Randomized quasi-Monte Carlo Research Articles

Deep learning based on randomized quasi-Monte Carlo method for solving linear Kolmogorov partial differential equation

Deep learning algorithms have been widely used to solve linear Kolmogorov partial differential equations (PDEs) in high dimensions, where the loss function is defined as a mathematical expectation. We propose to use the randomized quasi-Monte Carlo (RQMC) method instead of the Monte Carlo (MC) method for computing the loss function. In theory, we decompose the error from empirical risk minimization (ERM) into the generalization error and the approximation error. Notably, the approximation error is independent of the sampling methods. We prove that the convergence order of the mean generalization error for the RQMC method is O(n−1+ϵ) for arbitrarily small ϵ>0, while for the MC method it is O(n−1/2+ϵ) for arbitrarily small ϵ>0. Consequently, we find that the overall error for the RQMC method is asymptotically smaller than that for the MC method as n increases. Our numerical experiments show that the algorithm based on the RQMC method consistently achieves smaller relative L2 error than that based on the MC method.

Stratified sample tiling

The paper introduces a practical method for the construction of large-scale point sets for analysis of computer models. The constructed experimental design is useful for (probabilistic) integration, construction of approximation or a screening. The essence of the presented approach is the stratification of the design domain into an orthogonal grid of substrata and a subsequent tiling with tiles of points. If optimized, such tiles experience a major reduction of the number of degrees of freedom in the optimization process. That way, optimal or near-optimal point patterns can be feasibly identified and are further utilized for construction of larger point sets, thanks to the idea of self-similarity and structured space stratification The space-filling properties of the resulting point sets may be further enhanced by various “scrambling” strategies, which may remove the undesired sample collapsibility achieved via regular tiling. The performance of the constructed point sets is compared to Quasi Monte Carlo (QMC), Randomized Quasi Monte Carlo (RQMC) sequences, which are still today considered by engineers and even scientists as choices for variance reduction of numerical integration Further, the mentioned sampling strategies are compared in the terms of robustness when integrating a multivariate function with a localized feature. It is concluded that the proposed sampling approach reaches a superior performance in numerical integration and identification of function extremes as compared to sampling methods used by practicing researchers and engineers. Additionally, the reader is supplied with the open-access, ready-to-use implementation of the presented algorithm named SampleTiler.

Adaptive Particle Filter With Randomized Quasi-Monte Carlo Sampling for Unbalanced Distribution System State Estimation

Particle filter (PF) is an effective state estimation method for nonlinear unbalanced distribution systems with non-Gaussian noises. However, the computational inefficiency limits its real-time application in large-scale distribution systems. This paper proposes an adaptive PF (APF) method by using randomized quasi-Monte Carlo (RQMC) sampling to enhance computational efficiency. Firstly, an RQMC sampling method is introduced to enable uniform sampling based on the low discrepancy Sobol’ sequence. This even sampling accelerates the convergence rate of numerical integrations. Secondly, the standard PF is extended to be an RQMC-PF estimator by using the RQMC sampling instead of the MC sampling. Benefiting from the higher convergence rate, fewer particles are required to obtain precise state estimates, thus notably improving computation efficiency. Thirdly, to further make a trade-off between estimation accuracy and computational efficiency, the number of particles used in the RQMC-PF is adaptively determined with the root mean square of the estimated state covariance, yielding an APF. Test results on unbalanced distribution systems with different scales demonstrate that the proposed method can efficiently provide accurate estimates as well as achieve good scalability.

Unbiased MLMC-based Variational Bayes for Likelihood-Free Inference

Variational Bayes (VB) is a popular tool for Bayesian inference in statistical modeling. Recently, some VB algorithms are proposed to handle intractable likelihoods with applications such as approximate Bayesian computation. In this paper, we propose several unbiased estimators based on multilevel Monte Carlo (MLMC) for the gradient of Kullback-Leibler divergence between the posterior distribution and the variational distribution when the likelihood is intractable, but can be estimated unbiasedly. The new VB algorithm differs from the VB algorithms in the literature which usually render biased gradient estimators. Moreover, we incorporate randomized quasi-Monte Carlo (RQMC) sampling within the MLMC-based gradient estimators, which was known to provide a favorable rate of convergence in numerical integration. Theoretical guarantees for RQMC are provided in this new setting. Numerical experiments show that using RQMC in MLMC greatly speeds up the VB algorithm, and finds a better parameter value than some existing competitors do.

A General Numerical Algorithm for CDO Pricing Based on Single Factor Copula Framework and Nonhomogeneous Assumptions

In view of the fact that different factor Copula models are only applicable to different practical problems in collateralized debt obligations (CDO) market and that there is no semianalytical solution under nonhomogeneous assumptions to CDO pricing model, we designed a general numerical algorithm which was based on the framework of single factor Copula model and randomized quasi-Monte Carlo (RQMC) simulation method. We took two single factor Copula models as examples to conduct empirical study, in which the simulation results of RQMC and Monte Carlo (MC) simulation method were compared and analyzed based on variance changes. The result showed that the algorithm in this paper was not only applicable to general single factor Copula model but also very stable. So, it was a general and efficient numerical method to solve the problem of CDO pricing under nonhomogeneous assumptions.

Application of RQMC for CDO Pricing with Stochastic Correlations under Nonhomogeneous Assumptions

In consideration of that the correlation between any two assets of the asset pool is always stochastic in the actual market and that collateralized debt obligation (CDO) pricing models under nonhomogeneous assumptions have no semianalytic solutions, we designed a numerical algorithm based on randomized quasi‐Monte Carlo (RQMC) simulation method for CDO pricing with stochastic correlations under nonhomogeneous assumptions and took Gaussian factor copula model as an example to conduct experiments. The simulation results of RQMC and Monte Carlo (MC) method were compared from the perspective of variance changes. The results showed that this numerical algorithm was feasible, efficient, and stable for CDO pricing with stochastic correlation under nonhomogeneous assumptions. This numerical algorithm is expected to be extended to other factor Copula models for CDO pricing with stochastic correlations under nonhomogeneous assumptions.

Sensitivity estimation of conditional value at risk using randomized quasi-Monte Carlo

Conditional value at risk (CVaR) is a popular measure for quantifying portfolio risk. Sensitivity analysis of CVaR is common in risk management and gradient-based optimization algorithms. In this paper, we study the infinitesimal perturbation analysis estimator for CVaR sensitivity using randomized quasi-Monte Carlo (RQMC) simulation. RQMC has proved valuable in financial option pricing with a better rate of convergence compared to Monte Carlo sampling, but theoretical guarantees for this new application of RQMC shall be studied. To this end, we first prove that the RQMC-based estimator is strongly consistent under very mild conditions. Under some technical conditions, RQMC yields a mean error rate of O(n−1/2−1/(4d−2)+ϵ) for arbitrarily small ϵ>0, where d represents the dimension of RQMC points and n is the sample size. Some typical applications of CVaR sensitivity estimation are conducted to both show how the theoretical results can be applied, as well as to provide numerical results documenting the superiority of the RQMC estimator.

Stochastic Simulation of Mould Growth Performance of Wood-Frame Building Envelopes under Climate Change: Risk Assessment and Error Estimation

Previous studies have shown that the effects of climate change on building structures will increase the mould growth risk of the wood-frame building envelope in many circumstances. This risk can be controlled by wind-driven rain deflection, improving water tightness of the exterior facade, and improving cladding ventilation. However, the effectiveness of these risk mitigation strategies are subject to various uncertainties, such as the uncertainties of wall component properties and micro-climatic conditions. The objective of this paper is to apply stochastic hygrothermal simulation to evaluate the mould growth risk of a brick veneer-clad wood-frame wall with a drainage cavity under historical and future climatic conditions of Ottawa, a Canadian city located in a cold climate zone. An extensive literature review was conducted to quantify the range of stochastic variables including rain deposition factor, rain leakage moisture source, cladding ventilation rate and material properties of brick. The randomised Sobol sequence-based sampling method, one of the Randomized Quasi-Monte Carlo (RQMC) methods, was applied for risk assessment and error estimation. It was found that, under the climatic condition of Ottawa, limiting the amount of wind-driven rain to which walls are subjected is a more robust mitigation measure than improving cladding ventilation in controlling mould growth risk, the improving of water tightness of exterior façade is not as robust as wind-driven rain deflection and cladding ventilation, however, the reduction of rainwater penetration can reduce the mould growth risk at different levels of rain deposition factor and cladding ventilation rate.

Normal variance mixtures: Distribution, density and parameter estimation

Efficient algorithms for computing the distribution function, (log-)density function and for estimating the parameters of multivariate normal variance mixtures are introduced. For the evaluation of the distribution function, randomized quasi-Monte Carlo (RQMC) methods are utilized in a way that improves upon existing methods proposed for the special case of normal and t distributions. For evaluating the log-density function, an adaptive RQMC algorithm that similarly exploits the superior convergence properties of RQMC methods is introduced. This allows the parameter estimation task to be accomplished via an expectation–maximization-like algorithm where all weights and log-densities are numerically estimated. Numerical examples demonstrate that the suggested algorithms are quite fast. Even for high dimensions around 1000 the distribution function can be estimated with moderate accuracy using only a few seconds of run time. Also, even log-densities around −100 can be estimated accurately and quickly. An implementation of all algorithms presented is available in the R package nvmix (version ≥0.0.4).

A Quasi-Monte Carlo method to compute scattering effects in radiative heat transfer: Application to a sooted jet flame

The Monte-Carlo simulation of radiative heat transfer is known to be very accurate. Its convergence can be significantly improved using Randomized Quasi-Monte-Carlo (RQMC) methods that rely on low-discrepancy sequences. The RQMC approach derived recently to deal with thermal radiation is, however, limited to non-scattering media. The present work proposes a methodology aiming at extending this technique to scattering media based on the prior estimation of the low-discrepancy sequence dimension. Firstly, the method is tested on 3D homogeneous fields with various operating points based on the domain’s optical thickness and albedo. It is observed that, for a given number of generated rays, the error can be reduced by up to one order of magnitude. Secondly, the RQMC approach is combined with importance sampling to increase its efficiency further. The number of rays required is even lower, resulting in saving CPU time to reach a given error. The RQMC approach is then applied along with an accurate model for soot particles’ radiative properties: the Rayleigh-Debye-Gans for Fractal Aggregates (RDGFA) theory. The model assumes a complex morphological shape of particles contrary to Rayleigh theory that is valid for spherical particles only. Monte-Carlo simulations are performed on a fixed turbulent sooted flame field taken from coupled calculations with large-eddy simulation. The overall CPU cost is divided by a factor of 2 compared to a standard Monte Carlo calculation. The simulation allows for accurate quantification of soot scattering effects with RDGFA, which eventually appear small in this configuration. On the other hand, the use of Rayleigh theory strongly underpredicts the actual scattering impact.

Density Estimation by Randomized Quasi-Monte Carlo

We consider the problem of estimating the density of a random variable $X$ that can be sampled exactly by Monte Carlo (MC). We investigate the effectiveness of replacing MC by randomized quasi Monte Carlo (RQMC) or by stratified sampling over the unit cube, to reduce the integrated variance (IV) and the mean integrated square error (MISE) for kernel density estimators. We show theoretically and empirically that the RQMC and stratified estimators can achieve substantial reductions of the IV and the MISE, and even faster convergence rates than MC in some situations, while leaving the bias unchanged. We also show that the variance bounds obtained via a traditional Koksma-Hlawka-type inequality for RQMC are much too loose to be useful when the dimension of the problem exceeds a few units. We describe an alternative way to estimate the IV, a good bandwidth, and the MISE, under RQMC or stratification, and we show empirically that in some situations, the MISE can be reduced significantly even in high-dimensional settings.

Grouped Normal Variance Mixtures

Grouped normal variance mixtures are a class of multivariate distributions that generalize classical normal variance mixtures such as the multivariate t distribution, by allowing different groups to have different (comonotone) mixing distributions. This allows one to better model risk factors where components within a group are of similar type, but where different groups have components of quite different type. This paper provides an encompassing body of algorithms to address the computational challenges when working with this class of distributions. In particular, the distribution function and copula are estimated efficiently using randomized quasi-Monte Carlo (RQMC) algorithms. We propose to estimate the log-density function, which is in general not available in closed form, using an adaptive RQMC scheme. This, in turn, gives rise to a likelihood-based fitting procedure to jointly estimate the parameters of a grouped normal mixture copula jointly. We also provide mathematical expressions and methods to compute Kendall’s tau, Spearman’s rho and the tail dependence coefficient λ. All algorithms presented are available in the R package nvmix (version ≥ 0.0.5).

Primal–dual quasi-Monte Carlo simulation with dimension reduction for pricing American options

The pricing of American options is one of the most challenging problems in financial engineering due to the involved optimal stopping time problem, which can be solved by using dynamic programming (DP). But applying DP is not always practical, especially when the state space is high dimensional. However, the curse of dimensionality can be overcome by Monte Carlo (MC) simulation. We can get lower and upper bounds by MC to ensure that the true price falls into a valid confidence interval. During the recent decades, progress has been made in using MC simulation to obtain both the lower bound by least-squares Monte Carlo method (LSM) and the upper bound by duality approach. However, there are few works on pricing American options using quasi-Monte Carlo (QMC) methods, especially to compute the upper bound. For comparing the sample variances and standard errors in the numerical experiments, randomized QMC (RQMC) methods are usually used. In this paper, we propose to use RQMC to replace MC simulation to compute both the lower bound (by the LSM) and the upper bound (by the duality approach). Moreover, we propose to use dimension reduction techniques, such as the Brownian bridge, principal component analysis, linear transformation and the gradients based principle component analysis. We perform numerical experiments on American–Asian options and American max-call options under the Black–Scholes model and the variance gamma model, in which the options have the path-dependent feature or are written on multiple underlying assets. We find that RQMC in combination with dimension reduction techniques can significantly increase the efficiency in computing both the lower and upper bounds, resulting in better estimates and tighter confidence intervals of the true price than pure MC simulation.

Randomized quasi Monte Carlo methods for pricing of barrier options under fractional Brownian motion

Randomized quasi-Monte Carlo (RQMC) method is presented to compute the problem of a barrier option pricing. It is assumed that stock prices are modeled with a fractional Brownian motion (FBM). The FBM is a Gaussian process with dependent and stationary increments except H = ½. The FBM can model stock prices with short or long memory. We propose a trajectory generation technique based on fast Fourier transforms to simulate stock prices modeled by FBM. A stock price trajectory is utilized to predict pricing of barrier options. Barrier options are options whose payoff function depend on the stock prices during the option’s lifetime. Using the results of the stock price trajectory and RQMC method can be determined the price of a barrier option under FBM. We conclude that RQMC is an efficient technique for calculating the price of barrier options rather than a standard Monte Carlo (MC).

On the Convergence Rate of Randomized Quasi--Monte Carlo for Discontinuous Functions

This paper studies the convergence rate of randomized quasi--Monte Carlo (RQMC) for discontinuous functions, which are often of infinite variation in the sense of Hardy and Krause. It was previously known that the root mean square error (RMSE) of RQMC is only $o(n^{-1/2})$ for discontinuous functions. For certain discontinuous functions in $d$ dimensions, we prove that the RMSE of RQMC is $O(n^{-1/2-1/(4d-2)+\epsilon})$ for any $\epsilon>0$ and arbitrary $n$. If some discontinuity boundaries are parallel to some coordinate axes, the rate can be improved to $O(n^{-1/2-1/(4d_u-2)+\epsilon})$, where $d_u$ denotes the so-called irregular dimension, that is, the number of axes which are not parallel to the discontinuity boundaries. Moreover, this paper shows that the RMSE is $O(n^{-1/2-1/(2d)})$ for certain indicator functions.

Optimization of a Monte Carlo variance reduction method based on sensitivity derivatives

We propose an optimization technique for an efficient sampling method known as sensitivity derivative enhanced sampling (SDES). It has been shown in certain cases that SDES can bring no improvement over or even slow crude Monte Carlo sampling. Our proposed optimized version of SDES guarantees variance reduction and improved accuracy in estimates. The optimized SDES can also improve randomized quasi-Monte Carlo (RQMC) sampling, which converges at a higher rate compared to the Monte Carlo sampling. Numerical experiments are performed on three test cases including the generalized steady-state Burgers equation and the Korteweg–de Vries equation. The results show that the optimized SDES can improve crude Monte Carlo (or RQMC) and SDES by up to an order of magnitude. RQMC coupled with the optimized SDES provides the largest efficiency gains, which can be as high as 1800.

Parallel Randomized Quasi-Monte Carlo Simulation for Asian Basket Option Pricing

High-dimensional derivatives pricing, such as Asian basket options, poses great computational challenges in practice. In this paper, parallel Randomized Quasi-Monte Carlo (RQMC) simulation method is investigated to tackle this kind of intractable problems. By using the intrinsic nature of “embarrassingly parallel”, parallel RQMC algorithm for Asian basket option pricing is effectively implemented using MPI. Numerical experiments are performed on supercomputer DeepComp6800 and the parallel performance is then analyzed. Our parallel algorithm has exerted perfect performance and good scalability. Parallel computing has greatly improved the computational efficiency of derivatives pricing.

Estimation of the mixed logit likelihood function by randomized quasi-Monte Carlo

We examine the effectiveness of randomized quasi-Monte Carlo (RQMC) techniques to estimate the integrals that express the discrete choice probabilities in a mixed logit model, for which no closed form formula is available. These models are used extensively in travel behavior research. We consider popular RQMC constructions such as randomized Sobol’, Faure, and Halton points, but our main emphasis is on randomly-shifted lattice rules, for which we study how to select the parameters as a function of the considered class of integrands. We compare the effectiveness of all these methods and of standard Monte Carlo (MC) to reduce both the variance and the bias when estimating the log-likelihood function at a given parameter value. In our numerical experiments, randomized lattice rules (with carefully selected parameters) and digital nets are the best performers and they reduce the bias as much as the variance. With panel data, in our examples, the performance of all RQMC methods degrades rapidly when we simultaneously increase the dimension and the number of observations per individual.

Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands

We study a random sampling technique to approximate integrals $\int_{[0,1]^s}f(\mathbf{x})\,\mathrm{d}\mathbf{x}$ by averaging the function at some sampling points. We focus on cases where the integrand is smooth, which is a problem which occurs in statistics. The convergence rate of the approximation error depends on the smoothness of the function $f$ and the sampling technique. For instance, Monte Carlo (MC) sampling yields a convergence of the root mean square error (RMSE) of order $N^{-1/2}$ (where $N$ is the number of samples) for functions $f$ with finite variance. Randomized QMC (RQMC), a combination of MC and quasi-Monte Carlo (QMC), achieves a RMSE of order $N^{-3/2+\varepsilon}$ under the stronger assumption that the integrand has bounded variation. A combination of RQMC with local antithetic sampling achieves a convergence of the RMSE of order $N^{-3/2-1/s+\varepsilon}$ (where $s\ge1$ is the dimension) for functions with mixed partial derivatives up to order two. Additional smoothness of the integrand does not improve the rate of convergence of these algorithms in general. On the other hand, it is known that without additional smoothness of the integrand it is not possible to improve the convergence rate. This paper introduces a new RQMC algorithm, for which we prove that it achieves a convergence of the root mean square error (RMSE) of order $N^{-\alpha-1/2+\varepsilon}$ provided the integrand satisfies the strong assumption that it has square integrable partial mixed derivatives up to order $\alpha>1$ in each variable. Known lower bounds on the RMSE show that this rate of convergence cannot be improved in general for integrands with this smoothness. We provide numerical examples for which the RMSE converges approximately with order $N^{-5/2}$ and $N^{-7/2}$, in accordance with the theoretical upper bound.

Coupling from the past with randomized quasi-Monte Carlo

The coupling-from-the-past (CFTP) algorithm of Propp and Wilson permits one to sample exactly from the stationary distribution of an ergodic Markov chain. By using it n times independently, we obtain an independent sample from that distribution. A more representative sample can be obtained by creating negative dependence between these n replicates; other authors have already proposed to do this via antithetic variates, Latin hypercube sampling, and randomized quasi-Monte Carlo (RQMC). We study a new, often more effective, way of combining CFTP with RQMC, based on the array-RQMC algorithm. We provide numerical illustrations for Markov chains with both finite and continuous state spaces, and compare with the RQMC combinations proposed earlier.