Weighted integration over a hyperrectangle based on digital nets and sequences

Higher order digital nets are special classes of point sets for quasi-Monte Carlo rules which achieve the optimal convergence rate for numerical integration of smooth functions. An explicit construction of higher order digital nets was proposed by Dick, which is based on digitally interlacing in a certain way the components of classical digital nets whose number of components is a multiple ds of the dimension s. In this paper we give a fast computer search algorithm to find good classical digital nets suitable for interlaced components by using polynomial lattice point sets. We consider certain weighted Sobolev spaces of smoothness of arbitrarily high order, and derive an upper bound on the mean square worst-case error for digitally shifted higher order digital nets. Employing this upper bound as a quality criterion, we prove that the component-by-component construction can be used efficiently to find good polynomial lattice point sets suitable for interlaced components. Through this approach we are able to get some tractability results under certain conditions on the weights. Fast construction using the fast Fourier transform requires the construction cost of \(O(dsN \log N)\) operations using O(N) memory, where N is the number of points and s is the dimension. This implies a significant reduction in the construction cost as compared to higher order polynomial lattice point sets. Numerical experiments confirm that the performance of our constructed point sets often outperforms those of higher order digital nets with Sobol’ sequences and Niederreiter-Xing sequences used for interlaced components, indicating the usefulness of our algorithm.

We review the basic principles of quasi-Monte Carlo (QMC) methods, the randomizations that turn them into variance-reduction techniques, the integration error and variance bounds obtained in terms of QMC point set discrepancy and variation of the integrand, and the main classes of point set constructions: lattice rules, digital nets, and permutations in different bases. QMC methods are designed to estimate s-dimensional integrals, for moderate or large (perhaps infinite) values of s. In principle, any stochastic simulation whose purpose is to estimate an integral fits this framework, but the methods work better for certain types of integrals than others (e.g., if the integrand can be well approximated by a sum of low-dimensional smooth functions). Such QMC-friendly integrals are encountered frequently in computational finance and risk analysis. We summarize the theory, give examples, and provide computational results that illustrate the efficiency improvement achieved. This article is targeted mainly for those who already know Monte Carlo methods and their application in finance, and want an update of the state of the art on quasi-Monte Carlo methods.

A quasi-Monte Carlo solver for thermal radiation in participating media

constructions of lowdiscrepancy point sets and sequences, which are of great importance for quasi-Monte Carlo methods in multidimensional numerical integration, are based on the concept of (t,m,s)-nets and (t,s)-sequences. A detailed theory was developed in Niederreiter [9] (see also [10, Chapter 4] for surveys of this theory). So-called digital nets and sequences are of special interest due to the following two reasons. First, until now all construction methods for (t,m,s)nets and (t,s)-sequences which are relevant for applications in quasi-Monte Carlo methods are digital methods over certain rings. Second, digital (t,m,s)-nets behave extremely well for the numerical integration of functions which are representable by an in some sense rapidly converging multivariate Walsh series. In a series of papers, Larcher and several co-authors established lattice rules for the numerical integration of multivariate Walsh series by digital nets. We refer to [5] for a concise introduction in the field of Larcher’s lattice rules. 1.1. Definitions and notations. The concepts of (t,m,s)-nets and of (t,s)-sequences in a base b provide point sets of b m points, respectively infinite sequences, in the half-open s-dimensional unit cube I s := [0, 1) s ,

The sound fields generated by ultrasonic transducers are modeled using the quasi-Monte Carlo (QMC) method, which is found to overcome the conflict between accuracy and efficiency that occurs in existing wave field calculation methods. The RI equation, which is frequently used as a model equation in ultrasonic field calculation, is used here as an exact method and for comparison purposes. In the QMC method, the judgment sampling method and Halton sequence are used for pseudo-random sampling from the sound source, and then the sound field distributions are found by solving the integral solution using the sample mean. Numerical examples and results are presented when modeling unfocused, focused, and steered and focused beam fields. The accuracy and efficiency of the QMC method are discussed by comparing the results obtained using different modeling methods. The results show that the proposed method has a high level of efficiency due to the nature of the QMC algorithm and a high level of accuracy because no approximation is required. In addition, wave fields can be modeled with the QMC method as long as sound sources can be effectively pseudo-randomly sampled, allowing the proposed method to be applied to various types of transducers.

In this article, we evaluate a quasi-Monte Carlo (QMC) method with various low-discrepancy sequences (LDS) in illumination optical systems which are adopted in some commercial products, and clarify the method's effectiveness quantitatively. We assumed the evaluated systems were an illumination optical system with a perfectly diffusing surface, and we compared them against the theoretical irradiance distribution. The evaluation results indicate that the QMC method delivers higher asymptotic convergence rate than the MC method does, and there is little difference between each LDS. In evaluation of simple optical systems that can be boiled down to low-dimensional numerical integration problems, the QMC method was found to be extremely effective.

High-dimensional multivariate integration is a difficult problem for which the Monte Carlo method is often the only viable approach. This method provides an unbiased estimator of the integral, together with a probabilistic error estimate (e.g., in the form of a confidence interval). The aim of randomized quasi-Monte Carlo (QMC) methods is to provide lower-variance unbiased estimators, also with error estimates. This talk will concentrate on one class of randomized QMC methods: randomized lattice rules.We will explain how these methods fit into QMC methods in general and why they are interesting, how to choose their parameters, and how they can be used for medium and large-scale simulations. Numerical examples will be given to illustrate their effectiveness.

Digital nets with infinite digit expansions and construction of folded digital nets for quasi-Monte Carlo integration

Digital net properties of a polynomial analogue of Frolov's construction

The way of measuring diameter by use of measuring bow height and chord length is commonly adopted for the large diameter work piece. In the process of computing the diameter of large work piece, measurement uncertainty is an important parameter and is always employed to evaluate the reliability of the measurement results. Therefore, it is essential to present reliable methods to evaluate the measurement uncertainty, especially in precise measurement. Because of the limitations of low convergence and unstable results of the Monte-Carlo (MC) method, the quasi-Monte-Carlo (QMC) method is used to estimate the measurement uncertainty. The QMC method is an improvement of the ordinary MC method which employs highly uniform quasi random numbers to replace MC's pseudo random numbers. In the process of evaluation, first, more homogeneous random numbers (quasi random numbers) are generated based on Halton's sequence. Then these random numbers are transformed into the desired distribution random numbers. The desired distribution random numbers are used to simulate the measurement errors. By computing the simulation results, measurement uncertainty can be obtained. An experiment of cylinder diameter measurement and its uncertainty evaluation are given. In the experiment, the guide to the expression of uncertainty in measurement method, MC method, and QMC method are validated. The result shows that the QMC method has a higher convergence rate and more stable evaluation results than that of the MC method. Therefore, the QMC method can be applied effectively to evaluate the measurement uncertainty.

Capturing stochastic variabilities in pedestrian flows: A dynamic continuum modeling approach

Normal-transformation-based probabilistic load flow with correlated wind and load forecast errors

Quasi–Monte Carlo (QMC) methods are important numerical tools in the pricing and hedging of complex financial instruments. The effectiveness of QMC methods crucially depends on the discontinuity and the dimension of the problem. This paper shows how the two fundamental limitations can be overcome in some cases. We first study how path-generation methods (PGMs) affect the structure of the discontinuities and what the effect of discontinuities is on the accuracy of QMC methods. The insight is that the discontinuities can be QMC friendly (i.e., aligned with the coordinate axes) or not, depending on the PGM. The PGMs that offer the best performance in QMC methods are those that make the discontinuities QMC friendly. The structure of discontinuities can affect the accuracy of QMC methods more significantly than the effective dimension. This insight motivates us to propose a novel way of handling the discontinuities. The basic idea is to align the discontinuities with the coordinate axes by a judicious design of a method for simulating the underlying processes. Numerical experiments demonstrate that the proposed method leads to dramatic variance reduction in QMC methods for pricing options and for estimating Greeks. It also reduces the effective dimension of the problem. This paper was accepted by Assaf Zeevi, stochastic models and simulation.

We study equal weight numerical integration, or Quasi Monte Carlo (QMC) rules, for functions in a Sobolev space H s ( S d ) \mathbb {H}^s( \mathbb {S}^d) with smoothness parameter s > d / 2 s > d/2 defined over the unit sphere S d \mathbb {S}^d in R d + 1 \mathbb {R}^{d+1} . Focusing on N N -point configurations that achieve optimal order QMC error bounds (as is the case for efficient spherical designs), we are led to introduce the concept of QMC designs: these are sequences of N N -point configurations X N X_N on S d \mathbb {S}^d such that the worst-case error satisfies sup f ∈ H s ( S d ) , ‖ f ‖ H s ≤ 1 | 1 N ∑ x ∈ X N f ( x ) − ∫ S d f ( x ) d σ d ( x ) | = O ( N − s / d ) , N → ∞ , \begin{equation*} \sup _{\substack {f \in \mathbb {H}^s( \mathbb {S}^d ), \\ \| f \|_{\mathbb {H}^s} \leq 1}} \Bigg | \frac {1}{N} \sum _{\mathbf {x} \in X_N} f( \mathbf {x} ) - \int _{\mathbb {S}^d} f( \mathbf {x} ) \, \mathrm {d} \sigma _d( \mathbf {x} ) \Bigg | = \mathcal {O}\big ( N^{-s/d} \big ), \qquad N \to \infty , \end{equation*} with an implied constant that depends on the H s ( S d ) \mathbb {H}^s( \mathbb {S}^d ) -norm, but is independent of N N . Here σ d \sigma _d is the normalized surface measure on S d \mathbb {S}^d . We provide methods for generation and numerical testing of QMC designs. An essential tool is an expression for the worst-case error in terms of a reproducing kernel for the space H s ( S d ) \mathbb {H}^s( \mathbb {S}^d ) with s > d / 2 s > d/2 . As a consequence of this and a recent result of Bondarenko et al. on the existence of spherical designs with appropriate number of points, we show that minimizers of the N N -point energy for this kernel form a sequence of QMC designs for H s ( S d ) \mathbb {H}^s( \mathbb {S}^d ) . Furthermore, without appealing to the Bondarenko et al. result, we prove that point sets that maximize the sum of suitable powers of the Euclidean distance between pairs of points form a sequence of QMC designs for H s ( S d ) \mathbb {H}^s( \mathbb {S}^d ) with s s in the interval ( d / 2 , d / 2 + 1 ) {(d/2,d/2+1)} . For such spaces there exist reproducing kernels with simple closed forms that are useful for numerical testing of optimal order Quasi Monte Carlo integration. Numerical experiments suggest that many familiar sequences of point sets on the sphere (equal area points, spiral points, minimal [Coulomb or logarithmic] energy points, and Fekete points) are QMC designs for appropriate values of s s . For comparison purposes we show that configurations of random points that are independently and uniformly distributed on the sphere do not constitute QMC designs for any s > d / 2 s>d/2 . If ( X N ) (X_N) is a sequence of QMC designs for H s ( S d ) \mathbb {H}^s( \mathbb {S}^d) , we prove that it is also a sequence of QMC designs for H s ′ ( S d ) \mathbb {H}^{s’}( \mathbb {S}^d) for all s ′ ∈ ( d / 2 , s ) s’\in (d/2,s) . This leads to the question of determining the supremum of such s s (here called the QMC strength of the sequence), for which we provide estimates based on computations for the aforementioned sequences.

Quantile function or inverse cumulative distribution function (CDF) is heavily utilized in modelling, simulation, reliability analysis and random number generation. The use is often limited if the inversion method fails to estimate it from the cumulative distribution function. As a result, approximation becomes the other feasible option. The failure of the inversion method is often due to the intractable nature of the CDF of the distribution. Approximation may come in the form of series expansions, closed form or functional approximation, numerical algorithm and the closed form expression drafted in terms of the quantile function of another distribution. This work used the cubic spline interpolation to obtain the closed form of the inverse cumulative distribution function of the Nakagami-m distribution. Consequently, the closed form of the quantile function obtained for the selected parameters of the distribution serves as an approximation which compares favourably with the R software values. The result obtained was a significant improvement over some results surveyed from literature for four reasons. Firstly, the approximates produced better results in simulation as evidenced by the some values of the root mean square error of this work when compared with others. Secondly, the result obtained at the extreme tail of the distribution is better than others selected from the literature. Thirdly, the closed form estimates are easy to compute and save computation time. The closed form of the quantile function obtained in this work can be used in simulating Nakagami random variables which are used in modelling attenuation and fading channels in wireless communications and ultrasonic tissue characterization.