Most quasi-Monte Carlo research focuses on sampling from the unit cube. Many problems, especially in computer graphics, are defined via quadrature over the unit triangle. Quasi-Monte Carlo methods for the triangle have been developed by Pillands and Cools (2005) and by Brandolini et al. (2013). This paper presents two QMC constructions in the triangle with a vanishing discrepancy. The first is a version of the van der Corput sequence customized to the unit triangle. It is an extensible digital construction that attains a discrepancy below 12/sqrt(N). The second construction rotates an integer lattice through an angle whose tangent is a quadratic irrational number. It attains a discrepancy of O(log(N)/N) which is the best possible rate. Previous work strongly indicated that such a discrepancy was possible, but no constructions were available. Scrambling the digits of the first construction improves its accuracy for integration of smooth functions. Both constructions also yield convergent estimates for integrands that are Riemann integrable on the triangle without requiring bounded variation.
Most quasi-Monte Carlo research focuses on sampling from the unit cube. Many problems, especially in computer graphics, are defined via quadrature over the unit triangle. Quasi-Monte Carlo methods for the triangle have been developed by Pillands and Cools (2005) and by Brandolini et al. (2013). This paper presents two QMC constructions in the triangle with a vanishing discrepancy. The first is a version of the van der Corput sequence customized to the unit triangle. It is an extensible digital construction that attains a discrepancy below 12/sqrt(N). The second construction rotates an integer lattice through an angle whose tangent is a quadratic irrational number. It attains a discrepancy of O(log(N)/N) which is the best possible rate. Previous work strongly indicated that such a discrepancy was possible, but no constructions were available. Scrambling the digits of the first construction improves its accuracy for integration of smooth functions. Both constructions also yield convergent estimates for integrands that are Riemann integrable on the triangle without requiring bounded variation.
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.
Despite possessing the low-discrepancy property, the classical d-dimensional Halton sequence is known to exhibit poorly distributed projections whendbecomes 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.
This paper is a contemporary review of QMC (‘quasi-Monte Carlo’) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1] s , where s may be large, or even infinite. After a general introduction, the paper surveys recent developments in lattice methods, digital nets, and related themes. Among those recent developments are methods of construction of both lattices and digital nets, to yield QMC rules that have a prescribed rate of convergence for sufficiently smooth functions, and ideally also guaranteed slow growth (or no growth) of the worst-case error as s increases. A crucial role is played by parameters called ‘weights’, since a careful use of the weight parameters is needed to ensure that the worst-case errors in an appropriately weighted function space are bounded, or grow only slowly, as the dimension s increases. Important tools for the analysis are weighted function spaces, reproducing kernel Hilbert spaces, and discrepancy, all of which are discussed with an appropriate level of detail.
Quasi-Monte Carlo Beyond Hardy-Krause II: (1+ε)n Samples Suffice
Numerical integration studies how well one can estimate the integral of a function f over [0,1)d using n sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error Od(σSO(f)/n), where the smoothed-out variation σSO(f) can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires n2 i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant ε>0, we show that (1+ε)n i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.
Abstract. We study quasi-Monte Carlo (QMC) integration over the multidimensional unit cube in several weighted function spaces with different smoothness classes. We consider approximating the integral of a function by the median of several integral estimates under independent and random choices of the underlying QMC point sets (either linearly scrambled digital nets or infinite-precision polynomial lattice point sets). Even though our approach does not require any information on the smoothness and weights of a target function space as an input, we can prove a probabilistic upper bound on the worst-case error for the respective weighted function space, where the failure probability converges to 0 exponentially fast as the number of estimates increases. Our obtained rates of convergence are nearly optimal for function spaces with finite smoothness, and we can attain a dimension-independent super-polynomial convergence for a class of infinitely differentiable functions. This implies that our median-based QMC rule is universal in the sense that it does not need to be adjusted to the smoothness and the weights of the function spaces and yet exhibits the nearly optimal rate of convergence. Numerical experiments support our theoretical results.
High-dimensional integrals arise in a variety of areas, including quantum physics, the physics and chemistry of molecules, statistical mechanics and more recently, in financial applications. In order to approximate multidimensional integrals, one may use Monte Carlo methods in which the quadrature points are generated randomly or quasi-Monte Carlo methods, in which points are generated deterministically. One particular class of quasi-Monte Carlo methods for multivariate integration is represented by lattice rules. Lattice rules constructed throughout this thesis allow good approximations to integrals of functions belonging to certain weighted function spaces. These function spaces were proposed as an explanation as to why integrals in many variables appear to be successfully approximated although the standard theory indicates that the number of quadrature points required for reasonable accuracy would be astronomical because of the large number of variables. The purpose of this thesis is to contribute to theoretical results regarding the construction of lattice rules for multiple integration. We consider both lattice rules for integrals over the unit cube and lattice rules suitable for integrals over Euclidean space. The research reported throughout the thesis is devoted to finding the generating vector required to produce lattice rules that have what is termed a low weighted discrepancy . In simple terms, the discrepancy is a measure of the uniformity of the distribution of the quadrature points or in other settings, a worst-case error. One of the assumptions used in these weighted function spaces is that variables are arranged in the decreasing order of their importance and the assignment of weights in this situation results in so-called product weights . In other applications it is rather the importance of group of variables that matters. This situation is modelled by using function spaces in which the weights are general . In the weighted settings mentioned above, the quality of the lattice rules is assessed by the weighted discrepancy mentioned earlier. Under appropriate conditions on the weights, the lattice rules constructed here produce a convergence rate of the error that ranges from O(n−1/2) to the (believed) optimal O(n−1+δ) for any δ gt 0, with the involved constant independent of the dimension.