quasi-Monte Carlo Algorithms Research Articles

Energy Study of Monte Carlo and Quasi-Monte Carlo Algorithms for Solving Integral Equations

In the past few years the development of exascale computing technology necessitated to obtain an estimate for the energy consumption when large-scale problems are solved with different high-performance computing (HPC) systems. In this paper we study the energy efficiency of a class of Monte Carlo (MC) and Quasi-Monte Carlo (QMC) algorithms for a given integral equation using hybrid HPC systems. The algorithms are applied to solve quantum kinetic integral equations describing ultra-fast transport in quantum wire. We compare the energy performance of the algorithms using a GPU-based computer platform and CPU-based computer platform both with and without hyper-threading (HT) technology. We use SPRNG library and CURAND generator to produce parallel pseudo-random (PPR) sequences for the MC algorithms on CPU-based and GPU -based platforms, respectively. For our QMC algorithms Sobol and Halton sequences are used to produce parallel quasi-random (PQR) sequences.We compare the obtained results of the tested algorithms with respect to the given energy metric. The results of our study demonstrate the importance of taking into account not only scalability of the HPC intensive algorithms but also their energy efficiency. They also show the need for further optimisation of the QMC algorithms when GPU-based computing platforms are used.

On the Performance, Scalability and Sensitivity Analysis of a Large Air Pollution Model

Computationally efficient sensitivity analysis of a large-scale air pollution model is an important issue we focus on in this paper. Sensitivity studies play an important role for reliability analysis of the results of complex nonlinear models as those used in the air pollution modelling. There is a number of uncertainties in the input data sets, as well as in some internal coefficients, which determine the speed of the main chemical reactions in the chemical part of the model. These uncertainties are subject to our quantitative sensitivity study. Monte Carlo and quasi-Monte Carlo algorithms are used in this study.A large number of numerical experiments with some special modifications of the model must be carried out in order to collect the necessary input data for the particular sensitivity study. For this purpose we created an efficient high performance implementation SA-DEM, based on the MPI version of the package UNI-DEM. A large number of numerical experiments were carried out with SA-DEM on the IBM MareNostrum III at BSC - Barcelona, helped us to identify a severe performance problem with an earlier version of the code and to resolve it successfuly. The improved implementation appears to be quite efficient for that challenging computational problem, as our experiments show. Some numerical results with performance and scalability analysis of these results are presented in the paper.

Open type quasi-Monte Carlo integration based on Halton sequences in weighted Sobolev spaces

In this paper, we study quasi-Monte Carlo (QMC) integration in weighted Sobolev spaces. In contrast to many previous results the QMC algorithms considered here are of open type, i.e., they are extensible in the number of sample points without having to discard the samples already used. As the underlying integration nodes we consider randomized Halton sequences in prime bases p=(p1,…,ps) for which we study the root mean square (RMS) worst-case error. The randomization method is a p-adic shift which is based on p-adic arithmetic.In the class of open methods, the error bounds are optimal with respect to the order of magnitude of the number of sample nodes. Furthermore we obtain conditions on the coordinate weights under which the error bounds are independent of the dimension s. In terms of the field of Information-Based Complexity this means that the corresponding QMC rule achieves a strong polynomial tractability error bound. Our findings on the RMS worst-case error of randomized Halton sequences can be carried over to the RMS L2-discrepancy.Except for the p-adic shift our results are fully constructive and no search algorithms (such as the component-by-component algorithm) are required.

Scrambled Geometric Net Integration Over General Product Spaces

Quasi-Monte Carlo (QMC) sampling has been developed for integration over $[0,1]^s$ where it has superior accuracy to Monte Carlo (MC) for integrands of bounded variation. Scrambled net quadrature gives allows replication based error estimation for QMC with at least the same accuracy and for smooth enough integrands even better accuracy than plain QMC. Integration over triangles, spheres, disks and Cartesian products of such spaces is more difficult for QMC because the induced integrand on a unit cube may fail to have the desired regularity. In this paper, we present a construction of point sets for numerical integration over Cartesian products of $s$ spaces of dimension $d$, with triangles ($d=2$) being of special interest. The point sets are transformations of randomized $(t,m,s)$-nets using recursive geometric partitions. The resulting integral estimates are unbiased and their variance is $o(1/n)$ for any integrand in $L^2$ of the product space. Under smoothness assumptions on the integrand, our randomized QMC algorithm has variance $O(n^{-1 - 2/d} (\log n)^{s-1})$, for integration over $s$-fold Cartesian products of $d$-dimensional domains, compared to $O(n^{-1})$ for ordinary Monte Carlo.

Preparing input data for sensitivity analysis of an air pollution model by using high-performance supercomputers and algorithms

Sensitivity analysis is an important issue in large-scale mathematical modelling. We developed a novel 3-stage method for global sensitivity analysis of the Unified Danish Eulerian Model (UNI-DEM). This is a powerful large-scale air pollution model with an up-to-date high-performance software implementation. There is a number of uncertain internal parameters, especially in the chemistry–emission submodel, which are subject to our quantitative sensitivity study. Efficient Monte Carlo and quasi-Monte Carlo algorithms based on Sobol sequences are used in this study.A large number of numerical experiments with some special modifications of the model must be carried out in order to collect the necessary input data for the particular sensitivity study. For this purpose we created an efficient high performance implementation SA-DEM, based on the MPI version of the package UNI-DEM. A vast number of numerical experiments were carried out with SA-DEM on an IBM Blue Gene/P, the most powerful parallel supercomputer, at the time of the write-up of this paper, in Bulgaria. Even this powerful machine has some problems with the storage when SA-DEM is to be run with the refined (480×480) version of the mesh. The code was implemented with some enhancements on the IBM MareNostrum III at BSC — Barcelona, the most powerful parallel supercomputer in Spain. This implementation appears to be quite efficient for that challenging computational problem, as our experiments show. Some numerical results and performance analysis of these results will be presented in the paper.

From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules

In 1935 J.G. van der Corput introduced a sequence which has excellent uniform distribution properties modulo 1. This sequence is based on a very simple digital construction scheme with respect to the binary digit expansion. Nowadays the van der Corput sequence, as it was named later, is the prototype of many uniformly distributed sequences, also in the multi-dimensional case. Such sequences are required as sample nodes in quasi-Monte Carlo algorithms, which are deterministic variants of Monte Carlo rules for numerical integration. Since its introduction many people have studied the van der Corput sequence and generalizations thereof. This led to a huge number of results.On the occasion of the 125th birthday of J.G. van der Corput we survey many interesting results on van der Corput sequences and their generalizations. In this way we move from van der Corput’s ideas to the most modern constructions of sequences for quasi-Monte Carlo rules, such as, e.g., generalized Halton sequences or Niederreiter’s (t,s)-sequences.

Quasi-Monte Carlo methods for linear two-stage stochastic programming problems

Quasi-Monte Carlo (QMC) algorithms are studied for generating scenarios to solve two-stage linear stochastic programming problems. Their integrands are piecewise linear-quadratic, but do not belong to the function spaces considered for QMC error analysis. We show that under some weak geometric condition on the two-stage model all terms of their ANOVA decomposition, except the one of highest order, are continuously differentiable and second order mixed derivatives exist almost everywhere and belong to $$L_{2}$$L2. This implies that randomly shifted lattice rules may achieve the optimal rate of convergence $$O(n^{-1+\delta })$$O(n-1+?) with $$\delta \in (0,\frac{1}{2}]$$??(0,12] and a constant not depending on the dimension if the effective superposition dimension is less than or equal to two. The geometric condition is shown to be satisfied for almost all covariance matrices if the underlying probability distribution is normal. We discuss effective dimensions and techniques for dimension reduction. Numerical experiments for a production planning model with normal inputs show that indeed convergence rates close to the optimal rate are achieved when using randomly shifted lattice rules or scrambled Sobol' point sets accompanied with principal component analysis for dimension reduction.

Dimension Reduction for Pricing Options Under Multidimensional Lévy Processes

The aim of this study was to develop efficient quasi-Monte Carlo algorithms for pricing European derivative securities under multidimensional Levy models. In the paper, we first introduce the multidimensional generalized hyperbolic distribution as a normal variance–mean mixture. Using this distribution, we can model a multidimensional generalized Levy process as a subordinated Brownian motion. Under this process, we develop practically efficient dimension reduction methods that can enhance the numerical efficiency of the quasi-Monte Carlo method. The algorithms extend the generalized linear transformation method that was originally proposed for a univariate Levy process. We also propose hybrid types of dimension reduction methods in which the dimension reduction techniques are applied separately to the subordinator and the Brownian motion. Through numerical examples we demonstrate that the proposed method realizes a substantial gain in efficiency, relative to the naive Monte Carlo and quasi-Monte Carlo methods in the context of pricing average options.

Infinite-Dimensional Integration in Weighted Hilbert Spaces: Anchored Decompositions, Optimal Deterministic Algorithms, and Higher-Order Convergence

We study numerical integration of functions depending on an infinite number of variables. We provide lower error bounds for general deterministic linear algorithms and provide matching upper error bounds with the help of suitable multilevel algorithms and changing dimension algorithms. More precisely, the spaces of integrands we consider are weighted reproducing kernel Hilbert spaces with norms induced by an underlying anchored function space decomposition. Here the weights model the relative importance of different groups of variables. The error criterion used is the deterministic worst case error. We study two cost models for function evaluation which depend on the number of active variables of the chosen sample points, and two classes of weights, namely product and order-dependent (POD) weights and the newly introduced weights with finite active dimension. We show for these classes of weights that multilevel algorithms achieve the optimal rate of convergence in the first cost model while changing dimension algorithms achieve the optimal convergence rate in the second model. As an illustrative example, we discuss the anchored Sobolev space with smoothness parameter $\alpha$ and provide new optimal quasi-Monte Carlo multilevel algorithms and quasi-Monte Carlo changing dimension algorithms based on higher-order polynomial lattice rules.

基于准蒙特卡罗算法的测透镜焦距数据误差分析

基于准蒙特卡罗算法的测透镜焦距数据误差分析

A Novel Quota Sampling Algorithm for Generating Representative Random Samples given Small Sample Size

In this paper, a novel algorithm is proposed for sampling from discrete probability distributions using the probability proportional to size sampling method, which is a special case of Quota sampling method. The motivation for this study is to devise an efficient sampling algorithm that can be used in stochastic optimization problems -- when there is a need to minimize the sample size. Several experiments have been conducted to compare the proposed algorithm with two widely used sample generation methods, the Monte Carlo using inverse transform, and quasi-Monte Carlo algorithms. The proposed algorithm gave better accuracy than these methods, and in terms of time complexity it is nearly of the same order.

Advanced algorithms for multidimensional sensitivity studies of large-scale air pollution models based on Sobol sequences

In this paper advanced variance-based algorithms for global sensitivity analysis are studied. We consider efficient algorithms, such as Monte Carlo, quasi-Monte Carlo (QMC) and scrambled quasi-Monte Carlo algorithms based on Sobol sequences. Low discrepancy ΛΠτ Sobol sequences are considered as a basis. Two other approaches are also analyzed. The first one is an efficient Monte Carlo (MC) algorithm for multidimensional integration based on modified Sobol sequences (MCA-MSS) and proposed in an earlier work by some of the authors Dimov and Georgieva (2011) [28]. The second one is a randomized QMC algorithm proposed by Art Owen (1995) [20]. The procedure of randomization in the latter case is also known as Owen scrambling.The algorithms considered in this work are applied to sensitivity studies of air pollution levels calculated by the Unified Danish Eulerian Model (UNI-DEM) to some chemical reaction rates. UNI-DEM is chosen as a case study since it constitutes a typical large-scale mathematical model in which the chemical reactions are adequately presented. Extensive numerical experiments are performed to support the theoretical studies and to analyze applicability of algorithms under consideration to various classes of problems. Conclusions about the applicability and efficiency of the algorithms under consideration are drawn.

Statistical buffeting response of flexible bridges influenced by errors in aeroelastic loading estimation

This study discusses the development of an algorithm for predicting the “statistical” buffeting response of a long-span bridge using Monte Carlo (MC) and Quasi-Monte Carlo (QMC) methods. The proposed methodology, based on the standard multi-mode approach for bridge buffeting, was used to compute the power spectral density of the buffeting modal forces and to derive the dynamic bridge response in the case that measurement errors in the aeroelastic loading were anticipated. Closed-form solution, derived through standard numerical integration, was used to validate the MC and QMC algorithms. Preliminary simulations were used for validation of the algorithms and the optimal selection of the number of integration points.In the second part of the study, fragility curves and surfaces of the dynamic response were numerically derived by accounting for experimental errors in the estimation of the aeroelastic loading. A collection of these curves describes the conditional exceedance probability of representative indicators, associated with a specific feature of the dynamic response, at a given mean wind velocity and direction. A set of thresholds, derived from such indicators, was proposed to characterize the structural performance.

A Computational Approach to Financial Option Pricing Using Quasi Monte Carlo Methods via Variance Reduction Techniques

In this paper, we consider two types of pricing option in financial markets using quasi Monte Carlo algorithm with variance reduction procedures. We evaluate Asian-style and European-style options pricing based on Black-Scholes model. Finally, some numerical results presented.

Extensible hyperplane nets

Extensible (polynomial) lattice point sets have the property that the number N of points in the node set of a quasi-Monte Carlo algorithm may be increased while retaining the existing points. Explicit constructions for extensible (polynomial) lattice point sets have been presented recently by Niederreiter and Pillichshammer. It is the aim of this paper to establish extensibility for a powerful generalization of polynomial lattice point sets, the so-called hyperplane nets.

Contributors

Contributors

Quasi-Monte Carlo integration on the grid for sensitivity studies

In this paper we present error and performance analysis of quasi-Monte Carlo algorithms for solving multidimensional integrals (up to 100 dimensions) on the grid using MPI. We take into account the fact that the Grid is a potentially heterogeneous computing environment, where the user does not know the specifics of the target architecture. Therefore parallel algorithms should be able to adapt to this heterogeneity, providing automated load-balancing. Monte Carlo algorithms can be tailored to such environments, provided parallel pseudorandom number generators are available. The use of quasi-Monte Carlo algorithms poses more difficulties. In both cases the efficient implementation of the algorithms depends on the functionality of the corresponding packages for generating pseudorandom or quasirandom numbers. We propose efficient parallel implementation of the Sobol sequence for a grid environment and we demonstrate numerical experiments on a heterogeneous grid. To achieve high parallel efficiency we use a newly developed special grid service called Job Track Service which provides efficient management of available computing resources through reservations.

A functional-structural kiwifruit vine model integrating architecture, carbon dynamics and effects of the environment.

Functional-structural modelling can be used to increase our understanding of how different aspects of plant structure and function interact, identify knowledge gaps and guide priorities for future experimentation. By integrating existing knowledge of the different aspects of the kiwifruit (Actinidia deliciosa) vine's architecture and physiology, our aim is to develop conceptual and mathematical hypotheses on several of the vine's features: (a) plasticity of the vine's architecture; (b) effects of organ position within the canopy on its size; (c) effects of environment and horticultural management on shoot growth, light distribution and organ size; and (d) role of carbon reserves in early shoot growth. Using the L-system modelling platform, a functional-structural plant model of a kiwifruit vine was created that integrates architectural development, mechanistic modelling of carbon transport and allocation, and environmental and management effects on vine and fruit growth. The branching pattern was captured at the individual shoot level by modelling axillary shoot development using a discrete-time Markov chain. An existing carbon transport resistance model was extended to account for several source/sink components of individual plant elements. A quasi-Monte Carlo path-tracing algorithm was used to estimate the absorbed irradiance of each leaf. Several simulations were performed to illustrate the model's potential to reproduce the major features of the vine's behaviour. The model simulated vine growth responses that were qualitatively similar to those observed in experiments, including the plastic response of shoot growth to local carbon supply, the branching patterns of two Actinidia species, the effect of carbon limitation and topological distance on fruit size and the complex behaviour of sink competition for carbon. The model is able to reproduce differences in vine and fruit growth arising from various experimental treatments. This implies it will be a valuable tool for refining our understanding of kiwifruit growth and for identifying strategies to improve production.

Construction algorithms for higher order polynomial lattice rules

Higher order polynomial lattice point sets are special types of digital higher order nets which are known to achieve almost optimal convergence rates when used in a quasi-Monte Carlo algorithm to approximate high-dimensional integrals over the unit cube. The existence of higher order polynomial lattice point sets of “good” quality has recently been established, but their construction was not addressed. We use a component-by-component approach to construct higher order polynomial lattice rules achieving optimal convergence rates for functions of arbitrarily high smoothness and at the same time–under certain conditions on the weights–(strong) polynomial tractability. Combining this approach with a sieve-type algorithm yields higher order polynomial lattice rules adjusting themselves to the smoothness of the integrand up to a certain given degree. Higher order Korobov polynomial lattice rules achieve analogous results.

Quasi-Monte Carlo Methods for some Linear Algebra Problems. Convergence and Complexity

We present quasi-Monte Carlo analogs of Monte Carlo methods for some linear algebra problems: solving systems of linear equations, computing extreme eigenvalues, and matrix inversion. Reformulating the problems as solving integral equations with a special kernels and domains permits us to analyze the quasi-Monte Carlo methods with bounds from numerical integration. Standard Monte Carlo methods for integration provide a convergence rate of O(N^(−1/2)) using N samples. Quasi-Monte Carlo methods use quasirandom sequences with the resulting convergence rate for numerical integration as good as O((logN)^k)N^(−1)). We have shown theoretically and through numerical tests that the use of quasirandom sequences improves both the magnitude of the error and the convergence rate of the considered Monte Carlo methods. We also analyze the complexity of considered quasi-Monte Carlo algorithms and compare them to the complexity of the analogous Monte Carlo and deterministic algorithms.