Importance Sampling Parameters Research Articles

Automated importance sampling via optimal control for stochastic reaction networks: A Markovian projection–based approach

We propose a novel alternative approach to our previous work (Ben Hammouda et al., 2023) to improve the efficiency of Monte Carlo (MC) estimators for rare event probabilities for stochastic reaction networks (SRNs). In the same spirit of Ben Hammouda et al. (2023), an efficient path-dependent measure change is derived based on a connection between determining optimal importance sampling (IS) parameters within a class of probability measures and a stochastic optimal control formulation, corresponding to solving a variance minimization problem. In this work, we propose a novel approach to address the encountered curse of dimensionality by mapping the problem to a significantly lower-dimensional space via a Markovian projection (MP) idea. The output of this model reduction technique is a low-dimensional SRN (potentially even one dimensional) that preserves the marginal distribution of the original high-dimensional SRN system. The dynamics of the projected process are obtained by solving a related optimization problem via a discrete L2 regression. By solving the resulting projected Hamilton–Jacobi–Bellman (HJB) equations for the reduced-dimensional SRN, we obtain projected IS parameters, which are then mapped back to the original full-dimensional SRN system, resulting in an efficient IS-MC estimator for rare events probabilities of the full-dimensional SRN. Our analysis and numerical experiments reveal that the proposed MP-HJB-IS approach substantially reduces the MC estimator variance, resulting in a lower computational complexity in the rare event regime than standard MC estimators.

Adaptive Importance Sampling for Efficient Stochastic Root Finding and Quantile Estimation

Stochastic root-finding problems are fundamental in the fields of operations research and data science. However, when the root-finding problem involves rare events, crude Monte Carlo can be prohibitively inefficient. Importance sampling (IS) is a commonly used approach, but selecting a good IS parameter requires knowledge of the problem’s solution, which creates a circular challenge. In “Adaptive Importance Sampling for Efficient Stochastic Root Finding and Quantile Estimation,” He, Jiang, Lam, and Fu propose an adaptive IS approach to untie this circularity. The adaptive IS simultaneously estimates the root and the IS parameters, and can be embedded in sample average approximation–type algorithms and stochastic approximation–type algorithms. They provide theoretical analysis on strong consistency and asymptotic normality of the resulting estimators, and show the benefit of adaptivity from a worst-case perspective. They also provide specialized analyses on extreme quantile estimation under milder conditions.

Efficient and Unbiased Safety Test for Autonomous Driving Systems

Test the safety of Autonomous Driving Systems (ADS) with realistic traffic conditions is important to the insurance industry, legislators, and third-party technical services. However, the scarcity of risky driving events distributed in real-world driving often makes sampling inefficient. In this paper, we propose a unified and hierarchical testing framework for efficient and unbiased safety tests of ADS. We first extract the risk subspace from the Naturalistic Driving Data (NDD) with defined safety measures. Subsequently, we use the Gaussian Copula method to accurately model the subspace joint probability density function (PDF). We further formulate a Kriging model-based optimization problem for finding the appropriate Importance Sampling (IS) parameters, where the Kriging model is trained iteratively within limited computational resources. Eventually, the risky concrete scenarios library is generated using the obtained Importance Sampling parameters. With the simulation results skewered back to the original risk subspace, the ADS probability of failure (e.g., crash rate) can thus be accurately and unbiasedly estimated. Experiment results show that using 0.94% computational resources, the Kriging surrogate model captures 96.04% of ADS crashes with at least 90.25% precision. On crash rate estimation, our proposed method achieves a consistent result with that of Monte Carlo simulation, provided that the relative error does not exceed 10% while improving the testing efficiency by up to 6,539 times.

Learning-based importance sampling via stochastic optimal control for stochastic reaction networks

We explore efficient estimation of statistical quantities, particularly rare event probabilities, for stochastic reaction networks. Consequently, we propose an importance sampling (IS) approach to improve the Monte Carlo (MC) estimator efficiency based on an approximate tau-leap scheme. The crucial step in the IS framework is choosing an appropriate change of probability measure to achieve substantial variance reduction. This task is typically challenging and often requires insights into the underlying problem. Therefore, we propose an automated approach to obtain a highly efficient path-dependent measure change based on an original connection in the stochastic reaction network context between finding optimal IS parameters within a class of probability measures and a stochastic optimal control formulation. Optimal IS parameters are obtained by solving a variance minimization problem. First, we derive an associated dynamic programming equation. Analytically solving this backward equation is challenging, hence we propose an approximate dynamic programming formulation to find near-optimal control parameters. To mitigate the curse of dimensionality, we propose a learning-based method to approximate the value function using a neural network, where the parameters are determined via a stochastic optimization algorithm. Our analysis and numerical experiments verify that the proposed learning-based IS approach substantially reduces MC estimator variance, resulting in a lower computational complexity in the rare event regime, compared with standard tau-leap MC estimators.

Research on Accelerated Testing of Cut-In Condition of Electric Automated Vehicles Based on Monte Carlo Simulation

Electric automated vehicles are zero-emission, energy-saving, and environmentally friendly vehicles, and testing and verification is an important means to ensure their safety. Because of the scarcity of dangerous scenarios in natural driving roads, it is required to conduct accelerated tests and evaluations for electric automated vehicles. According to the scenario data of the natural road in cut-in conditions, we used the kernel density estimation method to calculate the probability distribution of the scenario parameters. Additionally, we used the Metropolis–Hastings algorithm to sample based on the probability distribution of the parameters, and the Euclidean distance was combined with the paired combination to accelerate the simulation test process. The critical scenarios were screened out by the safety indicator, and the feature distribution of the critical scenario parameters was analyzed based on the Euclidean distance clustering method, so as to design importance sampling parameters and carry out importance sampling. The study obtained the distribution characteristics of critical scenario parameters under cut-in conditions and found that the importance sampling method can accelerate the test under the condition of ensuring that the relative error is small, and the improved accelerated simulation method makes the overall calculation amount smaller.

Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation

In this work, we propose a smart idea to couple importance sampling and Multilevel Monte Carlo (MLMC). We advocate a per level approach with as many importance sampling parameters as the number of levels, which enables us to compute the different levels independently. The search for parameters is carried out using sample average approximation, which basically consists in applying deterministic optimisation techniques to a Monte Carlo approximation rather than resorting to stochastic approximation. Our innovative estimator leads to a robust and efficient procedure reducing both the discretization error (the bias) and the variance for a given computational effort. In the setting of discretized diffusions, we prove that our estimator satisfies a strong law of large numbers and a central limit theorem with optimal limiting variance, in the sense that this is the variance achieved by the best importance sampling measure (among the class of changes we consider), which is however non tractable. Finally, we illustrate the efficiency of our method on several numerical challenges coming from quantitative finance and show that it outperforms the standard MLMC estimator.

Adaptive importance sampling Monte Carlo simulation for general multivariate probability laws

We establish a parametric adaptive importance sampling variance reduction method for general multivariate probability laws. Employing the principle of bypass distributions makes it possible to develop adaptive algorithms without relying on particular properties of the target and proposal laws, both of which in the proposed framework are as general as the uniform law on the unit hypercube, without changing the sampling distribution at each iteration. We establish the asymptotic normality of the estimator of the desired mean and of the importance sampling parameter as the number of observations tends to infinity. Although implementation of the proposed methodology requires a small amount of initial work, it has the potential to yield substantial improvements in estimator efficiency in various general problem settings. To illustrate the applicability and effectiveness, we provide numerical results throughout, in which we apply exponential and normal bypass distributions, as well as demonstrate that well-known adaptive importance sampling formulations in the literature can be easily rewritten in the proposed framework.

A generalization of the Drude-Smith formula for magneto-optical conductivities in Faraday geometry

In this study, we generalize the impulse response approach and Poisson statistics proposed by Smith [Phys. Rev. B 64, 155106 (2001)] to evaluate the longitudinal and transverse magneto-optical conductivities in an electron gas system in Faraday geometry. Comparing with the standard Drude model, the coefficients an are introduced in the Drude-Smith formula to describe the backscattering or localization effect for the nth electronic scattering event. Such a formula can also be applied to study the elements of the dielectric function matrix in the presence of magnetic and radiation fields in electron gas systems. This theoretical work is primely motivated by recent experimental activities in measuring the real and imaginary parts of longitudinal and transverse magneto-optical conductivities in condensed matter materials and electronic devices using terahertz time-domain spectroscopy. We believe that the results obtained from this study can provide an appropriate theoretical tool in reproducing the experimental findings and in fitting with experimental data to determine the important sample and material parameters.

Characterization of material parameters of La0.33Pr0.34Ca0.33MnO3 thin film by terahertz time-domain spectroscopy

We present a systemic study of the terahertz (THz) optical conductivity of a strongly correlated La0.33Pr0.34Ca0.33MnO3 (LPCMO) thin film on a LaAlO3 substrate. The measurements are carried out by THz time-domain spectroscopy in the temperature regime from 15 to 105 K. The frequency-dependent optical conductivity in the metallic phase region of the samples exhibits a non-Drude-like response. We find that below 105 K, both the real and imaginary parts of the complex conductivity can be reproduced by the Drude–Smith model. The important sample and material parameters of the LPCMO thin film (such as the persistence of velocity, the ratio of carrier density to effective mass, and electronic scattering time) can be determined by fitting experimental data. The results obtained agree with those obtained from four-probe electrical transport measurements.

A Geometric Programming to Importance Sam-pling for Power System Reliability Evaluation

A novel Geometric Programming (GP) is presented in the first time by the optimization model of importance sampling parameters (ISP) in Variance Minimization (VM) for importance sampling (IS) of power system reliability evaluation. The key point of the proposed method is that the equality constraints of VM optimization model can be relaxed into inequalities because of its special structure, thus a new GP-VM convex optimization model can be built exactly to solve the difficulty of obtaining the optimal ISP. Numerical results of two test systems verify the effectiveness of the proposed method.

On sample average approximation algorithms for determining the optimal importance sampling parameters in pricing financial derivatives on Lévy processes

We formulate the problem of determining the optimal importance sampling measure change for pricing financial derivatives under Lévy processes as a parametric optimization problem, and propose a solution approach using sample average approximation (SAA) with Newton iteration to find the optimal parameters in the Esscher probability measure change. Theoretical results, such as convergence rate of the optimal solutions, are provided. A numerical example illustrates the effectiveness of the approach.

Simultaneous determination of sample thickness, tilt, and electron mean free path using tomographic tilt images based on Beer-Lambert law.

Cryo-electron tomography (cryo-ET) is an emerging technique that can elucidate the architecture of macromolecular complexes and cellular ultrastructure in a near-native state. Some important sample parameters, such as thickness and tilt, are needed for 3-D reconstruction. However, these parameters can currently only be determined using trial 3-D reconstructions. Accurate electron mean free path plays a significant role in modeling image formation process essential for simulation of electron microscopy images and model-based iterative 3-D reconstruction methods; however, their values are voltage and sample dependent and have only been experimentally measured for a limited number of sample conditions. Here, we report a computational method, tomoThickness, based on the Beer-Lambert law, to simultaneously determine the sample thickness, tilt and electron inelastic mean free path by solving an overdetermined nonlinear least square optimization problem utilizing the strong constraints of tilt relationships. The method has been extensively tested with both stained and cryo datasets. The fitted electron mean free paths are consistent with reported experimental measurements. The accurate thickness estimation eliminates the need for a generous assignment of Z-dimension size of the tomogram. Interestingly, we have also found that nearly all samples are a few degrees tilted relative to the electron beam. Compensation of the intrinsic sample tilt can result in horizontal structure and reduced Z-dimension of tomograms. Our fast, pre-reconstruction method can thus provide important sample parameters that can help improve performance of tomographic reconstruction of a wide range of samples.

Luminescence properties of dilute bismide systems

Systematic photoluminescence measurements on a series of GaBixAs1−x samples are analyzed theoretically using a fully microscopic approach. Based on sp3s⁎ tight-binding calculations, an effective k·p model is set up and used to compute the band structure and dipole matrix elements for the experimentally investigated samples. With this input, the photoluminescence spectra are calculated using a systematic microscopic approach based on the semiconductor luminescence equations. The detailed theory-experiment comparison allows us to quantitatively characterize the experimental structures and to extract important sample parameters.

Performance of solid and cellular structured fly ash geopolymers exposed to a simulated fire

This paper reports on the effects of simulated fire exposure on solid and low density, cellular structured geopolymers in order to assess their suitability for high temperature applications such as thermal barriers and fire resistant panels. Geopolymer mixes with designed Si:Al ratios of 2.5 were synthesised from three different class F fly ashes. Low density samples were produced by adding a small amount of metallic aluminium to the geopolymer slurry which reacted with the free NaOH to produce a cellular structure in the hardened material. Physical properties of the materials are presented as well as results from scale fire tests. An international standard fire curve (ISO 834) was used to simulate the heating conditions of a fire. Fire testing was conducted on 50mm thick panels with an exposure region of 200mm×200mm. Fire ratings of more than 90min were achieved with water content and resistance to shrinkage cracking identified as important sample parameters.

State-dependent doubly weighted stochastic simulation algorithm for automatic characterization of stochastic biochemical rare events

In recent years there has been substantial growth in the development of algorithms for characterizing rare events in stochastic biochemical systems. Two such algorithms, the state-dependent weighted stochastic simulation algorithm (swSSA) and the doubly weighted SSA (dwSSA) are extensions of the weighted SSA (wSSA) by H. Kuwahara and I. Mura [J. Chem. Phys. 129, 165101 (2008)]. The swSSA substantially reduces estimator variance by implementing system state-dependent importance sampling (IS) parameters, but lacks an automatic parameter identification strategy. In contrast, the dwSSA provides for the automatic determination of state-independent IS parameters, thus it is inefficient for systems whose states vary widely in time. We present a novel modification of the dwSSA--the state-dependent doubly weighted SSA (sdwSSA)--that combines the strengths of the swSSA and the dwSSA without inheriting their weaknesses. The sdwSSA automatically computes state-dependent IS parameters via the multilevel cross-entropy method. We apply the method to three examples: a reversible isomerization process, a yeast polarization model, and a lac operon model. Our results demonstrate that the sdwSSA offers substantial improvements over previous methods in terms of both accuracy and efficiency.

A comparison of cross-entropy and variance minimization strategies

The variance minimization (VM) and cross-entropy (CE) methods are two versatile adaptive importance sampling procedures that have been successfully applied to a wide variety of difficult rare-event estimation problems. We compare these two methods via various examples where the optimal VM and CE importance densities can be obtained analytically. We find that in the cases studied both VM and CE methods prescribe the same importance sampling parameters, suggesting that the criterion of minimizing the CE distance is very close, if not asymptotically identical, to minimizing the variance of the associated importance sampling estimator.

A comparison of cross-entropy and variance minimization strategies

The variance minimization (VM) and cross-entropy (CE) methods are two versatile adaptive importance sampling procedures that have been successfully applied to a wide variety of difficult rare-event estimation problems. We compare these two methods via various examples where the optimal VM and CE importance densities can be obtained analytically. We find that in the cases studied both VM and CE methods prescribe the same importance sampling parameters, suggesting that the criterion of minimizing the CE distance is very close, if not asymptotically identical, to minimizing the variance of the associated importance sampling estimator.

Weighted next reaction method and parameter selection for efficient simulation of rare events in biochemical reaction systems

The weighted stochastic simulation algorithm (wSSA) recently developed by Kuwahara and Mura and the refined wSSA proposed by Gillespie et al. based on the importance sampling technique open the door for efficient estimation of the probability of rare events in biochemical reaction systems. In this paper, we first apply the importance sampling technique to the next reaction method (NRM) of the stochastic simulation algorithm and develop a weighted NRM (wNRM). We then develop a systematic method for selecting the values of importance sampling parameters, which can be applied to both the wSSA and the wNRM. Numerical results demonstrate that our parameter selection method can substantially improve the performance of the wSSA and the wNRM in terms of simulation efficiency and accuracy.

State-dependent biasing method for importance sampling in the weighted stochastic simulation algorithm

The weighted stochastic simulation algorithm (wSSA) was developed by Kuwahara and Mura [J. Chem. Phys. 129, 165101 (2008)] to efficiently estimate the probabilities of rare events in discrete stochastic systems. The wSSA uses importance sampling to enhance the statistical accuracy in the estimation of the probability of the rare event. The original algorithm biases the reaction selection step with a fixed importance sampling parameter. In this paper, we introduce a novel method where the biasing parameter is state-dependent. The new method features improved accuracy, efficiency, and robustness.

A framework for adaptive Monte Carlo procedures

Adaptive Monte Carlo methods are recent variance reduction techniques. In this work, we propose a mathematical setting which greatly relaxes the assumptions needed by for the adaptive importance sampling techniques presented in [Arouna, Monte Carlo Methods Appl. 10: 1–24, 2004, Arouna, The Journal of Computational Finance 7: Winter 2003/2004, Su and Fu, Journal of Computational Finance 5: 27–50, 2002, Vázquez-Abad and Dufresne, Accelerated simulation for pricing asian options: 1493–1500, IEEE Computer Society Press, 1998]. We establish the convergence and asymptotic normality of the adaptive Monte Carlo estimator under local assumptions which are easily verifiable in practice. We present one way of approximating the optimal importance sampling parameter using a randomly truncated stochastic algorithm. Finally, we apply this technique to some examples of valuation of financial derivatives.