Order Taylor Polynomial Research Articles

TaylorNet: A novel approach for spectral filter learning on graph data

Graphs are indispensable data structures used to model interconnected nodes and their relationships. However, devising filters for graph data is challenging. Previous approaches, including BernNet, utilized Bernstein polynomials to approximate spectral graph convolutions but suffered from lengthy training times compared to alternatives. To address this, we propose TaylorNet, a novel graph neural network built upon rigorous mathematical principles and theoretical foundations. TaylorNet provides an efficient approach for learning arbitrary graph spectral filters. Our methodology revolves around utilizing Kth order Taylor polynomials to estimate filters on the normalized Laplacian operator of graphs. By carefully setting the coefficients of these polynomials, we achieve effective filter design. Additionally, TaylorNet personalizes filter learning for specific graphs by leveraging observed graph data and signals to determine these coefficients. Extensive experimentation demonstrates TaylorNet’s ability to learn arbitrary spectral filters, surpassing current state-of-the-art methods, particularly for handling large graphs. Its exceptional performance extends to real-world graph modeling tasks, highlighting its versatility and applicability. For interested researchers, the complete implementation of TaylorNet is available at https://github.com/12chen20/TaylorNet.

Maximum A Posteriori Estimation of Hamiltonian Systems with High Order Taylor Polynomials

Maximum A Posteriori Estimation of Hamiltonian Systems with High Order Taylor Polynomials

Taylor’s theorem: A new perspective for neural tensor networks

Neural tensor networks have been widely used in a large number of natural language processing tasks such as conversational sentiment analysis, named entity recognition and knowledge base completion. However, the mathematical explanation of neural tensor networks remains a challenging problem, due to the bilinear term. According to Taylor’s theorem, a kth order differentiable function can be approximated by a kth order Taylor polynomial around a given point. Therefore, we provide a mathematical explanation of neural tensor networks and also reveal the inner link between them and feedforward neural networks from the perspective of Taylor’s theorem. In addition, we unify two forms of neural tensor networks into a single framework and present factorization methods to make the neural tensor networks parameter-efficient. Experimental results bring some valuable insights into neural tensor networks.

On the global convergence of a fast Halley’s family to solve nonlinear equations

The purpose of this paper is to suggest an approach for increasing the convergence speed of Halley’s method to solve a non-linear equation. This approach is based on the second order Taylor polynomial and on Halley’s formula. By applying it a certain number of times, we obtain a new family of methods. The originality of this family is manifested in the fact that all its sequences are generated from one exceptional formula that depends on a natural integer parameter p. In addition, under certain conditions, the convergence speed of its sequences increases with p. The convergence analysis shows that the order of convergence of all proposed methods is three. A study on their global convergence is carried out. To illustrate the performance of this family, several numerical comparisons are made with other third and higher order methods.

Theoretical and numerical analysis of key sub-grid quantities' effect on filtered Eulerian drag force

An expression of the filtered Eulerian drag force is proposed based on the second order Taylor polynomial approximation of a microscopic drag coefficient. Theoretical computations of the expression at low particle Reynolds numbers show that four sub-grid quantities, the gas drift velocity, the solid drift velocity, the scalar variance of solid volume fraction and a third-order moment, are significant for an accurate estimation of the filtered drag force. Various drag models with different combinations of the four sub-grid quantities are proposed and their coefficients are optimized based on fine-grid two fluid model results. The gas drift velocity is demonstrated to be the most important marker in predicting the filtered drag force. The model with the solid drift velocity and the scalar variance of the solid volume fraction is found to be promising in computational fluid dynamic-discrete element model simulations due to the ready availability of the two sub-grid quantities.

Fast and accurate dynamic synchrophasor estimator for protection applications

This paper proposes a new synchrophasor estimation algorithm capable of speeding up the rate of measurements. It reduces the computational complexity and meets the accuracy requirements of the P-class phasor measurement unit (PMU) defined by the IEEE Std C37.118.1™-2011 and its amendment IEEE Std C37.118.1a™-2014. The proposed estimator is constructed by inserting the second order Taylor polynomial of signal model in the modulated sliding discrete Fourier transform (MSDFT) expression, taking advantage of the dynamic model of the Taylor series expansion and the low computational complexity of the MSDFT. The proposed technique is found to be as accurate as the traditional one (based on joint application of Taylor's second order polynomial and the DFT) with higher reporting rates and it satisfies all the tests listed by the standard as well as the influence of wideband noise and decaying DC component.

Heat flow and quantitative differentiation

For every Banach space (Y,\|\cdot\|_Y) that admits an equivalent uniformly convex norm we prove that there exists c=c(Y)\in (0,\infty) with the following property. Suppose that n\in \mathbb N and that X is an n -dimensional normed space with unit ball B_X . Then for every 1 -Lipschitz function f:B_X\to Y and for every \varepsilon \in (0,1/2] there exists a radius r\ge \exp(-1/\varepsilon^{cn}) , a point x\in B_X with x+rB_X\subset B_X , and an affine mapping \Lambda:X\to Y such that \|f(y)-\Lambda(y)\|_Y\le \varepsilon r for every y\in x+rB_X . This is an improved bound for a fundamental quantitative differentiation problem that was formulated by Bates, Johnson, Lindenstrauss, Preiss and Schechtman (1999), and consequently it yields a new proof of Bourgain's discretization theorem (1987) for uniformly convex targets. The strategy of our proof is inspired by Bourgain's original approach to the discretization problem, which takes the affine mapping \Lambda to be the first order Taylor polynomial of a time- t Poisson evolute of an extension of f to all of X and argues that, under appropriate assumptions on f , there must exist a time t\in (0,\infty) at which \Lambda is (quantitatively) invertible. However, in the present context we desire a more stringent conclusion, namely that \Lambda well-approximates f on a macroscopically large ball, in which case we show that for our argument to work one cannot use the Poisson semigroup. Nevertheless, our strategy does succeed with the Poisson semigroup replaced by the heat semigroup. As a crucial step of our proof, we establish a new uniformly convex-valued Littlewood–Paley–Stein \mathcal{G} -function inequality for the heat semigroup; influential work of Martínez, Torrea and Xu (2006) obtained such an inequality for subordinated Poisson semigroups but left the important case of the heat semigroup open. As a byproduct, our proof also yields a new and simple approach to the classical Dorronsoro theorem (1985) even for real-valued functions.

Improved asymptotic predictions for the effective slip over a corrugated topography

• Significantly improved asymptotic predictions on effective slip in comparison to literature. • Systematic study on convergence radii of asymptotic series. • Novel use of series improvement and acceleration techniques. • Novel use of spectral analysis and symbolic computation. Accurate analytical prediction of the effective slip boundary condition in shear-driven Stokes flows directed longitudinally and transversely to a one-dimensional sinusoidal no-slip topography is explored. First, the domain perturbation technique is extended through spectral analysis and symbolic computations to obtain polynomial approximations (Taylor polynomials) of arbitrary specifiable order for the effective slip length. However, when assessed for numerical accuracy against fully-resolved numerical simulations using the finite-element-method, higher order Taylor polynomials give progressively inferior predictions in comparison to lower-order ones, unless the product of amplitude and wave-number is restricted below unity. From Domb–Sykes plots, the reason for the poor accuracy of higher order Taylor polynomials is assessed to be the finite convergence radii, approximately equaling unity, of the asymptotic power series for both longitudinal and transverse flows. For either of the flows, application of Euler transformation to the expansion parameter provide polynomial-form approximations that are accurate for amplitude values exceeding the convergence radius. The slow convergence of the Euler-transformed series can be remedied through Shanks transformation, at the cost of losing the convenience of closed forms. Finally, Padé approximants are shown to provide even more accurate but still closed-form alternatives to polynomials that work accurately at amplitudes much exceeding the above-identified convergence radii.

Investigating vegetation water dynamics and drought using Metop ASCAT over the North American Grasslands

In this study, we examined the ASCAT backscatter data from Metop-A from 2007 to 2016 to characterize spatial and temporal variability in the vegetation parameters of the TU Wien Soil Moisture Retrieval approach (TUW SMR) across the North American Grasslands. The vegetation parameters are the slope and curvature of a second order Taylor polynomial used to describe the incidence angle dependence of backscatter σ°. A recent development allows the vegetation parameters to be determined dynamically using the local slope values within a prescribed temporal window. Seasonal, interannual and diurnal variations in the vegetation parameters were found to vary across grassland cover types, reflecting variations in soil moisture availability and growing season length. While the slope has always been considered a measure of vegetation density, our results show that curvature also contains information about vegetation. Drought events in 2011 and 2012 resulted in extensive negative σ40∘ and soil moisture anomalies during the maximum biomass period. Contiguous anomalies in slope and curvature were observed where the severity and persistence of the drought were enough to impact vegetation. Observed diurnal differences in slope and curvature suggest that daily moisture transport within the vegetation influences the relative dominance of scattering from the vegetation and soil surface.

An Approximate Approach for Systems of Singular Volterra Integral Equations Based on Taylor Expansion

In this article, an extended Taylor expansion method is proposed to estimate the solution of linear singular Volterra integral equations systems. The method is based on combining the m-th order Taylor polynomial of unknown functions at an arbitrary point and integration method, such that the given system of singular integral equations is converted into a system of linear equations with respect to unknown functions and their derivatives. The required solutions are obtained by solving the resulting linear system. The proposed method gives a very satisfactory solution, which can be performed by any symbolic mathematical packages such as Maple, Mathematica, etc. Our proposed approach provides a significant advantage that the m-th order approximate solutions are equal to exact solutions if the exact solutions are polynomial functions of degree less than or equal to m. We present an error analysis for the proposed method to emphasize its reliability. Six numerical examples are provided to show the accuracy and the efficiency of the suggested scheme for which the exact solutions are known in advance.

Implicit surrogate models for trust region based methods

In this paper, theoretical and practical aspects of Taylor based surrogate models for trust region (TR) optimization are discussed. We propose two surrogate models: one of them based on convex combinations of first order Taylor polynomials while the other one is based on that of quadratic Taylor expansions. During optimization steps of TR methods, the proposed models are built as follows: samples from the cost function are taken about the current solution and then, a linear or a quadratic Taylor polynomial is built about each sampled point. All Taylor expansions are convexly combined by using radial basis functions in order to propose a surrogate model for the optimization step. Theoretically, we prove that, the proposed models are fully linear or fully quadratic depending on whether they are built based on convex combinations of linear or quadratic Taylor expansions, respectively. This guaranties that, regardless the initial guess of the iterative process, the optimization method will converge to local solutions where at least first order optimality conditions are satisfied. Discussions on how to solve the surrogate optimization problems among iterations are presented as well. Experimental results are carried out making use of a sequential data assimilation problem. The results reveal that, the use of our proposed models in such context can improve the quality of analysis states in terms of root-mean-square-error values while uncertainties can be decreased across different assimilation cycles.

Hydroassets portfolio management for intraday electricity trading from a discrete time stochastic optimization perspective

Hydro storage system optimization is becoming one of the most challenging tasks in energy finance. While currently the state-of-the-art of the commercial software in the industry implements mainly linear models, we would like to introduce risk aversion and a generic utility function. At the same time, we aim to develop and implement a computational efficient algorithm, which is not affected by the curse of dimensionality and does not utilize subjective heuristics to prevent it. For the short term power market we propose a simultaneous solution for both dispatch and bidding problems. Following the Blomvall and Lindberg (Eur J Oper Res 143(2):452–461, 2002) interior point model, we set up a stochastic multiperiod optimization procedure by means of a “bushy”recombining tree that provides fast computational results. Inequality constraints are packed into the objective function by the logarithmic barrier approach and the utility function is approximated by its second order Taylor polynomial. The optimal solution for the original problem is obtained as a diagonal sequence where the first diagonal dimension is the parameter controlling the logarithmic penalty and the second one is the parameter for the Newton step in the construction of the approximated solution. Optimal intraday electricity trading and water values for hydroassets as shadow prices are computed. The algorithm is implemented in Mathematica.

Portfolio Management and Stochastic Optimization in Discrete Time: An Application to Intraday Electricity Trading and Water Values for Hydroassets

Hydro storage system optimization is becoming one of the most challenging task in Energy Finance. Following the Blomvall and Lindberg (2002) interior point model, we set up a stochastic multiperiod optimization procedure by means of a ''bushy'' recombining tree that provides fast computational results. Inequality constraints are packed into the objective function by the logarithmic barrier approach and the utility function is approximated by its second order Taylor polynomial. The optimal solution for the original problem is obtained as a diagonal sequence where the first diagonal dimension is the parameter controlling the logarithmic penalty and the second is the parameter for the Newton step in the construction of the approximated solution. Optimimal intraday electricity trading and water values for hydroassets are computed. The algorithm is implemented in Mathematica.

Dealing with uncertainties in angles-only initial orbit determination

A method to deal with uncertainties in initial orbit determination (IOD) is presented. This is based on the use of Taylor differential algebra (DA) to nonlinearly map uncertainties from the observation space to the state space. When a minimum set of observations is available, DA is used to expand the solution of the IOD problem in Taylor series with respect to measurement errors. When more observations are available, high order inversion tools are exploited to obtain full state pseudo-observations at a common epoch. The mean and covariance of these pseudo-observations are nonlinearly computed by evaluating the expectation of high order Taylor polynomials. Finally, a linear scheme is employed to update the current knowledge of the orbit. Angles-only observations are considered and simplified Keplerian dynamics adopted to ease the explanation. Three test cases of orbit determination of artificial satellites in different orbital regimes are presented to discuss the feature and performances of the proposed methodology.

Note on Friedrichs' inequality in N-star-shaped domains

We establish an explicit constant in the Friedrichs inequality‖u−1|S|∫Su(x)dx‖Lp(Ω)≤C(Ω,S,B)‖∇u‖Lp(Ω) for u∈W1,p(Ω), 1≤p<∞, a domain Ω⊂Rn star-shaped with respect to a convex set B, and a measurable subset S⊂Ω with |S|>0, n≥2. This result will be generalized to so-called N-star-shaped domains which can be written as union of N domains star-shaped with respect to convex sets. A second generalization concerns the use of higher order moments when replacing the integral mean of u over S by the mean of higher order Taylor polynomials. The proofs are direct and elementary.

Curvature weight method of solving the point reactor neutron kinetic equations

The point kinetic equations are the system of a couple stiff ordinary differential equations. Many studies have focused on the development of more advanced and efficient methods of solving the equations, such as the high order Taylor polynomials method, the Haar wavelet operational method, the fractional point-neutron kinetic model method, the basis function method, the homotopy analysis method, and other methods. Most of these methods are successful in some specific problems, but still have, more or less, disadvantages. For example, the accuracy of the Haar wavelet operational method is limited by the collocation points, and it needs more computing time for a high precision. Aiming at the requirements that some numerical calculation results must have the higher precision and only the positive error in the nuclear reactor safety engineering and ship reactor for the maneuverability, in this paper we try to look for a new numerical method to satisfy that the calculation value is slightly higher than the real value when the actual curve is upward convex or downward concave, and the error is not greater than that by the Euler and improved Euler method. The new method is so-called the curvature weight (CW) method, which is based on the curvature circle method and considers the contributions of two curvatures at the interval step point to the average curvature inside the interval step. Using the decoupling method to remove the stiffness of equations and the instantaneous jump approximation to derive the neutron differential equations, the first and second derivative of neutron density are obtained. Then the CW method is used to solve the point reactor neutron kinetic equations, and thus obtaining the numerical solution. Compared with the results by the Euler and improved Euler method, the numerical calculation results by the CW method are always higher than the real value, and the calculation accuracy and speed are improved significantly. When this new method is used to solve the point reactor neutron differential equations with the step and linear reactivity inserted into the subcritical reactor, the numerical results which satisfy the requirements of positive calculation error and high precision can be obtained quickly. After improving the calculation step length, the precision reduction by the CW method is significantly lower than that by the Euler and improved Euler method. So the CW method can greatly shorten the total computing time, and it is also effective for most of differential equation systems.

Nonlinear analysis of position estimate in global navigation satellite systems

The paper presents a nonlinear analysis of position estimation based on a global navigation satellite system. A classical problem formulation and iterative solution that results in the weighted least squares estimate of the receiver state are assumed. The analysis employs the Taylor's theorem to express the nonlinear measurement model using the first order Taylor polynomial at the state estimate and the Lagrange form of the remainder. A sensitivity analysis of the Jacobian matrix pseudoinverse is performed, and an upper bound on the size of the Lagrange remainder is derived using eigenstructure of the Hessian matrix. The results obtained show that both the sensitivity of the pseudoinverse and the size of the quadratic term are not significant, and thus the linear approximation commonly used to derive stochastic properties of the state estimate is reasonable. Although this result has been experimentally confirmed by numerous successful applications, this analysis can serve as a more rigorous basis when the design procedures for a safety critical system have to be satisfied.

An Analytic Approach for Stochastic Differential Utility for Endowment and Production Economies

In this paper, an analytic approach is applied to approximate stochastic differential utility for both endowment and production economies. This utility function satisfies a second order nonlinear differential equation and is analytic near the stationary mean of the state variable. The radius of convergence of its power series representation is at lease two standard deviations of the stationary distribution of the state variable around its mean. The main contribution of this paper is to use a higher order Taylor polynomial to accurately approximate the utility function, which confirms the empirical properties of the pricing kernel and equity premium on wealth.

High order optimal control of space trajectories with uncertain boundary conditions

A high order optimal control strategy is proposed in this work, based on the use of differential algebraic techniques. In the frame of orbital mechanics, differential algebra allows to represent, by high order Taylor polynomials, the dependency of the spacecraft state on initial conditions and environmental parameters. The resulting polynomials can be manipulated to obtain the high order expansion of the solution of two-point boundary value problems. Since the optimal control problem can be reduced to a two-point boundary value problem, differential algebra is used to compute the high order expansion of the solution of the optimal control problem about a reference trajectory. Whenever perturbations in the nominal conditions occur, new optimal control laws for perturbed initial and final states are obtained by the mere evaluation of polynomials. The performances of the method are assessed on lunar landing, rendezvous maneuvers, and a low-thrust Earth–Mars transfer.

A note on ‘Taylor series approach to fuzzy multiple objective linear fractional programming’

Toksari, in his paper ‘Taylor series approach to fuzzy multiobjective linear fractional programming’, published in Information Sciences 178 (2008), proposed a new method to solve multiple objective linear fractional programming problems and declared that it is more effective when compare to previous methods. He constructed fuzzy goals and then used first order Taylor-polynomials to approximate the corresponding linear fractional membership functions by linear functions. Aggregating the linear functions Toksari obtained a crisp linear programming problem and claimed that it is ‘equal’ to the fuzzy fractional one. In this paper, we indicate the fallacy that arises by using Taylor approximation and propose an improved method that guarantees the efficiency of the solution it provides. Both practical applications from Toksari’s paper are recalled to show that the improvements we suggest are effective.