Accurate Quadrature Research Articles

Accurate Low Complexity Quadrature Angular Diversity Aperture Receiver for Visible Light Positioning.

Despite the many potential applications of an accurate indoor positioning system (IPS), no universal, readily available system exists. Much of the IPS research to date has been based on the use of radio transmitters as positioning beacons. Visible light positioning (VLP) instead uses LED lights as beacons. Either cameras or photodiodes (PDs) can be used as VLP receivers, and position estimates are usually based on either the angle of arrival (AOA) or the strength of the received signal. Research on the use of AOA with photodiode receivers has so far been limited by the lack of a suitable compact receiver. The quadrature angular diversity aperture receiver (QADA) can fill this gap. In this paper, we describe a new QADA design that uses only three readily available parts: a quadrant photodiode, a 3D-printed aperture, and a programmable system on a chip (PSoC). Extensive experimental results demonstrate that this design provides accurate AOA estimates within a room-sized test chamber. The flexibility and programmability of the PSoC mean that other sensors can be supported by the same PSoC. This has the potential to allow the AOA estimates from the QADA to be combined with information from other sensors to form future powerful sensor-fusion systems requiring only one beacon.

Amplitude Control Loop to Compensate for I/Q Mismatch in an Accurate Quadrature LC Oscillator

In this paper, a novel parallel-coupled quadrature LC oscillator (PC-QO) is presented that can automatically eliminate imbalances of output phase and amplitude resulting from mismatches in the LC tanks. The method of using unequal coupling factors in a parallel-coupled quadrature oscillator is the base of this design; canceling the phase and amplitude errors takes place by tuning the imbalanced coupling factor using tuneable tail currents. First, the proposed circuit senses the amplitude error and then adjusts the coupling factors according to the situation. We show that using the inversely proportional coupling factors can eliminate the phase and amplitude error simultaneously. In other words, this design uses the amplitude control loop to cancel the phase and amplitude error. The circuit has been simulated using TSMC 0.18 CMOS practical model to confirm the high accuracy of the analysis and capability of the canceling [Formula: see text]/[Formula: see text] mismatch technique. The simulation results show that the phase noise of the proposed quadrature voltage controlled oscillator (QVCO) is −123.9 dBc/Hz at 1[Formula: see text]MHz offset from 4.4[Formula: see text]GHz operation frequency. The total power consumption of the QVCO is 3.4[Formula: see text]mW.

Computing vibrational spectra using a new collocation method with a pruned basis and more points than basis functions: Avoiding quadrature.

We present a new collocation method for computing the vibrational spectrum of a polyatomic molecule. Some form of quadrature or collocation is necessary when the potential energy surface does not have a simple form that simplifies the calculation of the potential matrix elements required to do a variational calculation. With quadrature, better accuracy is obtained by using more points than basis functions. To achieve the same advantage with collocation, we introduce a collocation method with more points than basis functions. Critically important, the method can be used with a large basis because it is incorporated into an iterative eigensolver. Previous collocation methods with more points than functions were incompatible with iterative eigensolvers. We test the new ideas by computing energy levels of molecules with as many as six atoms. We use pruned bases but expect the new method to be advantageous whenever one uses a basis for which it is not possible to find an accurate quadrature with about as many points as there are basis functions. For our test molecules, accurate energy levels are obtained even using non-optimal, simple, equally spaced points.

Efficient and accurate quadrature methods of Fourier integrals with a special oscillator and weak singularities

The recent article (J. Math. Anal. Appl. 494 (2021), Article number: 124448) presented an asymptotic Filon-type method for computing the oscillatory integral with a special oscillator and weak singularities, ∫0bxα(b−x)βf(x)eiωxrdx,−1<α,β≤0,0<b<+∞,ω∈R,r∈N+. In this article, we propose and analyze two different efficient and accurate quadrature methods for this singularly oscillatory integral. First, we give a two-point Taylor interpolation method by using a two-point Taylor polynomial instead of f(x). In addition, we propose a more efficient contour integration method. By exploiting the Taylor polynomial of the function f at x=0, and then based on the additivity of the integration interval, we change the considered integral into two integrals. One integral can be efficiently computed by the contour integration method based on Cauchy Residue Theorem and generalized Gaussian-Laguerre quadrature rule. The other integral can be explicitly calculated by special functions. Specifically, we perform the rigorous error analysis of the proposed methods and obtain asymptotic error estimates in inverse powers of the frequency parameter ω. Ultimately, the proposed methods are compared with the asymptotic Filon-type method given in this work (J. Math. Anal. Appl. 494 (2021), Article number: 124448) and the modified Filon-type method. At the same computational cost, the two-point Taylor interpolation method and the asymptotic Filon-type method have a very close accuracy level. Their accuracy is higher than that of the modified Filon-type method, and the precision of the contour integration method is much higher than that of the asymptotic Filon-type method, the modified Filon-type method, and the two-point Taylor interpolation method. We verify error analyses of the proposed methods by experimental results. Numerical experiments can also verify the efficiency and precision of the proposed methods.

Highly accurate quadrature schemes for singular integrals in energetic BEM applied to elastodynamics

We consider an exterior linear elastodynamics problem with vanishing initial conditions and Dirichlet datum on the scatterer. We convert the Navier Equation, governing the wave behaviour, into two space–time Boundary Integral Equations (BIEs) whose solution is approximated by the energetic Boundary Element Method (BEM). To apply this technique, we have to set the BIEs in a weak form related to the energy of the differential problem solution at the final time instant of analysis. After the space–time discretization of the weak formulation, we have to deal with double space–time integrals, with a weakly singular kernel depending on primary and secondary wave speeds and multiplied by Heaviside functions. The main purpose of this work is the analysis of these peculiar integrals and the study of suitable quadrature schemes for their approximation.

Mortar-based Entropy-Stable Discontinuous Galerkin Methods on Non-conforming Quadrilateral and Hexahedral Meshes

High-order entropy-stable discontinuous Galerkin (DG) methods for nonlinear conservation laws reproduce a discrete entropy inequality by combining entropy conservative finite volume fluxes with summation-by-parts (SBP) discretization matrices. In the DG context, on tensor product (quadrilateral and hexahedral) elements, SBP matrices are typically constructed by collocating at Lobatto quadrature points. Recent work has extended the construction of entropy-stable DG schemes to collocation at more accurate Gauss quadrature points (Chan et al. in SIAM J Sci Comput 41(5):A2938–A2966, 2019) . In this work, we extend entropy-stable Gauss collocation schemes to non-conforming meshes. Entropy-stable DG schemes require computing entropy conservative numerical fluxes between volume and surface quadrature nodes. On conforming tensor product meshes where volume and surface nodes are aligned, flux evaluations are required only between “lines” of nodes. However, on non-conforming meshes, volume and surface nodes are no longer aligned, resulting in a larger number of flux evaluations. We reduce this expense by introducing an entropy-stable mortar-based treatment of non-conforming interfaces via a face-local correction term, and provide necessary conditions for high-order accuracy. Numerical experiments for the compressible Euler equations in two and three dimensions confirm the stability and accuracy of this approach.

Convergence of the MPIE Galerkin MoM Thin Wire Formulation

Convergence studies of the Method of Moments solution of thin wires have long been handicapped by the issue of conditional convergence: when the segment length approaches the wire radius, the approximations typically made result in exponentially diverging solutions. In this communication, it is shown that it is the approximation of the singularity which is the main cause of this problem. Using the mixed potential integral equation, paralleling the classic RWG formulation for surfaces, the formulation in this communication approximates the surface (as opposed to equivalent filamentary) current, and applies Galerkin testing on the wire surface. The use of contemporary singularity-canceling integral transforms for highly efficient and accurate quadrature is demonstrated. Results are presented for typical thin-wire dipoles for both radiation and scattering problems. It is shown that with a rigorous treatment of the singularity, a far more comprehensive convergence study may be undertaken than previously demonstrated. Furthermore, the results shown justify the many years of successful use in practice of simple source models with modest mesh requirements.

Zeta correction: a new approach to constructing corrected trapezoidal quadrature rules for singular integral operators

A high-order accurate quadrature rule for the discretization of boundary integral equations (BIEs) on closed smooth contours in the plane is introduced. This quadrature can be viewed as a hybrid of the spectral quadrature of Kress (Math. Comput. Model. 15(3-5), 229–243 1991) and the locally corrected trapezoidal quadrature of Kapur and Rokhlin (SIAM J. Numer. Anal. 34(4), 1331–1356, 1997). The new technique combines the strengths of both methods, and attains high-order convergence, numerical stability, ease of implementation, and compatibility with the “fast” algorithms (such as the Fast Multipole Method or Fast Direct Solvers). Important connections between the punctured trapezoidal rule and the Riemann zeta function are introduced, which enable a complete convergence analysis and lead to remarkably simple procedures for constructing the quadrature corrections. The paper reports a detailed comparison between the new method and the methods of Kress, of Kapur and Rokhlin, and of Alpert (SIAM J. Sci. Comput. 20(5), 1551–1584, 1999).

Negative Binomial mixed models estimated with the maximum likelihood method can be used for longitudinal RNAseq data.

Time-course RNAseq experiments, where tissues are repeatedly collected from the same subjects, e.g. humans or animals over time or under several different experimental conditions, are becoming more popular due to the reducing sequencing costs. Such designs offer the great potential to identify genes that change over time or progress differently in time across experimental groups. Modelling of the longitudinal gene expression in such time-course RNAseq data is complicated by the serial correlations, missing values due to subject dropout or sequencing errors, long follow up with potentially non-linear progression in time and low number of subjects. Negative Binomial mixed models can address all these issues. However, such models under the maximum likelihood (ML) approach are less popular for RNAseq data due to convergence issues (see, e.g. [1]). We argue in this paper that it is the use of an inaccurate numerical integration method in combination with the typically small sample sizes which causes such mixed models to fail for a great portion of tested genes. We show that when we use the accurate adaptive Gaussian quadrature approach to approximate the integrals over the random-effects terms, we can successfully estimate the model parameters with the maximum likelihood method. Moreover, we show that the boostrap method can be used to preserve the type I error rate in small sample settings. We evaluate empirically the small sample properties of the test statistics and compare with state-of-the-art approaches. The method is applied on a longitudinal mice experiment to study the dynamics in Duchenne Muscular Dystrophy. Contact:s.tsonaka@lumc.nlRoula Tsonaka is an assistant professor at the Medical Statistics, Department of Biomedical Data Sciences, Leiden University Medical Center. Her research focuses on statistical methods for longitudinal omics data. Pietro Spitali is an assistant professor at the Department of Human Genetics, Leiden University Medical Center. His research focuses on the identification of biomarkers for neuromuscular disorders.

Evaluating crystallographic likelihood functions using numerical quadratures

Intensity-based likelihood functions in crystallographic applications have the potential to enhance the quality of structures derived from marginal diffraction data. Their usage, however, is complicated by the ability to efficiently compute these target functions. Here, a numerical quadrature is developed that allows the rapid evaluation of intensity-based likelihood functions in crystallographic applications. By using a sequence of change-of-variable transformations, including a nonlinear domain-compression operation, an accurate, robust and efficient quadrature is constructed. The approach is flexible and can incorporate different noise models with relative ease.

Evaluating crystallographic likelihood functions using numerical quadratures.

Intensity-based likelihood functions in crystallographic applications have the potential to enhance the quality of structures derived from marginal diffraction data. Their usage, however, is complicated by the ability to efficiently compute these target functions. Here, a numerical quadrature is developed that allows the rapid evaluation of intensity-based likelihood functions in crystallographic applications. By using a sequence of change-of-variable transformations, including a nonlinear domain-compression operation, an accurate, robust and efficient quadrature is constructed. The approach is flexible and can incorporate different noise models with relative ease.

Accurate quadrature of nearly singular line integrals in two and three dimensions by singularity swapping

The numerical method of Helsing and co-workers evaluates Laplace and related layer potentials generated by a panel (composite) quadrature on a curve, efficiently and with high-order accuracy for arbitrarily close targets. Since it exploits complex analysis, its use has been restricted to two dimensions (2D). We first explain its loss of accuracy as panels become curved, using a classical complex approximation result of Walsh that can be interpreted as “electrostatic shielding” of a Schwarz singularity. We then introduce a variant that swaps the target singularity for one at its complexified parameter preimage; in the latter space the panel is flat, hence the convergence rate can be much higher. The preimage is found robustly by Newton iteration. This idea also enables, for the first time, a near-singular quadrature for potentials generated by smooth curves in 3D, building on recurrences of Tornberg–Gustavsson. We apply this to accurate evaluation of the Stokes flow near to a curved filament in the slender body approximation. Our 3D method is several times more efficient (both in terms of kernel evaluations, and in speed in a C implementation) than the only existing alternative, namely, adaptive integration.

A Wideband Frequency Divider With Programmable Odd/Even Division Factors and Quadrature/Symmetrical Outputs

An injection-locked frequency divider (ILFD) with odd/even division factor and wide locking range over process and temperature variations is presented. To enhance the locking range of the frequency divider and generate accurate quadrature with 50% duty cycle outputs, a comprehensive analysis of the frequency divider to find the optimal phase sequence for two, three and five division ratios is provided. Generating quadrature and symmetrical outputs with simultaneous realization of even and odd division ratios at low power is the main challenge that is addressed in this paper as the power consumption of existing quadrature circuits is an order of magnitude higher than that of the proposed ILFD. The ILFD is designed and simulated in a 65 nm CMOS technology and the analytical results verified by the circuit simulation results. Post layout simulation results show that the locking range of the divider without any tuning is from 5.4 GHz to 8.5 GHz, 1.1 GHz to 6.1 GHz and 0.1 GHz to 5.3 GHz in divide-by-five, three and two mode for 0 dBm injection signal level, respectively. The phase deviation of the quadrature outputs for dividers is less than 1.8° and the deviation of duty cycle from 50%, is less than 0.4%.

A Quadrature PLL With Phase Mismatch Calibration for 32GS/s Time-Interleaved ADC

This article presents an accurate quadrature phase-locked loop (PLL) with quadrature phase mismatch calibration for 32 GS/s analog-to-digital converter (ADC). Due to the mismatches of clock distribution in layout and variations of the active devices, the quadrature phase of the sampling clock is significantly deteriorated. To solve the problem, a novel quadrature divider with phase calibration is induced in PLL loop. Moreover, a theoretical model of the quadrature divider is proposed to predict the performance and potential ability for phase calibration. Based on the theoretical and model analysis, the proposed PLL can realize accurate quadrature phase for high-speed real-time sampling system. The output frequency of PLL is 8 GHz with quadrature phases for 32 GS/s sampling rate. The proposed clock can realize 8-bit signal to noise ratio requirement with 16 GHz bandwidth. The proposed PLL was fabricated in 65-nm CMOS process with 28 mW dc power consumption under 1.2 V supply voltage. Testing results show that the phase noise of the clock is −127 dBc/Hz @10 MHz-offset frequency when the sampling speed is 27.4 GS/s. With proposed methods, the range of the phase error calibration is around ±20°.

The series expansions and Gauss-Legendre rule for computing arbitrary derivatives of the Beta-type functions

The beta-type functions play an important role in many applied sciences. The partial derivatives of the beta function and the incomplete beta function are integrals involving algebraic and logarithmic endpoint singularities. In this paper, some series expansions for these beta-type functions are found, which are easily used to evaluate these functions with prescribed precision. On the other hand, an accurate Gauss-Legendre quadrature formula is designed to compute these beta-type functions and their partial derivatives based on the Puiseux series for the integrands at their singularities. Some numerical examples confirm the high accuracy and high efficiency of the two algorithms, and also show that the algorithms can be used to effectively evaluate the generalized beta-type functions.

Approximation to Logarithmic-Cauchy Type Singular Integrals with Highly Oscillatory Kernels

In this paper, a fast and accurate numerical Clenshaw-Curtis quadrature is proposed for the approximation of highly oscillatory integrals with Cauchy and logarithmic singularities, ⨍ − 1 1 f ( x ) log ( x − α ) e i k x x − t d x , t ∉ ( − 1 , 1 ) , α ∈ [ − 1 , 1 ] for a smooth function f ( x ) . This method consists of evaluation of the modified moments by stable recurrence relation and Cauchy kernel is solved by steepest descent method that transforms the oscillatory integral into the sum of line integrals. Later theoretical analysis and high accuracy of the method is illustrated by some examples.

Double exponential transformation for computing three-center nuclear attraction integrals

ABSTRACTThree-center nuclear attraction integrals, which arise in density functional and ab initio calculations, are one of the most time-consuming computations involved in molecular electronic structure calculations. Even for relatively small systems, millions of these laborious calculations need to be executed. Highly efficient and accurate methods for evaluating molecular integrals are therefore all the more vital in order to perform the calculations necessary for large systems. When using a basis set of B functions, an analytical expression for the three-center nuclear attraction integrals can be derived via the Fourier transform method. However, due to the presence of the highly oscillatory semi-infinite spherical Bessel integral, the analytical expression still remains problematic. By applying the S transformation, the spherical Bessel integral can be converted into a much more favorable sine integral. In the present work, we then apply two types of double exponential transformations to the resulting sine integral, which leads to a highly efficient and accurate quadrature formula. This method facilitates the approximation of the molecular integrals to a high predetermined accuracy, while still keeping the calculation times low. The fast convergence properties analyzed in the numerical section illustrate the advantages of the method.

Mathematical Foundation of the Collocation Boundary Element Method: Consistent Formulation, Convergence Theorem and Accurate Numerical Quadrature

Mathematical Foundation of the Collocation Boundary Element Method: Consistent Formulation, Convergence Theorem and Accurate Numerical Quadrature

Fokker-Planck equation driven by asymmetric Lévy motion

Non-Gaussian L\'evy noises are present in many models for understanding underlining principles of physics, finance, biology and more. In this work, we consider the Fokker-Planck equation(FPE) due to one-dimensional asymmetric L\'evy motion, which is a nonlocal partial differential equation. We present an accurate numerical quadrature for the singular integrals in the nonlocal FPE and develop a fast summation method to reduce the order of the complexity from $O(J^2)$ to $O(J\log J)$ in one time-step, where $J$ is the number of unknowns. We also provide conditions under which the numerical schemes satisfy maximum principle. Our numerical method is validated by comparing with exact solutions for special cases. We also discuss the properties of the probability density functions and the effects of various factors on the solutions, including the stability index, the skewness parameter, the drift term, the Gaussian and non-Gaussian noises and the domain size.

Quadrature for parabolic Galerkin BEM with moving surfaces

The successful implementation of the Galerkin Boundary Element Method hinges on the accurate and effective quadrature of the influence coefficients. For parabolic boundary integral operators quadrature must be performed in space and time where integrals have singularities when source- and evaluation points coincide. For problems where the surface is fixed, the time integration can be performed analytically, but for moving geometries numerical quadrature in space and time must be used. For this case a set of transformations is derived that render the singular space–time integrals into smooth integrals that can be treated with standard tensor product Gauss quadrature rules. This methodology can be applied to the heat equation and to transient Stokes flow.