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Mathematics and Education in Mathematics

Deterministic algorithm for optimising the direction numbers of the Sobol’ sequence

Emanouil I. Atanassov

Published Jan 1, 2021• 0 citations
OpenAlex

Abstract

The quasi-Monte Carlo (QMC) methods use deterministic, usually low-discrepancy sequences, to improve the order of convergence when compared with Monte Carlo methods dealing with the same problem. Although this improved convergence depends on the smoothness properties of the functions under consideration and sometimes cannot be established theoretically, good practical results were obtained in many areas, notably in Financial Mathematics. Most of the QMC methods do not depend on particular properties of the sequences used beyond their good distribution in a suitable multidimensional space. Many families of low-discrepancy sequences have been studies, with the family of the Sobol sequences being not only one of the oldest but also the most popular in practical applications. Since these sequences have many free parameters, it is desirable to have methods to obtain a good set of parameters, suitable for a wide class of problems. In this work we propose a fully deterministic method for optimisation of the direction numbers of the Sobol sequences that is flexible in taking into account smoothness properties of the potential sub-integral functions. We demonstrate how by implementing this algorithm on modern GPUs it is possible to obtain good sets of direction numbers within a reasonable timeframe. The numerical tests demonstrate the significant advantage in terms of accuracy that is observed, which makes the corresponding QMC methods even more competitive.

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The quasi-Monte Carlo (QMC) methods use deterministic, usually low-discrepancy sequences, to improve the order of convergence when compared with Monte Carlo methods dealing with the same problem. Although this improved convergence depends on the smoothness properties of the functions under consideration and sometimes cannot be established theoretically, good practical results were obtained in many areas, notably in Financial Mathematics. Most of the QMC methods do not depend on particular properties of the sequences used beyond their good distribution in a suitable multidimensional space. Many families of low-discrepancy sequences have been studies, with the family of the Sobol sequences being not only one of the oldest but also the most popular in practical applications. Since these sequences have many free parameters, it is desirable to have methods to obtain a good set of parameters, suitable for a wide class of problems. In this work we propose a fully deterministic method for optimisation of the direction numbers of the Sobol sequences that is flexible in taking into account smoothness properties of the potential sub-integral functions. We demonstrate how by implementing this algorithm on modern GPUs it is possible to obtain good sets of direction numbers within a reasonable timeframe. The numerical tests demonstrate the significant advantage in terms of accuracy that is observed, which makes the corresponding QMC methods even more competitive.
Despite possessing the low-discrepancy property, the classical d-dimensional Halton sequence is known to exhibit poorly distributed projections whendddbecomes even moderately large. This, in turn, often implies bad performance when implemented in quasi-Monte Carlo (QMC) methods in comparison to, for example, the Sobol’ sequence. As an attempt to eradicate this issue, we propose an adapted Halton sequence built by integer and irrational based van der Corput sequences and show empirically improved performance with respect to the accuracy of estimates in numerical integration and simulation. In addition, for the first time, a scrambling algorithm is proposed for irrational based digital sequences.
Monte Carlo (MQ) method is a powerful tool to approximate high dimensional integrals. The disadvantage of ordinary MQ is the slow convergence rate by cause of the essential randomness of this method. Computations of this convergence can lead to pure time consuming. Quasi-Monte Carlo (QMC) method yields considerably better results since this method is a deterministic alternative which uses Low-Discrepancy sequences instead of random samples . There are a multiplicity of big open problems in QMC-methods, problems partly arising from applications and partly arising from theory. QMC are developed to integrate over unite cube, where it has much more accuracy than MC for integrands of bounded variation. Integration over more general spaces such as triangles, disks and Cartesian products of such spaces is more challenging for QMC. Nevertheless in real-world applications various problems are defined over such spaces. The aim of this thesis is to provide a survey of a solution of such problems of numerical integration defined over non-cubical spaces. We present QMC and randomised QMC (RQMC) constructions in the triangle with a vanishing discrepancy based on the recent work of Basu 2016. The QMC construction is a version of the Van der Corput-Halton sequences specially made to the unit triangle. The attraction of scrambled net is replication based error estimation for QMC with slightly the same accuracy as QMC, and for smooth enough integrands.
Low-discrepancy point sets and digital sequences underpin quasi-Monte Carlo (QMC) methods for high-dimensional integration. We cast two long-standing QMC design problems as program synthesis and solve them with an LLM-guided evolutionary loop that mutates and selects code under task-specific fitness: (i) constructing finite 2D/3D point sets with low star discrepancy, and (ii) choosing Sobol' direction numbers that minimize randomized QMC error on downstream integrands. Our two-phase procedure combines constructive code proposals with iterative numerical refinement. On finite sets, we rediscover known optima in small 2D cases and set new best-known 2D benchmarks for N >= 40, while matching most known 3D optima up to the proven frontier (N <= 8) and reporting improved 3D benchmarks beyond. On digital sequences, evolving Sobol' parameters yields consistent reductions in randomized quasi-Monte Carlo (rQMC) mean-squared error for several 32-dimensional option-pricing tasks relative to widely used Joe--Kuo parameters, while preserving extensibility to any sample size and compatibility with standard randomizations. Taken together, the results demonstrate that LLM-driven evolutionary program synthesis can automate the discovery of high-quality QMC constructions, recovering classical designs where they are optimal and improving them where finite-N structure matters. Data and code are available at https://github.com/hockeyguy123/openevolve-star-discrepancy.git.
Quasi-Monte Carlo methods are a variant of ordinary Monte Carlo methods that employ highly uniform quasirandom numbers in place of Monte Carlo's pseudorandom numbers. Monte Carlo methods offer statistical error estimates; however, while quasi-Monte Carlo has a faster convergence rate than normal Monte Carlo, one cannot obtain error estimates from quasi-Monte Carlo sample values by any practical way. A recently proposed method, called randomized quasi-Monte Carlo methods, takes advantage of Monte Carlo and quasi-Monte Carlo methods. Randomness can be brought to bear on quasirandom sequences through scrambling and other related randomization techniques in randomized quasi-Monte Carlo methods, which provide an elegant approach to obtain error estimates for quasi-Monte Carlo based on treating each scrambled sequence as a different and independent random sample. The core of randomized quasi-Monte Carlo is to find an effective and fast algorithm to scramble (randomize) quasirandom sequences. This dissertation surveys research on algorithms and implementations of scrambled quasirandom sequences and proposes some new algorithms to improve the quality of scrambled quasirandom sequences. Besides obtaining error estimates for quasi-Monte Carlo, scrambling techniques provide a natural way to parallelize quasirandom sequences. This scheme is especially suitable for distributed or grid computing. By scrambling a quasirandom sequence we can produce a family of related quasirandom sequences. Finding one or a subset of optimal quasirandom sequences within this family is an interesting problem, as such optimal quasirandom sequences can be quite useful for quasi-Monte Carlo. The process of finding such optimal quasirandom sequences is called the derandomization of a randomized (scrambled) family. We summarize aspects of this technique and propose some new algorithms for finding optimal sequences from the Halton, Faure and Sobol sequences. Finally we explore applications of derandomization.
We study a Monte Carlo algorithm that is based on a specific (randomly shifted and dilated) lattice point set. The main result of this paper is that the mean squared error for a given compactly supported, square-integrable function is bounded by n−1/2n^{-1/2}n−1/2 times the L2L_2L2​-norm of the Fourier transform outside a region around the origin, where nnn is the expected number of function evaluations. As corollaries we obtain the order of convergence for the Sobolev spaces HpsH^s_pHps​ with isotropic, anisotropic or mixed smoothness for all values of the parameters. This proves, in particular, that the optimal order of convergence in the latter case is n−s−1/2n^{-s-1/2}n−s−1/2 for p≥2p\ge2p≥2, which is, in contrast to the case of deterministic algorithms, independent of the dimension. This shows that Monte Carlo algorithms can improve the order by more than n−1/2n^{-1/2}n−1/2 for a whole class of practically important function classes. All results carry over to functions defined on the unit cube without boundary conditions.
We study quasi–Monte Carlo algorithms based on low discrepancy sequences for multivariate integration. We consider the problem of how the minimal number of function evaluations needed to reduce the worst-case error from its initial error by a factor of ε \varepsilon depends on ε − 1 {\varepsilon }^{-1} and the dimension s s . Strong tractability means that it does not depend on s s and is bounded by a polynomial in ε − 1 {\varepsilon }^{-1} . The least possible value of the power of ε − 1 {\varepsilon }^{-1} is called the ε \varepsilon -exponent of strong tractability. Sloan and Woźniakowski established a necessary and sufficient condition of strong tractability in weighted Sobolev spaces, and showed that the ε \varepsilon -exponent of strong tractability is between 1 and 2. However, their proof is not constructive . In this paper we prove in a constructive way that multivariate integration in some weighted Sobolev spaces is strongly tractable with ε \varepsilon -exponent equal to 1, which is the best possible value under a stronger assumption than Sloan and Woźniakowski’s assumption. We show that quasi–Monte Carlo algorithms using Niederreiter’s ( t , s ) (t,s) -sequences and Sobol sequences achieve the optimal convergence order O ( N − 1 + δ ) O(N^{-1+\delta })