Quasi-random Research Articles

On The Classification of Ls-Sequences

Abstract This paper addresses the question whether the LS-sequences constructed in [Car12] yield indeed a new family of low-discrepancy sequences. While it is well known that the case S = 0 corresponds to van der Corput sequences, we prove here that the case S = 1 can be traced back to symmetrized Kronecker sequences and moreover that for S ≥ 2 none of these two types occurs anymore. In addition, our approach allows for an improved discrepancy bound for S = 1 and L arbitrary.

A parametric model order reduction technique for inverse viscoelastic material identification

Viscoelastic materials are mostly used to passively suppress structural vibrations. The theoretical analysis and design of such system require the knowledge of damping properties which are highly frequency-dependent and proper mathematical models for this dependency. The fractional derivative model is attractive as very few empirical parameters are required. These parameters can be identified through inverse procedures, either by fitting the frequency dependent constitutive properties from a dedicated dynamical mechanical analyzer experiment, or by fitting response curves, e.g. frequency response functions (FRFs) by using a dedicated numerical model. The optimization process requires frequently iterative prediction of the FRFs of large-scale full order model and furthermore each model inversion is expensive. To speed up numerical simulations, a parametric model order reduction technique is introduced. In the material parameter search space, quasi-random sequences are chosen and divided into two disjoint sets: a sample set is used to construct a reduced order model (ROM); while a validation set is used to assess its performance. A global orthonormal basis can then be constructed by non-weighted singular value decomposition on all local bases. Since the parameter- and frequency-dependency can be suitably preserved, the generated single ROM in conjunction with optimization algorithms is very useful to identify the material parameters of viscoelastic damping. The versatility and efficiency of the present procedure are demonstrated through a number of validation cases.

A fast scanning ion conductance microscopy imaging method using compressive sensing and low-discrepancy sequences.

A scanning ion conductance microscope (SICM) is a multifunctional, high-resolution imaging technique whose non-contact nature makes it very suitable for imaging of biological samples such as living cells in a physiological environment. However, a drawback of hopping/backstep mode of SICM is its relatively slow imaging speed, which seriously restricts the study on the dynamic process of biological samples. This paper presents a new undersampled scanning method based on Compressed Sensing (CS-based scanning mode) theory to solve extended acquisition time issues in the hopping/backstep mode. Compressive sensing can break through the limit of the Nyquist sampling theorem and sample the original sparse/compressible signal at a rate lower than the Nyquist frequency. In the CS-based scanning mode, three sampling patterns, including the random sampling pattern and two kinds of sampling patterns produced by low-discrepancy sequences, were employed as the measurement locations to obtain the undersampled data with different undersampling ratios. Also TVAL3 (Total Variation Augmented Lagrangian ALternating-direction ALgorithm) was then utilized as a reconstruction algorithm to reconstruct the undersampled data. Compared with the nonuniform sampling points of random patterns at a low undersampling ratio, low-discrepancy sequences can produce a more uniform distribution point. Three types of samples with different complexity of topography were scanned by SICM using the conventional hopping/backstep mode and CS-based undersampled scanning mode. The comparisons of the imaging speed and quality with two scanning modes illustrate that the CS-based scanning mode can effectively speed up SICM imaging speed while not sacrificing the image quality. Also low-discrepancy sampling patterns can achieve a better reconstruction performance than that of the random sampling pattern under the same undersampling ratio.

Efficient global sensitivity analysis with correlated variables

Existing methods for the computation of global sensitivity indices are challenged by both number of input-output samples required and the presence of dependent or correlated variables. First, a methodology is developed to increase the efficiency of sensitivity computations with independent variables by incorporating optimal space-filling quasi-random sequences into an existing importance sampling-based kernel regression sensitivity method. Two prominent situations where parameter correlations cannot be ignored, however, are (1) posterior distributions of calibrated parameters and (2) transient, coupled simulations. Therefore, the sensitivity methodology is generalized to dependent variables allowing for efficient post-calibration sensitivity analyses using input-output samples obtained directly from Bayesian calibration. These methods are illustrated using coupled, aerothermal simulations where it is observed that model errors and parameter correlations control the sensitivity estimates until coupling effects become dominant over time.

The generalized and modified Halton sequences in Cantor bases

This paper aims to generalize results that have appeared in Atanassov (Math Balk New Ser 18(1–2):15–32, 2004). We consider here variants of the Halton sequences in a generalized numeration system, called the Cantor expansion, with respect to arbitrary sequences of permutations of the Cantor base. We first show that they provide a wealth of low-discrepancy sequences by giving an estimate of (star) discrepancy bound of the generalized Halton sequence in bounded Cantor bases. Then we impose certain conditions on the sequences of permutations of the Cantor base which are analogous, but not straightforward, to the modified Halton sequence introduced by E.I. Atanassov. We show that this modified Halton sequence in Cantor bases attains a better estimate of the (star) discrepancy bound than the generalized Halton sequence in Cantor bases.

Nondeterministic dynamic stability assessment of Euler–Bernoulli beams using Chebyshev surrogate model

A novel nondeterministic dynamic stability assessment of Euler–Bernoulli beams using Chebyshev surrogate model is proposed to investigate the upper and lower bounds of the dynamic buckling responses in this paper. In the proposed approach, the Galerkin method in conjunction with the force equilibrium is used to obtain the unified implicitly nonlinear ordinary differential equation (ODE) and then this equation can be transformed into a series of ODEs at observation points for each interval variable. Thus, the explicit approximate performance function is constructed with regards to all the interval variables by using Chebyshev surrogate strategy. By combining with the low-discrepancy sequences initialized high-order nonlinear particle swarm optimization (LHNPSO) algorithm, the computational cost is drastically reduced. The comprehensive computational method provides a unified analytical framework to model interval uncertainty for dynamic buckling analysis, which is competent to investigate the extremely upper and lower bounds of structural behaviors. Additionally, the validity, accuracy, as well as the applicability of the proposed computational approach are rigorously investigated by comparing the outputs (i.e., critical buckling load, time of onset of buckling, the corresponding axial shortening displacement and transverse displacement) of the proposed approach with that of Quasi-Monte-Carlo Simulation (QMCS) method for various boundary conditions with interval inputs (i.e., Young's modulus, initial imperfections and loading velocity). The nondeterministic dynamic buckling assessment methodology can help optimization design of beam-type structures under time-dependent loads faster, efficient and flexible.

Interval exchange transformations and low-discrepancy

In Masur (Ann Math 115(1):169–200, 1982) and Veech (J Anal Math 33:222–272, 1978), it was proved independently that almost every interval exchange transformation is uniquely ergodic. The Birkhoff ergodic theorem implies that these maps mainly have uniformly distributed orbits. This raises the question under which conditions the orbits yield low-discrepancy sequences. The case of $$n=2$$ intervals corresponds to circle rotation, where conditions for low-discrepancy are well-known. In this paper, we give corresponding conditions in the case $$n=3$$ . Furthermore, we construct infinitely many interval exchange transformations with low-discrepancy orbits for $$n \ge 4$$ . We also show that these examples do not coincide with LS-sequences if $$S \ge 2$$ .

The power of online thinning in reducing discrepancy

Consider an infinite sequence of independent, uniformly chosen points from $$[0,1]^d$$ . After looking at each point in the sequence, an overseer is allowed to either keep it or reject it, and this choice may depend on the locations of all previously kept points. However, the overseer must keep at least one of every two consecutive points. We call a sequence generated in this fashion a two-thinning sequence. Here, the purpose of the overseer is to control the discrepancy of the empirical distribution of points, that is, after selecting n points, to reduce the maximal deviation of the number of points inside any axis-parallel hyper-rectangle of volume A from nA. Our main result is an explicit low complexity two-thinning strategy which guarantees discrepancy of $$O(\log ^{2d+1} n)$$ for all n with high probability [compare with $$\Theta (\sqrt{n\log \log n})$$ without thinning]. The case $$d=1$$ of this result answers a question of Benjamini. We also extend the construction to achieve the same asymptotic bound for ( $$1+\beta $$ )-thinning, a set-up in which rejecting is only allowed with probability $$\beta $$ independently for each point. In addition, we suggest an improved and simplified strategy which we conjecture to guarantee discrepancy of $$O(\log ^{d+1} n)$$ [compare with $$\theta (\log ^d n)$$ , the best known construction of a low discrepancy sequence]. Finally, we provide theoretical and empirical evidence for our conjecture, and provide simulations supporting the viability of our construction for applications.

Variance Reduction of Ordinary Monte-Carlo Estimates with the Brownian-Bridge Path Construction

It is well known that the Brownian-bridge path construction is usually very effective at reducing the variance of quasi Monte-Carlo estimates, with some exceptions – cliquet options in particular. It will however not change the variance of the value of a financial derivative contract obtained by a standard Monte-Carlo simulation, where pseudo-random numbers are used instead of quasi-random numbers (or low discrepancy sequences such as the Sobol sequence). We will see here that the Brownian-bridge path construction can still significantly reduce the variance of the Theta estimate, that is the sensitivity towards time of a financial derivative contract, even without relying on quasi-random numbers.

Toward Energy-Efficient Stochastic Circuits Using Parallel Sobol Sequences

Stochastic computing (SC) often requires long stochastic sequences and, thus, a long latency to achieve accurate computation. The long latency leads to an inferior performance and low energy efficiency compared with most conventional binary designs. In this paper, a type of low-discrepancy sequences, the Sobol sequence, is considered for use in SC. Compared to the use of pseudorandom sequences generated by linear feedback shift registers (LFSRs), the use of Sobol sequences improves the accuracy of stochastic computation with a reduced sequence length. The inherent feature in Sobol sequence generators enables the parallel implementation of random number generators with an improved performance and hardware efficiency. In particular, the underlying theory is formulated and circuit design is proposed for an arbitrary level of parallelization in a power of 2. In addition, different strategies are implemented for parallelizing combinational and sequential stochastic circuits. The hardware efficiency of the parallel stochastic circuits is measured by energy per operation (EPO), throughput per area (TPA), and runtime. At a similar accuracy, the $8{\times}$ parallel stochastic circuits using Sobol sequences consume approximately 1% of the EPO of the conventional LFSR-based nonparallelized circuits. Meanwhile, an average of 70 (up to 89) times improvements in TPA and less than 1% runtime are achieved. A sorting network is implemented for a median filter (MF) as an application. For a similar image processing quality, a higher energy efficiency is obtained for an $8{\times}$ parallelized stochastic MF compared with its binary counterpart.

An improved NSGA-II based control allocation optimisation for aircraft longitudinal automatic landing system

ABSTRACTIn this paper, an Improved Nondominated Sorting Genetic Algorithm II is proposed for control allocation optimisation in the field of aircraft automatic landing. The initial chromosomes are generated by using a quasi-random sequence to get a better initial searching ability. The adaptive crossover operator and mutation operator are proposed for the dynamic searching area by adaptively relocating the target according to the current searching results. The corresponding online boundary adjustment strategy is created to maintain the robustness of the algorithm during the whole searching process. The control relationships between the elevator and engine channels are presented and a set of feasible solutions are chosen as a reasonable control range for the ALS design. Finally, a 6 Degrees of Freedom (6DoF) rigid model of the F/A-18 with external wind perturbation is used as a test bed to demonstrate the feasibility of the proposed method.

Remarks on randomization of quasi-random numbers

Abstract In this paper we discuss estimation of the quasi-Monte Carlo methods error in the case of calculation of high-order integrals. Quasi-random Halton sequences are considered as a special case. Randomization of these sequences by the random shift method turns out to lead to well-known random quadrature formulas with one free node. Some new properties of such formulas are pointed out. The subject is illustrated by a number of numerical examples.

Sequences with Low‐Discrepancy Blue‐Noise 2‐D Projections

AbstractDistributions of samples play a very important role in rendering, affecting variance, bias and aliasing in Monte‐Carlo and Quasi‐Monte Carlo evaluation of the rendering equation. In this paper, we propose an original sampler which inherits many important features of classical low‐discrepancy sequences (LDS): a high degree of uniformity of the achieved distribution of samples, computational efficiency and progressive sampling capability. At the same time, we purposely tailor our sampler in order to improve its spectral characteristics, which in turn play a crucial role in variance reduction, anti‐aliasing and improving visual appearance of rendering. Our sampler can efficiently generate sequences of multidimensional points, whose power spectra approach so‐called Blue‐Noise (BN) spectral property while preserving low discrepancy (LD) in certain 2‐D projections.In our tile‐based approach, we perform permutations on subsets of the original Sobol LDS. In a large space of all possible permutations, we select those which better approach the target BN property, using pair‐correlation statistics. We pre‐calculate such “good” permutations for each possible Sobol pattern, and store them in a lookup table efficiently accessible in runtime. We provide a complete and rigorous proof that such permutations preserve dyadic partitioning and thus the LDS properties of the point set in 2‐D projections. Our construction is computationally efficient, has a relatively low memory footprint and supports adaptive sampling. We validate our method by performing spectral/discrepancy/aliasing analysis of the achieved distributions, and provide variance analysis for several target integrands of theoretical and practical interest.

A quasi-Monte Carlo implementation of the ziggurat method

Abstract The ziggurat method is a fast random variable generation method introduced by Marsaglia and Tsang in a series of papers. We discuss how the ziggurat method can be implemented for low-discrepancy sequences, and present algorithms and numerical results when the method is used to generate samples from the normal and gamma distributions.

Stable and efficient retrospective 4D-MRI using non-uniformly distributed quasi-random numbers

The purpose of this work is the development of a robust and reliable three-dimensional (3D) Cartesian imaging technique for fast and flexible retrospective 4D abdominal MRI during free breathing. To this end, a non-uniform quasi random (NU-QR) reordering of the phase encoding (ky–kz) lines was incorporated into 3D Cartesian acquisition. The proposed sampling scheme allocates more phase encoding points near the k-space origin while reducing the sampling density in the outer part of the k-space. Respiratory self-gating in combination with SPIRiT-reconstruction is used for the reconstruction of abdominal data sets in different respiratory phases (4D-MRI). Six volunteers and three patients were examined at 1.5 T during free breathing. Additionally, data sets with conventional two-dimensional (2D) linear and 2D quasi random phase encoding order were acquired for the volunteers for comparison. A quantitative evaluation of image quality versus scan times (from 70 s to 626 s) for the given sampling schemes was obtained by calculating the normalized mutual information (NMI) for all volunteers. Motion estimation was accomplished by calculating the maximum derivative of a signal intensity profile of a transition (e.g. tumor or diaphragm).The 2D non-uniform quasi-random distribution of phase encoding lines in Cartesian 3D MRI yields more efficient undersampling patterns for parallel imaging compared to conventional uniform quasi-random and linear sampling. Median NMI values of NU-QR sampling are the highest for all scan times. Therefore, within the same scan time 4D imaging could be performed with improved image quality.The proposed method allows for the reconstruction of motion artifact reduced 4D data sets with isotropic spatial resolution of 2.1 × 2.1 × 2.1 mm3 in a short scan time, e.g. 10 respiratory phases in only 3 min. Cranio-caudal tumor displacements between 23 and 46 mm could be observed.NU-QR sampling enables for stable 4D-MRI with high temporal and spatial resolution within short scan time for visualization of organ or tumor motion during free breathing. Further studies, e.g. the application of the method for radiotherapy planning are needed to investigate the clinical applicability and diagnostic value of the approach.

QUASI-MONTE CARLO METHODS FOR COMPUTATION OF MULTIDIMENSIONAL INTEGRALS RELATED TO BAYESIAN MODELS IN INTERNATIONAL MIGRATION FORECASTING

The paper addresses selected methodological aspects of international migration forecasting. The new methods based on the Bayesian statistics have been recently developed. A fundamental problem in Bayesian statistics is the accurate evaluation of multidimensional integrals. A comprehensive experimental study based on Faure and Hammersley low discrepancy sequences and Fibonacci based lattice rule has been done. The numerical tests show that the stochastic algorithms under consideration are efficient tool for computing multidimensional integrals. It is important in order to obtain a more accurate and reliable interpretation of the results in Bayesian statistics which is a foundation in international migration forecasting.

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

Frolov's cubature formula on the unit hypercube has been considered important since it attains an optimal rate of convergence for various function spaces. Its integration nodes are given by shrinking a suitable full rank Z-lattice in Rd and taking all points inside the unit cube. The main drawback of these nodes is that they are hard to find computationally, especially in high dimensions. In such situations, quasi-Monte Carlo (QMC) rules based on digital nets have proven to be successful. However, there is still no construction known that leads to QMC rules which are optimal in the same generality as Frolov's.In this paper we investigate a polynomial analog of Frolov's cubature formula, which we expect to be important in this respect. This analog is defined in a field of Laurent series with coefficients in a finite field. A similar approach was previously studied in [M. B. Levin. Adelic constructions of low discrepancy sequences. Online Journal of Analytic Combinatorics. Issue 5, 2010.].We show that our construction is a (t,m,d)-net, which also implies bounds on its star-discrepancy and the error of the corresponding cubature rule. Moreover, we show that our cubature rule is a QMC rule, whereas Frolov's is not, and provide an algorithm to determine its integration nodes explicitly.To this end we need to extend the notion of (t,m,d)-nets to fit the situation that the points can have infinite digit expansion and develop a duality theory. Additionally, we adapt the notion of admissible lattices to our setting and prove its significance.

Quasi-Monte Carlo application in CGE systematic sensitivity analysis

ABSTRACTThe uncertainty and robustness of Computable General Equilibrium (CGE) models can be assessed by conducting a Systematic Sensitivity Analysis (SSA). Different methods have been used in the literature for SSA of CGE models such as Gaussian Quadrature and Monte Carlo methods. This article explores the use of Quasi-random Monte Carlo methods based on the Halton and Sobol’ sequences as means to improve the efficiency over regular Monte Carlo SSA, thus reducing the computational requirements of the SSA. The findings suggest that by using low-discrepancy sequences, the number of simulations required by the regular MC SSA methods can be notably reduced, hence lowering the computational time required for SSA of CGE models.

Quasi-Monte Carlo integration on manifolds with mapped low-discrepancy points and greedy minimal Riesz s-energy points

In this paper we consider two sets of points for Quasi-Monte Carlo integration on two-dimensional manifolds. The first is the set of mapped low-discrepancy sequence by a measure preserving map, from a rectangle U⊂R2 to the manifold. The second is the greedy minimal Riesz s-energy points extracted from a suitable discretization of the manifold. Thanks to the Poppy-seed Bagel Theorem we know that the classes of points with minimal Riesz s-energy, under suitable assumptions, are asymptotically uniformly distributed with respect to the normalized Hausdorff measure. They can then be considered as quadrature points on manifolds via the Quasi-Monte Carlo (QMC) method. On the other hand, we do not know if the greedy minimal Riesz s-energy points are a good choice to integrate functions with the QMC method on manifolds. Through theoretical considerations, by showing some properties of these points and by numerical experiments, we attempt to answer to these questions.

A Comparative Study on Barrier Option Pricing using Antithetic and Quasi Monte-Carlo Simulations

Monte-Carlo simulations have been utilized greatly in the pricing of derivative securities. Over the years, several variance reduction techniques have been developed to curb the instability, as well as, increase the simulation e?ciencies of the Monte-Carlo methods. Our approach in this research work will consider the use of antithetic variate techniques to estimate the fair prices of barrier options. Next, we use the quasi-Monte Carlo method, together with Sobol sequence to estimate the values of the same option. An extended version of the Black-Scholes model will serve as basis for the exact prices of these exotic options. The resulting simulated prices will be compared to the exact prices. The research concludes by showing some results which proves that when random numbers are generated via low discrepancy sequences in contrast to the normal pseudo-random numbers, a more efficient simulation method is ensued. This is further applicable in pricing complex derivatives without closed formsolutions.