Accumulation Of Numerical Errors Research Articles

A new very high-order finite-difference method for linear stability analysis and bi-orthogonal decomposition of hypersonic boundary layer flow

Precisely predicting laminar-turbulence transition locations is essential for improvements in hypersonic vehicle design related to flow control and heat protection. Currently state-of-the-art eN prediction method requires the evaluation of discrete normal modes F and S for the growth rate of instability wave. Meanwhile, in receptivity studies, both the discrete and continuous modes, including acoustic, entropy, and vorticity modes, contribute to the generation of the initial disturbance. The purpose of this paper is to introduce a new very high-order numerical method to accurately compute these normal modes with finite-difference on a non-uniform grid. Currently, numerical methods to obtain these normal modes include two major approaches, the boundary value problem approach and the initial value problem approach. The boundary value approach used by Malik (1990) [17] deploys fourth-order finite difference and spectral collocation methods to solve a boundary value problem for linear stability theory (LST). Nonetheless, Malik's presentation only demonstrated the computation of discrete modes, but not the continuous modes essential for conducting modal analysis on receptivity data. To obtain the continuous spectrum for his multimode decomposition framework, Tumin (2007) [16] relies on an initial value approach based on the Runge Kutta scheme with the Gram-Schmidt orthonormalization. However, the initial value approach is a local method that does not give a global evaluation of the eigenvalue spectra of discrete modes. Furthermore, Gram-Schmidt orthonormalization, which can be error-prone in implementation, is required at every step of the integration to minimize the accumulation of numerical errors. To overcome the drawbacks of these two approaches, this paper improves the boundary value approach by introducing a new general very high-order finite difference method for both discrete and continuous modes eigenfunctions. This general high-order finite difference method is based on a non-uniform grid method proposed by Zhong and Tatineni (2003) [22]. Under the finite difference framework, discrete and continuous modes can be obtained by imposing proper freestream asymptotic boundary conditions based on the freestream fundamental solution behavior. This asymptotic boundary condition is used for obtaining both discrete and continuous modes that have both distinct (acoustic) and similar (vorticity and entropy) eigenvalues. Extensive verification of the new method has been carried out by comparing the computed discrete and continuous modes. Subsequently, the discrete and continuous modes obtained with this finite difference method are essential for the bi-orthogonal decomposition, which holds promising potential in obtaining an accurate evaluation of receptivity coefficients. The result of the bi-orthogonal decomposition for a hypersonic boundary layer flow over a flat plate is verified by comparing with existing results. Ultimately, the bi-orthogonal decomposition using the eigenfunctions has been applied to a case of freestream receptivity simulation for an axis-symmetric hypersonic flow over a blunt nose cone with modal contributions computed as coefficients for receptivity analysis.

Spherical harmonic coefficients of isotropic polynomial functions with applications to gravity field modeling

Various aspects of gravity field modeling rely upon analytical mathematical functions for calculating spherical harmonic coefficients. Such functions allow quick and efficient evaluation of cumbersome convolution integrals defined on the sphere. In this work, we present a new analytical method for determining spherical harmonic coefficients of isotropic polynomial functions. This method in computationally flexible and efficient, since it makes use of recurrence relations. Also, its use is universal and could be extended to piecewise polynomials and polynomials with compact support. Our numerical investigation of the proposed method shows that certain recurrence relations lose accuracy as the order of implemented polynomials increases because of accumulation of numerical errors. Propagation of these errors could be mitigated by hybrid methods or using extended precision arithmetic. We demonstrate the relevance of our method in gravity field modeling and discuss two areas of application. The first one is the design of B-spline windows and filter kernels for the low-pass filtering of gravity field functionals (e.g., GRACE Follow-On monthly geopotential solutions). The second one is the calculation of spherical harmonic coefficients of isotropic polynomial covariance functions.

Effectiveness of KS Elements for Satellite Orbit Predictions With Earth’s Gravity and Drag Force

This study is meant for understanding the efficiency of KS regular equations for satellite orbit prediction. The analysis is mainly concentrated on Low-Earth satellite orbit, where the forces due to non spherical nature of the Earth and atmospheric drag, plays important role. The classical Newtonian equations of motions for which solutions are unstable, cannot be used for long term prediction of orbit. Hence, in literature many transformations have emerged to stabilize the equations of motion either to reduce the accumulation of local numerical errors or allowing to use a large integration step sizes, or both in the transformed space. KS transformation introduced by Kustaanheimo and Stiefel, regularize the nonlinear equation of motion to linear differential equations of a harmonic oscillator of constant frequency. In this paper a detailed analysis on KS differential equations is carried out by including perturbations of Earth’s gravity and atmospheric drag for orbit predictions. The recurrence relations of associated Legendre polynomial and its derivatives are used for including higher order Earth’s gravity terms. To know the effectiveness of the theory, results are compared with the orbit of the actual space object, CZ-2D R/B, for 7 days duration. The comparison shows that the KS theory is one of the best theories for orbit predictions available at present.

Orbit Predictions Using KS Elements With Earth’s Flattening

Predicting the orbit of a satellite is a fundamental requirement in many areas of aerospace, such as mission planning, satellite geodesy, re-entry prediction, maneuver planning, collision avoidance, formation flying, etc. For near-Earth orbits, the forces due to the non spherical nature of the Earth and atmospheric drag plays an important role and are mainly responsible to bring the satellite back to the Earth. Thus inclusion of the effect of these perturbing forces becomes important for precise orbit computation of near-Earth orbits. Many transformations have emerged in the literature to stabilize the equations of motion either to reduce the accumulation of local numerical errors or allowing the use of large integration step sizes, or both in the transformed space. One such transformation is known as KS transformation by Kustaanheimo and Stiefel [1], who regularized the nonlinear Kepler equation of motion and reduced it into linear differential equations of a harmonic oscillator of constant frequency. In this paper a detailed analysis is carried out for orbit prediction using KS differential equations by including the non spherical gravitational potential of the Earth as the perturbing force. Earths zonal and tesseral harmonics, which produce bands of constant deviation from the spherical field along lines of latitude and longitude, are included to precisely model the Earth’s gravitational potential. Higher order Earth’s flattening terms are included by utilizing the recurrence relations of associated Legendre polynomial and its derivatives. To know the effectiveness of the theory, the results are compared with the real IRS-1A satellite data and the results of other existing solutions for a long duration of nearly a month time. The comparison shows that the KS method provides one of the most and accurate techniques for orbit prediction available at present.

Does joint impedance improve dynamic leg simulations with explicit and implicit solvers?

The nervous system predicts and executes complex motion of body segments actuated by the coordinated action of muscles. When a stroke or other traumatic injury disrupts neural processing, the impeded behavior has not only kinematic but also kinetic attributes that require interpretation. Biomechanical models could allow medical specialists to observe these dynamic variables and instantaneously diagnose mobility issues that may otherwise remain unnoticed. However, the real-time and subject-specific dynamic computations necessitate the optimization these simulations. In this study, we explored the effects of intrinsic viscoelasticity, choice of numerical integration method, and decrease in sampling frequency on the accuracy and stability of the simulation. The bipedal model with 17 rotational degrees of freedom (DOF)-describing hip, knee, ankle, and standing foot contact-was instrumented with viscoelastic elements with a resting length in the middle of the DOF range of motion. The accumulation of numerical errors was evaluated in dynamic simulations using swing-phase experimental kinematics. The relationship between viscoelasticity, sampling rates, and the integrator type was evaluated. The optimal selection of these three factors resulted in an accurate reconstruction of joint kinematics (err < 1%) and kinetics (err < 5%) with increased simulation time steps. Notably, joint viscoelasticity reduced the integration errors of explicit methods and had minimal to no additional benefit for implicit methods. Gained insights have the potential to improve diagnostic tools and accurize real-time feedback simulations used in the functional recovery of neuromuscular diseases and intuitive control of modern prosthetic solutions.

Reducing the low-wavenumber dispersion error by building the Lagrange dual problem with a powerful local restriction

Abstract The broadband finite-difference (FD) coefficients computed by a cost function have been widely applied to suppress of numerical dispersion. Under the same conditions, the FD coefficients with small low-wavenumber dispersion error will produce a more accurate numerical solution in the long-time seismic wave simulation. Thus, how to reduce the low-wavenumber dispersion error becomes a crucial problem. According to the research into the zero-point position at the dispersion curve for three types of cost function, we found that the more zero points concentrate on the low-wavenumber region, the less the dispersion error. Therefore, the concentration of zero points is a good way to reduce dispersion error, which can be implemented by modified wavenumbers of zero points. Then, we design a Lagrange dual problem with a restriction based on the modified wavenumbers. The requirements for constructing the Lagrange dual problem are the optimization function and restricted condition, where the former is based on the dispersion relation, and the latter comprises the modified wavenumbers. The solution of this optimization problem, calculated by the dual ascent method, affords a less low-wavenumber dispersion error than the solution yielded by the alternating direction method of multipliers (ADMM). The theoretical analysis and numerical modeling suggest that the proposed method is superior to the existing optimal FD coefficients in reducing numerical error accumulation in low-frequency simulation.

Robust Zero-Watermarking of Color Medical Images Using Multi-Channel Gaussian-Hermite Moments and 1D Chebyshev Chaotic Map.

Copyright protection of medical images is a vital goal in the era of smart healthcare systems. In recent telemedicine applications, medical images are sensed using medical imaging devices and transmitted to remote places for screening by physicians and specialists. During their transmission, the medical images could be tampered with by intruders. Traditional watermarking methods embed the information in the host images to protect the copyright of medical images. The embedding destroys the original image and cannot be applied efficiently to images used in medicine that require high integrity. Robust zero-watermarking methods are preferable over other watermarking algorithms in medical image security due to their outstanding performance. Most existing methods are presented based on moments and moment invariants, which have become a prominent method for zero-watermarking due to their favorable image description capabilities and geometric invariance. Although moment-based zero-watermarking can be an effective approach to image copyright protection, several present approaches cannot effectively resist geometric attacks, and others have a low resistance to large-scale attacks. Besides these issues, most of these algorithms rely on traditional moment computation, which suffers from numerical error accumulation, leading to numerical instabilities, and time consumption and affecting the performance of these moment-based zero-watermarking techniques. In this paper, we derived multi-channel Gaussian–Hermite moments of fractional-order (MFrGHMs) to solve the problems. Then we used a kernel-based method for the highly accurate computation of MFrGHMs to solve the computation issue. Then, we constructed image features that are accurate and robust. Finally, we presented a new zero-watermarking scheme for color medical images using accurate MFrGHMs and 1D Chebyshev chaotic features to achieve lossless copyright protection of the color medical images. We performed experiments where their outcomes ensure the robustness of the proposed zero-watermarking algorithms against various attacks. The proposed zero-watermarking algorithm achieves a good balance between robustness and imperceptibility. Compared with similar existing algorithms, the proposed algorithm has superior robustness, security, and time computation.

Simulation of Quantum Many-Body Dynamics with Tensor Processing Units: Floquet Prethermalization

Tensor processing units (TPUs) are specialized hardware accelerators developed by Google to support large-scale machine-learning tasks but they can also be leveraged to accelerate and scale other linear-algebra-intensive computations. In this paper, we demonstrate the usage of TPUs for massively parallel classical simulations of quantum many-body dynamics on long time scales. We apply our methods to study the phenomenon of Floquet prethermalization, i.e., exponentially slow heating in quantum spin chains subject to high-frequency periodic driving. We simulate the dynamics of L=34 qubits for over 105 Floquet periods, corresponding to circuits with 4×106 nearest-neighbor two-qubit gates. The circuits simulated have no additional symmetries and represent a pure-state evolution in the full 2L-dimensional Hilbert space. This is achieved by distributing the computation over 128 TPU cores. On that size TPU cluster, we find speed-ups in wall-clock run time of 230 times and 15 times when compared to reference CPU and single-graphics-processing-unit (GPU) simulations, respectively, for shorter-time 30-qubit simulations that can be handled by all three platforms. We study the computational cost of the simulations, as a function of both the number of qubits and the number of TPU cores used, up to our maximum capacity of L=40 qubits, which requires a “full pod” of 2048 TPU cores with tens of terabytes of memory in total. For these simulations, an eight-TPU-core machine is comparable to a single A100 GPU and thus the full TPU pod is comparable to a machine with hundreds of top-of-the-line GPUs. However, the TPU pod is more energy and cost efficient and readily accessible (via Google Cloud), unlike such large many-GPU configurations. We also study the accumulation of numerical error as a function of circuit depth in very deep circuits. Our work demonstrates that TPUs can offer significant advantages for state-of-the-art simulations of quantum many-body dynamics.Received 20 November 2021Accepted 20 April 2022DOI:https://doi.org/10.1103/PRXQuantum.3.020331Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum simulationPhysical SystemsFloquet systemsTechniquesNumerical techniquesQuantum InformationStatistical PhysicsCondensed Matter, Materials & Applied Physics

Adaptive partitioning molecular dynamics using an extended Hamiltonian approach.

A recently proposed extended Hamiltonian approach to switching interaction potentials is generalized to enable adaptive partitioning molecular dynamics simulations. Switching is performed along a fictitious classical degree of freedom whose value determines the mixing ratio of the two potentials on a time scale determined by its associated mass. We propose to choose this associated fictitious mass adaptively so as to ensure a constant time scale for all switching processes. For different model systems, including a harmonic oscillator and a Lennard-Jones fluid, we investigate the window of switching time scales that guarantees the conservation of the extended Hamiltonian for a large number of switching events. The methodology is first applied in the microcanonical ensemble and then generalized to the canonical ensemble using a Nosé-Hoover chain thermostat. It is shown that the method is stable for thousands of consecutive switching events during a single simulation, with constant temperature and a conserved extended Hamiltonian. A slight modification of the original Hamiltonian is introduced to avoid accumulation of small numerical errors incurred after each switching process.

Dynamic response of asymmetric bodies assuming a rocking behaviour

The aim of the proposed work is to clarify the dynamic behaviour of asymmetric bodies in the case of rocking behaviour. The proposed algorithm is based on Housner theory and the integration of the equations of motion is evaluated with the Runge–Kutta–Fehlberg (RKF45) method. The significant accumulation of numerical errors is evaluated during the integration between two consecutive acceleration points where the rotation sign changes owing to the adoption of an equation that is wrong. An iterative calculation is developed to reduce numerical error depending on a specific tolerance and, therefore, the algorithm accuracy is investigated. In addition, the reduction of energy due to the impact at the point of change of the sign of rotation is analysed by means of a restitution coefficient. It is important to note that this is highly dependent on the rigid block properties, and experimental programmes are required to calibrate the restitution coefficient properly. Sensitivity analyses are performed in order to clarify the effects of convergence parameters and tolerances on the results, due to high numerical instabilities. Furthermore, fragility curves are retrieved based on far-field and near-field sets of accelerograms proposed by the Applied Technology Council in 2008, ATC-63, and a critical discussion is provided concerning the most efficient intensity measure parameters.

On the computational solution of vector-density based continuum dislocation dynamics models: A comparison of two plastic distortion and stress update algorithms

Continuum dislocation dynamics models of mesoscale plasticity consist of dislocation transport-reaction equations coupled with crystal mechanics equations. The coupling between these two sets of equations is such that dislocation transport gives rise to the evolution of plastic distortion (strain), while the evolution of the latter fixes the stress from which the dislocation velocity field is found via a mobility law. Earlier solutions of these equations employed a staggered solution scheme for the two sets of equations in which the plastic distortion was updated via time integration of its rate, as found from Orowan's law. In this work, we show that such a direct time integration scheme can suffer from accumulation of numerical errors. We introduce an alternative scheme based on field dislocation mechanics that ensures consistency between the plastic distortion and the dislocation content in the crystal. The new scheme is based on calculating the compatible and incompatible parts of the plastic distortion separately, and the incompatible part is calculated from the current dislocation density field. Stress field and dislocation transport calculations were implemented within a finite element based discretization of the governing equations, with the crystal mechanics part solved by a conventional Galerkin method and the dislocation transport equations by the least squares method. A simple test was first performed to show the accuracy of the two schemes for updating the plastic distortion, which shows that the solution method based on field dislocation mechanics is more accurate. This method then was used to simulate an austenitic steel crystal under uniaxial loading and multiple slip conditions. By considering dislocation interactions caused by junctions, a hardening rate similar to discrete dislocation dynamics simulation results was obtained. The simulations show that dislocations exhibit some self-organized structures as the strain is increased.

An analysis of software aging in cloud environment

Cloud Computing is the environment in which several virtual machines (VM) run concurrently on physical machines. The cloud computing infrastructure hosts multiple cloud service segments that communicate with each other using the interfaces. This creates distributed computing environment. During operation, the software systems accumulate errors or garbage that leads to system failure and other hazardous consequences. This status is called software aging. Software aging happens because of memory fragmentation, resource consumption in large scale and accumulation of numerical error. Software aging degrads the performance that may result in system failure. This happens because of premature resource exhaustion. This issue cannot be determined during software testing phase because of the dynamic nature of operation. The errors that cause software aging are of special types. These errors do not disturb the software functionality but target the response time and its environment. This issue is to be resolved only during run time as it occurs because of the dynamic nature of the problem. To alleviate the impact of software aging, software rejuvenation technique is being used. Rejuvenation process reboots the system or re-initiates the softwares. This avoids faults or failure. Software rejuvenation removes accumulated error conditions, frees up deadlocks and defragments operating system resources like memory. Hence, it avoids future failures of system that may happen due to software aging. As service availability is crucial, software rejuvenation is to be carried out at defined schedules without disrupting the service. The presence of Software rejuvenation techniques can make software systems more trustworthy. Software designers are using this concept to improve the quality and reliability of the software. Software aging and rejuvenation has generated a lot of research interest in recent years. This work reviews some of the research works related to detection of software aging and identifies research gaps.

Accelerated spin dynamics using deep learning corrections

Theoretical models capture very precisely the behaviour of magnetic materials at the microscopic level. This makes computer simulations of magnetic materials, such as spin dynamics simulations, accurately mimic experimental results. New approaches to efficient spin dynamics simulations are limited by integration time step barrier to solving the equations-of-motions of many-body problems. Using a short time step leads to an accurate but inefficient simulation regime whereas using a large time step leads to accumulation of numerical errors that render the whole simulation useless. In this paper, we use a Deep Learning method to compute the numerical errors of each large time step and use these computed errors to make corrections to achieve higher accuracy in our spin dynamics. We validate our method on the 3D Ferromagnetic Heisenberg cubic lattice over a range of temperatures. Here we show that the Deep Learning method can accelerate the simulation speed by 10 times while maintaining simulation accuracy and overcome the limitations of requiring small time steps in spin dynamic simulations.

Метод прогнозирования стабилизированного по энергии движения космического аппарата на основе дифференциально-тейлеровских преобразований

Для прогнозирования движения космических аппаратов предложен численно-аналитический метод интегрирования стабилизированного методом Баумгарта дифференциального уравнения орбитального движения космического аппарата. Стабилизация дифференциального уравнения движения методом Баумгарта осуществляется по энергии космического аппарата для уменьшения влияния неустойчивости по Ляпунову на накопление численных погрешностей интегрирования дифференциального уравнения, что эффективно при долгосрочном числовом прогнозировании движения космических аппаратов. Интегрирование стабилизированного уравнения проводится на основе дифференциально-тейлоровских преобразований. Рассмотрены вычислительные схемы с постоянным шагом и порядком интегрирования, а также схемы с адаптацией за шагом и порядком интегрирования. Для адаптивных схем приведены результаты прогнозирования движения космических аппаратов по критерию «точность-вычислительная сложность» для заданной относительной погрешности интегрирования за фазовыми переменными интегрированиями и энергией космического аппарата. Показано, что оба варианта требуют задания различных внутренних параметров адаптации, однако имеют соразмерную эффективность. Предложено использование разработанного метода интегрирования стабилизированных по энергии уравнений для прогнозирования движения космических аппаратов ближнего космоса в Гринвицкой прямоугольной системе координат.

A comprehensive assessment of heat loss mechanisms on the propagation of lean, premixed ethylene-oxygen flames in millimeter-scale tubes

Heat loss plays an important role on the flame propagation characteristics of premixed gas mixtures in small tubes. However, stringent spatial and temporal resolutions required to simulate these phenomena cause numerical investigations to resort to single-step global reaction mechanisms and ignore radiative losses. Both these restrictions are relaxed in this study where a skeletal 10-step mechanism and an optically thin radiation model are employed to study flame propagation in lean (Φ = 0.2) premixed ethylene/oxygen mixtures in millimeter-scale tubes (1 mm and 2 mm diameters). First, convergence in the spatial and temporal resolutions were ascertained and found to compare well with the characteristic reaction zone lengths and timescales determined from detailed reaction mechanisms. The accumulation of numerical errors over the simulation time frame was determined to be less than 0.06%. Radiative losses reduced the flame propagation velocities by 73% and 51% in the 1 mm diameter and 2 mm diameter tubes respectively and made the flames less concave. The flame velocities were moderately affected by the thermal boundary conditions (adiabatic versus isothermal walls). The radiant fractions were in the range 0.12–0.17. The slow CO oxidation reactions cause CO2 concentrations to be lower than those predicted from single-step mechanisms.

Verification of an improved equation-free projective integration method for neoclassical plasma-profile evolution in tokamak geometry

A brute-force, long-time gyrokinetic simulation of plasma profile evolution in magnetic fusion devices is not desirable due to large computational resource requirements and a possible accumulation of numerical error. The equation-free projective integration method of Keverekidis et al. [Commun. Math. Sci. 1(4), 715–762 (2003)] is one of the outstanding candidates in projecting micro-scale simulations to a longer timescale. However, its application to tokamak plasma has not been fruitful due to the appearance of spurious transient oscillations in the lifting process, which are present when the kinetic simulations are initialized with a simplified model distribution function and which make the kinetic simulations to deviate from the desired paths. In this work, a kinetically informed lifting algorithm is added to the equation-free projective integration method, which is then verified in the electrostatic gyrokinetic particle-in-cell code XGCa [R. Hager and C. S. Chang, Phys. Plasmas 23, 042503 (2016)] for a neoclassical ion heat transport problem with adiabatic electrons. This new lifting operator is demonstrated to control spurious transients, enabling an over four-times reduction in the overall computing time in the time-evolution of the ion temperature profile in an axisymmetric toroidal plasma. Further reduction in the computing time is found to be limited due to the stability properties of the linear least squares projective integrator.

Precise Integration Symplectic Analytical Singular Element for Cracks Analysis Under Transient Thermal Conduction

Numerical modeling of mechanical behavior of cracks under transient thermal conduction involves solving an initial value problem (IVP) and two boundary value problems (BVPs). Both of the BVPs have a singularity issue. Drawbacks such as numerical error accumulation and high computational expense of existing numerical approaches should be overcome. This contribution intends to build a unified framework with highly efficiency and accuracy for the numerical modeling of cracks under thermal shock. The precise integration method (PIM) and the symplectic analytical singular element (SASE) have been demonstrated to be favorable alternatives for each problem, i.e., the PIM for solving the IVP and SASE for the BVP. However, it is found that these two methods cannot be combined directly. In order to incorporate the SASEs into the PIM, the existing SASEs are reformulated for the thermal shock cracks analysis. Details of the mathematical derivations are provided. The validity of the proposed method is demonstrated through numerical examples.

Laplace-based computation of transient profiles along transmission lines including time-varying and non-linear elements

In this paper, a combination of a transmission line model defined in the q-s domain (spatial frequency-temporal frequency), the two-dimensional Laplace transform and the principle of superposition is applied to compute transient voltage and current profiles along transmission systems including time-dependent and non-linear elements. The detailed internal information provided by these profiles can be very useful in applications such as insulation coordination design, fault detection and location. The method presented in this paper has significant advantages over the computation of transient profiles using EMTP-type software, given that the latter requires the subdivision of the line, which is time consuming and prone to numerical error accumulation. Additionally, since the line model is defined in the frequency domain, it is possible to include the frequency dependence of the electrical parameters in a very straightforward manner. In order to evidence the accuracy and versatility of the proposed method, several applications examples are presented, and the computed results are compared with those obtained using ATP.

Low-Complexity Time-Domain DBP Based on Random Step-Size and Partitioned Quantization

We propose and experimentally validate a low-complexity time-domain (TD) digital backpropagation (DBP) algorithm for fiber nonlinearity compensation, targeting an optimized hardware implementation. To counteract the coherent accumulation of numerical quantization errors between DBP steps, we propose a random step-size distribution along the optical link (with $\pm$ 5% interval around the optimal step-size). In addition, to further reduce the average quantization bit precision requirements, we propose a partitioned quantization technique, enabling to quantize the finite-impulse response (FIR) filter tail coefficients with significantly lower precision. The proposed low-complexity DBP algorithm is experimentally demonstrated over a 2592 km long-haul wavelength division multiplexing transmission system with 21 $\times$ 32 GBaud PM-16QAM optical channels. Employing the proposed step-size randomization together with dual-time-slot quantization, we demonstrate penalty-free operation at an average of $\sim$ 4 b per FIR coefficient, leading to a 60% complexity reduction when compared to the standard TD-DBP implementation.

Development and Assessment of Fully Automated and Globally Transitive Geometric Morphometric Methods, With Application to a Biological Comparative Dataset With High Interspecific Variation.

Automated geometric morphometric methods are promising tools for shape analysis in comparative biology, improving researchers' abilities to quantify variation extensively (by permitting more specimens to be analyzed) and intensively (by characterizing shapes with greater fidelity). Although use of these methods has increased, published automated methods have some notable limitations: pairwise correspondences are frequently inaccurate and pairwise mappings are not globally consistent (i.e., they lack transitivity across the full sample). Here, we reassess the accuracy of published automated methods-cPDist (Boyer et al. Proc Nat Acad Sci 108 () 18221-18226) and auto3Dgm (Boyer et al.: Anat Rec 298 () 249-276)-and evaluate several modifications to these methods. We show that a substantial percentage of alignments and pairwise maps between specimens of dissimilar geometries were inaccurate in the study of Boyer et al. (Proc Nat Acad Sci 108 () 18221-18226), despite a taxonomically partitioned variance structure of continuous Procrustes distances. We show these inaccuracies are remedied using a globally informed methodology within a collection of shapes, rather than relying on pairwise comparisons (c.f. Boyer et al.: Anat Rec 298 () 249-276). Unfortunately, while global information generally enhances maps between dissimilar objects, it can degrade the quality of correspondences between similar objects due to the accumulation of numerical error. We explore a number of approaches to mitigate this degradation, quantify their performance, and compare the generated pairwise maps (and the shape space characterized by these maps) to a "ground truth" obtained from landmarks manually collected by geometric morphometricians. Novel methods both improve the quality of the pairwise correspondences relative to cPDist and achieve a taxonomic distinctiveness comparable to auto3Dgm. Anat Rec, 301:636-658, 2018. © 2017 Wiley Periodicals, Inc.