Truncated Taylor Series Expansion Research Articles

Scientific Deep Machine Learning Concepts for the Prediction of Concentration Profiles and Chemical Reaction Kinetics: Consideration of Reaction Conditions.

Emerging concepts from scientific deep machine learning such as physics-informed neural networks (PINNs) enable a data-driven approach for the study of complex kinetic problems. We present an extended framework that combines the advantages of PINNs with the detailed consideration of experimental parameter variations for the simulation and prediction of chemical reaction kinetics. The approach is based on truncated Taylor series expansions for the underlying fundamental equations, whereby the external variations can be interpreted as perturbations of the kinetic parameters. Accordingly, our method allows for an efficient consideration of experimental parameter settings and their influence on the concentration profiles and reaction kinetics. A particular advantage of our approach, in addition to the consideration of univariate and multivariate parameter variations, is the robust model-based exploration of the parameter space to determine optimal reaction conditions in combination with advanced reaction insights. The benefits of this concept are demonstrated for higher-order chemical reactions including catalytic and oscillatory systems in combination with small amounts of training data. All predicted values show a high level of accuracy, demonstrating the broad applicability and flexibility of our approach.

A stable Q-compensated reverse-time migration method using a modified fractional viscoacoustic wave equation

Seismic wave propagation inside the earth is usually accompanied by amplitude dissipation and velocity dispersion. It is essential to accurately characterize these effects and make the corresponding compensations in seismic reverse-time migration (RTM) and the following interpretation. Due to the approximation of [Formula: see text] in the derivation, the conventional fractional viscoacoustic wave equation in strongly attenuative media is unsatisfactory. This paper develops a modified viscoacoustic wave equation by combining the relationship between angular frequency and complex wavenumber. The numerical simulation demonstrates its advantage of high accuracy in describing constant- Q effects, especially in strongly attenuative media with small Q values. A truncated Taylor-series expansion algorithm and a pseudospectral method are developed to solve this equation during wave propagation. To address the issue of numerical instability in Q-compensated RTM ( Q-RTM), a stabilized technology with regularization is further developed. Unlike the conventional filtering method implemented in the wavenumber domain, our approach only adds an explicit regularization term, which can be directly computed in the space domain. Furthermore, this regularization depends on velocities and quality factors, significantly improving its applicability in complicated structures. Noise-free data tests demonstrate that our Q-RTM method can effectively correct phase distortion, compensate for amplitude attenuation, and produce high-resolution migrated images without any cutoff-frequency artifacts caused by filtering. Noisy data experiments further verify the antinoise performance and robustness of our method.

A Functional Data Approach for Continuous-Time Analysis Subject to Modeling Discrepancy under Infill Asymptotics

Parametric continuous-time analysis often entails derivations of continuous-time models from predefined discrete formulations. However, undetermined convergence rates of frequency-dependent parameters can result in ill-defined continuous-time limits, leading to modeling discrepancy, which impairs the reliability of fitting and forecasting. To circumvent this issue, we propose a simple solution based on functional data analysis (FDA) and truncated Taylor series expansions. It is demonstrated through a simulation study that our proposed method is superior—compared with misspecified parametric methods—in fitting and forecasting continuous-time stochastic processes, while the parametric method slightly dominates under correct specification, with comparable forecast errors to the FDA-based method. Due to its generally consistent and more robust performance against possible misspecification, the proposed FDA-based method is recommended in the presence of modeling discrepancy. Further, we apply the proposed method to predict the future return of the S&P 500, utilizing observations extracted from a latent continuous-time process, and show the practical efficacy of our approach in accurately discerning the underlying dynamics.

Higher-order implicit shock-capturing scheme based on linearization of implicit fluxes for the Euler equations

PurposeA higher-order implicit shock-capturing scheme is presented for the Euler equations based on time linearization of the implicit flux vector rather than the residual vector.Design/methodology/approachThe flux vector is linearized through a truncated Taylor-series expansion whose leading-order implicit term is an inner product of the flux Jacobian and the vector of differences between the current and previous time step values of conserved variables. The implicit conserved-variable difference vector is evaluated at cell faces by using the reconstructed states at the left and right sides of a cell face and projecting the difference between the left and right states onto the right eigenvectors. Flux linearization also facilitates the construction of implicit schemes with higher-order spatial accuracy (up to third order in the present study). To enhance the diagonal dominance of the coefficient matrix and thereby increase the implicitness of the scheme, wave strengths at cell faces are expressed as the inner product of the inverse of the right eigenvector matrix and the difference in the right and left reconstructed states at a cell face.FindingsThe accuracy of the implicit algorithm at Courant–Friedrichs–Lewy (CFL) numbers greater than unity is demonstrated for a number of test cases comprising one-dimensional (1-D) Sod’s shock tube, quasi 1-D steady flow through a converging-diverging nozzle, and two-dimensional (2-D) supersonic flow over a compression corner and an expansion corner.Practical implicationsThe algorithm has the advantage that it does not entail spatial derivatives of flux Jacobian so that the implicit flux can be readily evaluated using Roe’s approximate Jacobian. As a result, this approach readily facilitates the construction of implicit schemes with high-order spatial accuracy such as Roe-MUSCL.Originality/valueA novel finite-volume-based higher-order implicit shock-capturing scheme was developed that uses time linearization of fluxes at cell interfaces.

Modified viscoelastic wavefield simulations in the time domain using the new fractional Laplacians

Abstract Accurately characterizing seismic attenuation effects on wave propagations is crucially important for structure interpretation and reservoir evaluation. The conventional fractional viscoelastic wave equation is not satisfactory on accuracy for small Q values. To solve this issue, we derive a novel fractional viscoelastic wave equation by combining an accurate relationship between angular frequency and complex wavenumber. The dissipation- and dispersion-dominated wave equations are also derived to simulate the amplitude-dissipation and phase-dispersion characteristics. The truncated Taylor-series expansion (TE) algorithm is developed to approximate the mixed-domain operators. After that, the generalized pseudospectral approach can be directly used to solve the new wave equation. In addition, an accurate viscoelastic wave equation constructed by the fractional time derivatives is used to calculate reference solutions to evaluate the accuracy of the new expression. Modelling results indicate that the newly proposed viscoelastic wave equation using the new fractional Laplacians is more accurate than the conventional one, especially in a small Q medium (i.e. QP = QS = 5). Furthermore, we also examine the accuracy of the TE approximation with a series of Q values. A homogeneous model and the modified BP2004 viscoelastic model are used to investigate the accuracy of viscoelastic wave propagations using the TE algorithm. All modelling results fully demonstrate the performance of the newly proposed viscoelastic wave equation and numerical algorithm.

Error analysis of the truncated Taylor series expansion method for computing matrix exponential

Error analysis of the truncated Taylor series expansion method for computing matrix exponential

A novel and effective way to impose boundary conditions and to mitigate the surface effect in state‐based Peridynamics

AbstractPeridynamics is a nonlocal continuum theory capable of modeling effectively crack initiation and propagation in solid bodies. However, the nonlocal nature of this theory is the cause of two main problems near the boundary of the body: an undesired stiffness fluctuation, the so‐called surface effect, and the difficulty of defining a rational method to properly impose the boundary conditions. The surface effect is analyzed analytically and numerically in the present paper in a state‐based peridynamic model. The authors propose a modified fictitious node method based on an extrapolation with a truncated Taylor series expansion. Furthermore, a rational procedure to impose the boundary conditions is defined with the aid of the fictitious nodes. In particular, Neumann boundary conditions are implemented via the peridynamic concept of force flux. The accuracy of the proposed method is assessed by means of several numerical examples for a state‐based peridynamic model: with respect to the peridynamic model adopting no corrections, the results are significantly improved even if low values of the truncation order for the Taylor expansion are chosen.

Derivation of the differential continuity equation in an introductory engineering fluid mechanics course

The differential continuity equation is elegantly derived in advanced fluid mechanics textbooks using the divergence theorem of Gauss, where the surface integral of the mass flux flowing out of a finite control volume is replaced by the volume integral of the divergence of the mass flux within the control volume. To avoid the need for introducing the Gauss divergence theorem in an introductory fluid mechanics course, introductory textbooks in fluid mechanics have opted to use a more simple approach, which depends on the consideration of an infinitesimal control volume and the use of Taylor series expansion. This approach, however, involves a first order truncation of the Taylor series expansion and the use of approximate equality signs which may imply to undergraduate students that the derived continuity equation is an approximate equation. The present study proposes an alternative derivation of the differential continuity equation using a finite control volume and is based on the simple concept of the antiderivative function and the fundamental theorem of calculus. The proposed derivation eliminates the need to formally introduce the Gauss divergence theorem in an introductory engineering fluid mechanics course while avoiding the use of truncated Taylor series expansion and approximate equality signs, hence providing a more simple and sound understanding of the derivation of the differential continuity equation to undergraduate engineering students.

Accurate Real-time CNC Curve Interpolators Based Upon Richardson Extrapolation

Real-time CNC interpolators achieving a constant or variable feedrate V along a parametric curve r(ξ) are usually based on truncated Taylor series expansions defining the time-dependence of the curve parameter ξ. Since the feedrate should be specified as a function of a physically meaningful variable (such as time t, path arc length s, or curvature κ) rather than ξ, successive applications of the differentiation chain rule are necessary to determine Taylor series coefficients beyond the linear term. The closed-form expressions for the higher-order coefficients are increasingly cumbersome to derive and implement, and consequently error-prone. To address this issue, the use of Richardson extrapolation as a simple means to compute rapidly convergent approximations to the higher-order coefficients is investigated herein. The methodology is demonstrated in the context of (1) an arc-length-dependent feedrate for cornering motions; (2) direct real-time offset curve interpolation; and (3) a curvature-dependent feedrate. All of these examples admit simple implementations that circumvent the need for tedious symbolic calculations of higher-order coefficients, and are compatible with real-time controllers with millisecond sampling intervals.

QSW_MPI: A framework for parallel simulation of quantum stochastic walks

QSW_MPI is a Python package developed for time-series simulation of continuous-time quantum stochastic walks. This model allows for the study of Markovian open quantum systems in the Lindblad formalism, including a generalisation of the continuous-time random walk and continuous-time quantum walk. Consisting of a Python interface accessing parallelised Fortran libraries utilising sparse data structures, QSW_MPI is scalable to massively parallel computers, which makes possible the simulation of a wide range of walk dynamics on directed and undirected graphs of arbitrary complexity. Program summaryProgram Title: QSW_MPICPC Library link to program files:http://dx.doi.org/10.17632/r8jsx4xxgv.1Licensing provisions: GPLv3Programming language: Python 3 + Fortran 2003External routines/libraries: NumPy [1], SciPy [2], MPI for Python [3], h5py[4]Nature of problem: QSW_MPI provides a framework for the simulation of quantum stochastic walks on arbitrary graphs (directed/undirected, weighted/unweighted).Solution method: A parallel distributed-memory implementation of the matrix exponential via a truncated Taylor series expansion with scaling and squaring [5].Additional comments including restrictions and unusual features: QSW_MPI will provide support for the simulation of multiple quantum walkers in a future version.

A Novel Approximation Methodology and Its Efficient VLSI Implementation for the Sigmoid Function

In this brief, a novel approximation method and its optimized hardware implementation are proposed for the sigmoid function used in Deep Neural Networks (DNNs). Based on piecewise approximation and truncated Taylor series expansion, the proposed method achieves very good approximation with low complexity while exploiting data representation with powers of two. In addition, by analyzing gradients of the sigmoid function, a small trick is introduced to improve the approximation precision. Furthermore, to reduce the hardware complexity and shorten the critical path, sampled values of the function are generated with simple logical-mapping. It is shown that the proposed approximation schemes can be implemented with purely combinational logic and the sigmoid function can be computed in one clock cycle. The experimental results demonstrate that the mean absolute errors are at the order of 1 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> . Compared with prior arts, the new design can obtain significant improvement in critical path with comparable performance.

Approximate Nonlinear Discrete-Time Models Based on B-Spline Functions

We consider the discretization of continuous-time nonlinear systems described by normal forms. In particular, we consider the case when the input to the system is generated by a B-spline hold device to obtain an approximate discrete-time model. It is shown that the corresponding sampled-data model and its accuracy (measured in terms of the local truncation error) depend on the smoothness of the input and on the applied integration strategy, namely, the truncated Taylor series expansion. Moreover, the sampling zero dynamics of the discrete-time model are asymptotically characterized as the sampling period goes to zero, and it is shown that these zero dynamics converge to the asymptotic sampling zeros of the linear case.

Numerical solution of the Kirchhoff-transformed Richards equation for simulating variably saturated flow in heterogeneous layered porous media

A new numerical method was developed to accurately and efficiently compute a solution of the nonlinear Richards equation with a layered soil. In the proposed method, the Kirchhoff integral transformation was applied. However, in the Kirchhoff integral transformation approach, the transformed Kirchhoff head has dyadic characteristics at the material interface between different soil types. To avoid the dyadic characteristics at the material interface, a truncated Taylor series expansion was applied to the Kirchhoff head at the material interface and so the Kirchhoff head was replaced with a single pressure head value at the material interface. Accordingly, through the Taylor series expansion, a set of algebraic equations in the one-dimensional control volume finite difference discretized system formed a tridiagonal matrix system. Through a series of numerical experiments, the new method was compared to other numerical methods to determine its superiority. The results clearly demonstrated that the approach was not only more computationally efficient, but also more accurate and robust than other numerical methods. Computational performance was greatly enhanced with the proposed method, and which could be used to simulate complicated heterogeneous flow at a large-scale watershed or regional scale.

On Some Sufficiency-Type Global Stability Results for Time-Varying Dynamic Systems with State-Dependent Parameterizations

This paper formulates sufficiency-type global stability and asymptotic stability results for, in general, nonlinear time-varying dynamic systems with state-trajectory solution-dependent parameterizations. The stability proofs are based on obtaining sufficiency-type conditions which guarantee that either the norms of the solution trajectory or alternative interval-type integrals of the matrix of dynamics of the higher-order than linear terms do not grow faster than their available supremum on the preceding time intervals. Some extensions are also given based on the use of a truncated Taylor series expansion of chosen truncation order with multiargument integral remainder for the dynamics of the differential system.

An algebraic approach to on-line signal denoising and derivatives estimation

In this paper, a new algebraic approach to the on-line signal derivatives estimation is proposed. The proposed approach is based on the conversion of the truncated Taylor series expansion to the set of linearly independent equations regarding the signal derivatives. The nonhomogeneous parts of the obtained set of equations are convolution integrals, which can be transformed to the stable linear state-space filter realization. The proposed algebraic estimator provides stable convergence without the need for periodic re-initialization, as in the case of the conventional algebraic estimators. In contrast to the Taylor series-based tracking differentiators, the proposed estimator also provides an estimation of the arbitrary number of the higher-order signal derivatives. In addition, the tuning of the estimator parameters does not depends on the filter dimension. The efficiency of the proposed estimator is illustrated by the simulation examples and experimental results related to the monitoring of the surgical drilling process.

A Constrained Scheme for High Precision Downward Continuation of Potential Field Data

To further improve the accuracy of the downward continuation of potential field data, we present a novel constrained scheme in this paper combining the ideas of the truncated Taylor series expansion, the principal component analysis, the iterative continuation and the prior constraint. In the scheme, the initial downward continued field on the target plane is obtained from the original measured field using the truncated Taylor series expansion method. If the original field was with particularly low signal-to-noise ratio, the principal component analysis is utilized to suppress the noise influence. Then, the downward continued field is upward continued to the plane of the prior information. If the prior information was on the target plane, it should be upward continued over a short distance to get the updated prior information. Next, the difference between the calculated field and the updated prior information is calculated. The cosine attenuation function is adopted to get the scope of constraint and the corresponding modification item. Afterward, a correction is performed on the downward continued field on the target plane by adding the modification item. The correction process is iteratively repeated until the difference meets the convergence condition. The accuracy of the proposed constrained scheme is tested on synthetic data with and without noise. Numerous model tests demonstrate that downward continuation using the constrained strategy can yield more precise results compared to other downward continuation methods without constraints and is relatively insensitive to noise even for downward continuation over a large distance. Finally, the proposed scheme is applied to real magnetic data collected within the Dapai polymetallic deposit from the Fujian province in South China. This practical application also indicates the superiority of the presented scheme.

Two-body perturbation theory versus first order perturbation theory: A comparison based on the square-well fluid.

We show that the Zwanzig first-order perturbation theory can be obtained directly from a truncated Taylor series expansion of a two-body perturbation theory and that such truncation provides a more accurate prediction of thermodynamic properties than the full two-body perturbation theory. This unexpected result is explained by the quality of the resulting approximation for the fluid radial distribution function. We prove that the first-order and the two-body perturbation theories are based on different approximations for the fluid radial distribution function. To illustrate the calculations, the square-well fluid is adopted. We develop an analytical expression for the two-body perturbed Helmholtz free energy for the square-well fluid. The equation of state obtained using such an expression is compared to the equation of state obtained from the first-order approximation. The vapor-liquid coexistence curve and the supercritical compressibility factor of a square-well fluid are calculated using both equations of state and compared to Monte Carlo simulation data. Finally, we show that the approximation for the fluid radial distribution function given by the first-order perturbation theory provides closer values to the ones calculated via Monte Carlo simulations. This explains why such theory gives a better description of the fluid thermodynamic behavior.

Approximate Gibbs algorithm for blind data detection in two‐way relay networks

This study investigates the blind data detection in two-way relay networks (TWRN) that employ amplify-and-forward (AF) relay strategy. To blindly detect the data in TWRN in the presence of uncertain time-frequency offsets and phase noise, the authors develop a new Bayesian-based approximate Gibbs algorithm based on truncated Taylor series expansion approximation. In addition, the authors exploit available constraint information on parameters of interest. The authors present three receivers based on three different parameter estimation approaches. The authors further discuss the implementation issue and present diagnostic convergence analysis. The authors’ numerical results demonstrate the performance and efficacy of their proposed algorithm.

Further Comments on Taylor Series Expansion of the Delay/Advance Operator

The limitations of the Taylor series approximations of the delayed variables have been documented by means of examples in several works. This note unifies these previous comments through a higher-level analysis of the issue. It is shown that the use of the truncated Taylor series expansion of the (inverse) advance operator does not feature the same drawbacks.

An extended harmonic balance method based on incremental nonlinear control parameters

A new formulation for calculating the steady-state responses of multiple-degree-of-freedom (MDOF) non-linear dynamic systems due to harmonic excitation is developed. This is aimed at solving multi-dimensional nonlinear systems using linear equations. Nonlinearity is parameterised by a set of ‘non-linear control parameters' such that the dynamic system is effectively linear for zero values of these parameters and nonlinearity increases with increasing values of these parameters. Two sets of linear equations which are formed from a first-order truncated Taylor series expansion are developed. The first set of linear equations provides the summation of sensitivities of linear system responses with respect to non-linear control parameters and the second set are recursive equations that use the previous responses to update the sensitivities. The obtained sensitivities of steady-state responses are then used to calculate the steady state responses of non-linear dynamic systems in an iterative process. The application and verification of the method are illustrated using a non-linear Micro-Electro-Mechanical System (MEMS) subject to a base harmonic excitation. The non-linear control parameters in these examples are the DC voltages that are applied to the electrodes of the MEMS devices.