation of rare events within the class of linear stochastic partial differential equations. We find that if a spectral gap of appropriate size exists, then one can identify a lower dimensional manifold where the rare event takes place. This allows one to build importance sampling changes of measures that perform provably well even pre-asymptotically (i.e. for small but non-zero size of the noise) without degrading in performance due to infinite dimensionality or due to long simulation time horizons. Simulation studies supplement and illustrate the theoretical results.

Finite-temperature time-dependent variation with multiple Davydov states.

The Dirac-Frenkel time-dependent variational approach with Davydov Ansätze is a sophisticated, yet efficient technique to obtain an accurate solution to many-body Schrödinger equations for energy and charge transfer dynamics in molecular aggregates and light-harvesting complexes. We extend this variational approach to finite temperature dynamics of the spin-boson model by adopting a Monte Carlo importance sampling method. In order to demonstrate the applicability of this approach, we compare calculated real-time quantum dynamics of the spin-boson model with that from numerically exact iterative quasiadiabatic propagator path integral (QUAPI) technique. The comparison shows that our variational approach with the single Davydov Ansätze is in excellent agreement with the QUAPI method at high temperatures, while the two differ at low temperatures. Accuracy in dynamics calculations employing a multitude of Davydov trial states is found to improve substantially over the single Davydov Ansatz, especially at low temperatures. At a moderate computational cost, our variational approach with the multiple Davydov Ansatz is shown to provide accurate spin-boson dynamics over a wide range of temperatures and bath spectral densities.

Bayesian model comparison with un-normalised likelihoods

Models for which the likelihood function can be evaluated only up to a parameter-dependent unknown normalizing constant, such as Markov random field models, are used widely in computer science, statistical physics, spatial statistics, and network analysis. However, Bayesian analysis of these models using standard Monte Carlo methods is not possible due to the intractability of their likelihood functions. Several methods that permit exact, or close to exact, simulation from the posterior distribution have recently been developed. However, estimating the evidence and Bayes’ factors for these models remains challenging in general. This paper describes new random weight importance sampling and sequential Monte Carlo methods for estimating BFs that use simulation to circumvent the evaluation of the intractable likelihood, and compares them to existing methods. In some cases we observe an advantage in the use of biased weight estimates. An initial investigation into the theoretical and empirical properties of this class of methods is presented. Some support for the use of biased estimates is presented, but we advocate caution in the use of such estimates.

Rare Event Simulation for Multiscale Diffusions in Random Environments

We consider systems of stochastic differential equations with multiple scales and small noise and assume that the coefficients of the equations are ergodic and stationary random fields. Our goal is to construct provably efficient importance sampling Monte Carlo methods that allow efficient computation of rare event probabilities or expectations of functionals that can be associated with rare events. Standard Monte Carlo algorithms perform poorly in the small noise limit, and hence fast simulations algorithms become relevant. The presence of multiple scales complicates the design and the analysis of efficient importance sampling schemes. An additional complication is the randomness of the environment. We construct explicit changes of measures that are proven to be logarithmic asymptotically efficient with probability one with respect to the random environment (i.e., in the quenched sense). Numerical simulations support the theoretical results.

Reliability assessment on prediction of pile bearing capacity

There are many prediction methods that can be used to determine the bearing capacity of pipe pile. However, most of these methods are highly sensitive to specific in situ conditions and hence usually result in significant error. Therefore, it is important to evaluate the reliability of different pile bearing capacity prediction methods. Based on reliability theory, a computer code is developed in this paper to calculate reliability. Random variable statistic properties and corresponding reliabilities for different prediction methods are investigated and analysed. Results show that the reliability index for logarithmic normal distribution is higher than normal distribution. Analysis results demonstrate that the checking point method (JC method) method has a slightly lower efficiency than the Monte-Carlo importance sampling method and the Monte-Carlo direct sampling method. The Monte-Carlo importance sampling method has a higher efficiency than the Monte-Carlo direct sampling method. The Monte-Carlo importance sampling method can be used in major engineering projects. It is concluded that the clustering centre method has higher reliability in predicting pile bearing capacity than the Eslami and Fellenius method, Almeida method and Powell method.

Computational Methods for a Class of Network Models

In the following article, we provide an exposition of exact computational methods to perform parameter inference from partially observed network models. In particular, we consider the duplication attachment model that has a likelihood function that typically cannot be evaluated in any reasonable computational time. We consider a number of importance sampling (IS) and sequential Monte Carlo (SMC) methods for approximating the likelihood of the network model for a fixed parameter value. It is well-known that, for IS, the relative variance of the likelihood estimate typically grows at an exponential rate in the time parameter (here this is associated with the size of the network); we prove that, under assumptions, the SMC method will have relative variance that can grow only polynomially. In order to perform parameter estimation, we develop particle Markov chain Monte Carlo algorithms to perform Bayesian inference. Such algorithms use the aforementioned SMC algorithms within the transition dynamics. The approaches are illustrated numerically.

Quantum nuclear dynamics in the photophysics of diamondoids

The unusual electronic properties of diamondoids, the nanoscale relatives of diamond, make them attractive for applications ranging from drug delivery to field emission displays. Identifying the fundamental origin of these properties has proven highly challenging, with even the most advanced quantum many-body calculations unable to reproduce measurements of a quantity as ubiquitous as the optical gap. Here, by combining first-principles calculations and Importance Sampling Monte Carlo methods, we show that the quantum dynamics of carbon nuclei is key to understanding the electronic and optical properties of diamondoids. Quantum nuclear effects dramatically modify the absorption lineshapes and renormalize the optical gaps. These findings allow us to formulate a complete theory of optical absorption in diamondoids, and establish the universal role of quantum nuclear dynamics in nanodiamond across the length scales.