Quasi-Monte Carlo Integration on the Unit Disk

This paper presents work on generation of specification-driven test cases based on quasirandom (low-discrepancy) sequences instead of pseudorandom numbers. This approach is novel in software testing. This enhanced uniformity of quasirandom sequences leads to faster generation of test cases covering all possibilities. We demonstrate by examples that quasirandom sequences can be a viable alternative to pseudorandom numbers in generating test cases. In this paper, we present a method that can generate test cases from a decision table specification more effectively via quasirandom numbers. Analysis of a simple problem in this paper shows that quasirandom sequences achieve better data than pseudorandom numbers, and have the potential to converge faster and so reduce the computational burden. The use of different quasirandom sequences for generating test cases is presented in this paper.

Front propagation is a challenge in numerical modeling, particularly for multi-fluid or multi-material systems requiring clear material separation. The GRU (Glimm Random Update) scheme, inspired by Glimm’s method, has been developed to handle sharp fronts on unstructured 2D/3D meshes. It achieves convergence rates numerically measured between 0 . 8 and 0 . 9 , which are higher than those of classical first-order schemes, typically limited to 0 . 5 in the presence of contact discontinuities. Previous work established convergence in probability for planar fronts with uniform velocity but relied on random sequences, whereas practical implementations use low-discrepancy sequences. This study aims to establish new theoretical convergence results for the GRU scheme for 1D non-uniform meshes and uniform front velocity conditions, extending beyond uniform meshes considered previously. More precisely, we prove convergence in probability of order close to one-half when using random sequences on non-uniform meshes, and exact first-order convergence in probability when using low-discrepancy sequences (deterministic) on uniform meshes. Partial results are also obtained for non-uniform meshes, which are of significant practical interest. These findings provide insights into the scheme’s behavior with both random and low-discrepancy sequences. Numerical tests are presented to illustrate the theoretical results, with applications to non-uniform meshes, highlighting the scheme’s practical relevance.

We present the first study on the approximability of geometric versions of the unique coverage problem and the minimum membership set cover problem. In the former problem, one is given a family of sets of elements from some universe and aims to select sets that maximize the number of elements contained in precisely one selected set. Unique Coverage has important applications in wireless networks and it is thus natural to consider it in a geometric setting. We use the well-known (unit) disk model for wireless networks and show that Unique Coverage remains NP-hard in this case and present a polynomial-time 1/18-approximation algorithm. This algorithm is extended to the budgeted low-coverage problem, where covering can element multiple times yields less profit and we have a fixed budget to 'buy' sets. We give an asymptotic FPTAS in case the disks have arbitrary size, but bounded ply. For the case that the geometric objects are arbitrary fat objects, we show that these problems are as hard to approximate as in the general case. In the minimum membership set cover problem, the goal is to cover all elements while minimizing the maximum number of sets in which any element is contained. For unit squares and unit disks, we show that the problem remains NP-hard and does not admit a polynomial-time approximation algorithm with ratio smaller than 2 unless P=NP. For unit squares, we give a 5-approximation algorithm for instances where the optimum objective value is bounded by a constant.

Energy efficiency presents a significant challenge for stochastic computing (SC) due to the long random binary bit streams required for accurate computation. In this paper, a type of low discrepancy (LD) sequences, the Sobol sequence, is considered for energy-efficient implementations of SC circuits. The use of Sobol sequences improves the output accuracy of a stochastic circuit with a reduced sequence length compared to the use of another type of LD sequences, the Halton sequence, and conventional linear feedback shift register (LFSR)-generated pseudorandom sequence. The use of Sobol sequences leads to a similar or higher accuracy than using Halton sequences for basic arithmetic operations. Sobol sequence generators cost less energy than the Halton counterparts when multiple random sequences are required in a circuit, thus the use of Sobol sequences can lead to a higher energy efficiency in an SC circuit than using Halton sequences.

We define quasirandom geometric networks using low-discrepancy sequences, such as Halton, Sobol, and Niederreiter. The networks are built in d dimensions by considering the d-tuples of digits generated by these sequences as the coordinates of the vertices of the networks in a d-dimensional I^{d} unit hypercube. Then, two vertices are connected by an edge if they are at a distance smaller than a connection radius. We investigate computationally 11 network-theoretic properties of two-dimensional quasirandom networks and compare them with analogous random geometric networks. We also study their degree distribution and their spectral density distributions. We conclude from this intensive computational study that in terms of the uniformity of the distribution of the vertices in the unit square, the quasirandom networks look more random than the random geometric networks. We include an analysis of potential strategies for generating higher-dimensional quasirandom networks, where it is know that some of the low-discrepancy sequences are highly correlated. In this respect, we conclude that up to dimension 20, the use of scrambling, skipping and leaping strategies generate quasirandom networks with the desired properties of uniformity. Finally, we consider a diffusive process taking place on the nodes and edges of the quasirandom and random geometric graphs. We show that the diffusion time is shorter in the quasirandom graphs as a consequence of their larger structural homogeneity. In the random geometric graphs the diffusion produces clusters of concentration that make the process more slow. Such clusters are a direct consequence of the heterogeneous and irregular distribution of the nodes in the unit square in which the generation of random geometric graphs is based on.

Comparison of Path Generation Methods for Monte Carlo Valuation of Single Underlying Derivative Securities

Optimisation of linear structures is a well-developed and highly relevant research field. In particular, for linear problems with differentiable objective functions, gradient-based methods are well suited. In engineering practice, structural nonlinear behaviour and imperfections different from ideal design often need to be considered. Optimised lightweight structures, such as long-span bridges and stadium roofs, are more sensitive to sudden failure through buckling. Imperfections or perturbations in geometry and loading can drastically reduce the buckling load of a structure compared to that of a perfect structure. It is, however, computationally expensive to analyse structural problems with nonlinearities and imperfections and to compute the derivatives of the respective objective functions, rendering gradient-based methods unsuitable. In the present work, we use a Bayesian optimisation technique with a Gaussian process as the surrogate model for the described structural problems with random imperfections. Bayesian optimisation is exceptionally well suited for expensive black-box objective functions with no analytical expressions. To rapidly determine the random buckling loads, we consider an extended system of equilibrium equations for directly computing the buckling loads, i.e. bifurcation and snap-through points on the load-displacement path. This approach is exceedingly efficient for repeatedly computing buckling loads for different imperfections. We use the complex-step derivative approximation to determine the required directional derivatives of matrices. A low discrepancy sequence, Sobol sequence, is adopted for sampling from imperfections

Recently P. R. Halmos [5] mentioned a question raised by S. V. Fomin as to whether it follows that two measure preserving transformations are conjugate if they are unitarily equivalent and possess the same entropy. The purpose of this work is to concoct a counterexample to this conjecture. Let X be the unit interval with Borel measurability and Lebesgue measure. Let XXX be the unit square and XXXXX the unit cube with the usual direct product measurabilities and measures. Finally let T., T,,a, and T^,a,4, denote the following measure preserving transformations defined respectively on the unit interval, unit square, and unit cube by

Random and quasirandom sequences: Numerical estimates of uniformity of distribution

In this paper we first investigate the use of Markov Chain Monte Carlo (MCMC) methods to attack classical ciphers. MCMC has previously been used to break simple substitution, transposition and substitution-transposition ciphers. Here, we improve the accuracy of obtained results by these algorithms and we show the performance of the algorithms using quasi random numbers such as Faure, Sobol and Niederreiter sequences.

Our paper proposes an implementable procedure for using the method of quasi-random sequences in software debug testing. In random testing, the sequence of tests (if considered as points in an -dimensional unit hypercube) will give rise to regions where there are clusters of points, as well as underpopulated regions. Quasi-random sequences, also known as low-discrepancy or low-dispersion sequences, are sequences of points in such a hypercube that are spread more evenly throughout. Based on the observation that program faults tend to lead to contiguous failure regions within a program's input domain, and that an even spread of random tests enhances the failure detection effectiveness for certain failure patterns, we examine the use of quasi-random sequences as a replacement for random sequences in automated testing. Because there are only a small number of quasi-random sequence generation algorithms, and each of them can only generate a small number of distinct sequences, the applicability of quasi-random sequences in testing real programs is severely restricted. To alleviate this problem, we examine the use of two types of randomized quasi-random sequences, which are quasi-random sequences permuted in a nondeterministic fashion in such a way as to retain their low discrepancy properties. We show that testing using randomized quasi-random sequences is often significantly more effective than random testing.

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.

A general concept for parallelizing quasi-Monte Carlo methods is introduced. By considering the distribution of computing jobs across a multiprocessor as an additional problem dimension, the straightforward application of quasi-Monte Carlo methods implies parallelization. The approach in fact partitions a single low-discrepancy sequence into multiple low-discrepancy sequences. This allows for adaptive parallel processing without synchronization, i.e. communication is required only once for the final reduction of the partial results. Independent of the number of processors, the resulting algorithms are deterministic, and generalize and improve upon previous approaches.

In this chapter we deal with numerical computation of integrals over the domain ℝ s (s >1) with respect to a positive weight function. For one-dimensional integrals Gauss Hermite formulas compute integrals with respect to a Gaussian weight with quite high accuracy but for high-dimensional integrals the effort increases exponentially. For integrals over the s-dimensional unit cube probabilistic methods like Monte Carlo (MC) are not affected by this so-called “curse of dimensions,” but the convergence rate is rather poor. If the integration nodes are not pure random points but special deterministic point sequences (the method is called Quasi Monte Carlo (QMC) due to this fact) this rate can be significantly improved. These lowdiscrepancy sequences appear also in the computation of integrals over ℝ s .

We present an integrable solution for the detection of defective sensor elements using sigma-delta-(/spl Sigma//spl Delta/)-modulation and a matched filter. The sensor element is stimulated using a pseudo random binary sequence (PRBS). The sensor signal is read out and the analog output is digitized using a /spl Sigma//spl Delta/-modulator. The binary pulse density stream of the /spl Sigma//spl Delta/-modulator is the output of the sensor system and thus should ideally contain the PRBS. A matched filter has the task of detecting the pseudo random sequence in the pulse density stream and its sampled output is compared to a threshold thus making it possible to judge the functionality of the sensor element. By evaluating the magnitude of the matched filter output it is also possible to measure the sensor sensitivity. We present a discrete solution of this method, but an integrated chip using a standard 1.2 /spl mu/m CMOS process has been designed and is being fabricated.