Local Taylor Expansion Research Articles

Об интервальном оценивании с высоким порядком минимума функции

The article addresses the problem of two-sided estimation of the minimum of a function of one real variable using interval methods. These methods are widely used for finding the ranges of functions, but their order of accuracy often does not exceed 3 for the functions of one variable and 2 for the case of many variables. The paper proposes a method for finding an interval estimate for the minimum of a function of one variable with any odd order greater than 2. The proposed technique uses so-called functional intervals, the boundaries of which are described by polynomials of even degrees obtained from the Taylor formula for the function under consideration. The article provides a derivation of interval estimates for the minimum, and also proves an estimate for the order of convergence of the width of the resulting interval with the decrease in the size of the domain of the function. An example is given in which estimating the minimum of a function reaches the theoretical order of convergence. The approach developed in this paper allows, in principle, to achieve any odd accuracy order in the interval estimation of the minimum of a function. In practice, this method is convenient to use for the third order of accuracy, since, on the one hand, we need to find an interval estimate for derivatives of high degrees, and, on the other hand, we need to find roots of polynomials of even degrees. The new approach is shown to be useful and practical, as it uses the local Taylor expansion and therefore behaves stably as the width of the function domain decreases. The new approach can be used to improve the performance of interval global optimization algorithms based on adaptive domain partitioning. An increase in the order of estimation accuracy entails the acceleration of these algorithms, as well as a decrease in the number of subintervals generated in the process of partitioning the domain.

An upwind generalized finite difference method for meshless solution of two-phase porous flow equations

This paper makes the first attempt to apply newly developed upwind GFDM for the meshless solution of two-phase porous flow equations. In the presented method, node cloud is used to flexibly discretize the computational domain, instead of complicated mesh generation. Combining with moving least square approximation and local Taylor expansion, spatial derivatives of oil-phase pressure at a node are approximated by generalized difference operators in the local influence domain of the node. By introducing the first-order upwind scheme of phase relative permeability, and combining the discrete boundary conditions, fully-implicit GFDM-based nonlinear discrete equations of the immiscible two-phase porous flow are obtained and solved by the nonlinear solver based on the Newton iteration method with the automatic differentiation, to avoid the additional computational cost and possible computational instability caused by sequentially coupled scheme. Two numerical examples are implemented to test the computational performances of the presented method. Detailed error analysis finds the two sources of the calculation error, and points out the significant effect of the symmetry or uniformity of the node allocation in the node influence domain on the accuracy of generalized difference operators, and the radius of the node influence domain should be small to achieve high calculation accuracy, which is a significant difference between the studied hyperbolic two-phase porous flow problem and the elliptic problems when GFDM is applied. In all, the upwind GFDM with the fully implicit nonlinear solver and related analysis about computational performances given in this work may provide a critical reference for developing a general-purpose meshless numerical simulator for porous flow problems.

Simulation of antiplane shear problems with multiple inclusions using the generalized finite difference method

In this paper, the generalized finite difference method is adopted for solving antiplane shear problems with multiple inclusions. This method is applied to domains in matrix and inclusions respectively in which the local Taylor expansion and moving least square method are applied to form the local system. And the final sparse system is obtained through satisfying the continuity and equilibrium conditions along the interface between matrix and inclusions. Two numerical examples containing several number of inclusions with various locations, shapes, sizes as well as different shear moduli are provided to assess the applicability and accuracy of the proposed method. It is verified that the proposed method can be used for problems when the ratio of shear moduli is extremely large. The comparison is made between the proposed method and reference solutions.

Separating Mesoscale and Submesoscale Flows from Clustered Drifter Trajectories

Drifters deployed in close proximity collectively provide a unique observational data set with which to separate mesoscale and submesoscale flows. In this paper we provide a principled approach for doing so by fitting observed velocities to a local Taylor expansion of the velocity flow field. We demonstrate how to estimate mesoscale and submesoscale quantities that evolve slowly over time, as well as their associated statistical uncertainty. We show that in practice the mesoscale component of our model can explain much first and second-moment variability in drifter velocities, especially at low frequencies. This results in much lower and more meaningful measures of submesoscale diffusivity, which would otherwise be contaminated by unresolved mesoscale flow. We quantify these effects theoretically via computing Lagrangian frequency spectra, and demonstrate the usefulness of our methodology through simulations as well as with real observations from the LatMix deployment of drifters. The outcome of this method is a full Lagrangian decomposition of each drifter trajectory into three components that represent the background, mesoscale, and submesoscale flow.

Polynomial Chaos Expansion for Parametric Problems in Engineering Systems: A Review

In engineering systems with uncertain parameters, it is crucial for system analysis and control to analyze the relationship between these uncertain parameters and system outputs (or states). Nevertheless, the acquisition of this parameter-output function, termed parametric problem in this article, is not easy because it is usually a complicated implicit nonlinear function. Polynomial chaos expansion (PCE), based on the generalized Fourier expansion, is a state-of-the-art method for both parametric problems and uncertainty quantification. Its basic idea is to approximate the implicit parameter-output function with a globally optimal explicit polynomial function, which remains accurate in the case of strong nonlinearity compared with the local Taylor expansion. This article provides a review of PCE and its applications in parametric problems. In terms of PCE theory, a rigorous and complete framework of PCE is established and a detailed review of typical PCE methods is presented. In terms of applications, two kinds of general parametric problems, namely parametric nonlinear algebraic equations and parametric differential equations, as well as their examples in some engineering fields, are introduced. Furthermore, some worthwhile future works are presented at the end of this article, which may facilitate the development of PCE.

Fractional Hermite interpolation for non-smooth functions

The interpolation of functions plays a fundamental role in numerical analysis. The highly accurate approximation of non-smooth functions is a challenge in science and engineering as traditional polynomial interpolation cannot characterize the singular features of these functions. This paper aims at designing a fractional Hermite interpolation for non-smooth functions based on the local fractional Taylor expansion and at deriving the corresponding explicit formula and its error remainder. We also present a piecewise hybrid Hermite interpolation scheme, a combination of fractional Hermite interpolation and traditional Hermite interpolation. Some numerical examples are presented to show the high accuracy of the fractional Hermite interpolation method.

Positivity-preserving sixth-order implicit finite difference weighted essentially non-oscillatory scheme for the nonlinear heat equation

This paper presents a class of semi-implicit finite difference weighted essentially non-oscillatory (WENO) schemes for solving the nonlinear heat equation. For the discretization of second-order spatial derivatives, a sixth-order modified WENO scheme is directly implemented. This scheme preserves the positivity principle and rejects spurious oscillations close to non-smooth points. In order to admit large time steps, a class of implicit Runge–Kutta methods is used for the temporal discretization. The implicit parts of these methods are linearized in time by using the local Taylor expansion of the flux. The stability analysis of the semi-implicit WENO scheme with 3-stages form is provided. Finally, some comparative results for one-, two- and three-dimensional PDEs are included to illustrate the effectiveness of the proposed approach.

The ‘hanging flag’ problem: on the heaving motion of a thin filament in the limit of small flexural stiffness

We investigate the fluid–structure interaction of a vertically hanging filament immersed in uniform incompressible high Reynolds number flow. The filament is subject to small-amplitude harmonic heaving at its upstream edge, and to a gravity-induced (‘hanging chain’) tension force. We focus on the limit of small bending rigidity to examine the differences between a highly elastic beam (where bending rigidity is small but finite) and a membrane (where bending rigidity vanishes). The problem is analysed by means of thin airfoil theory, in conjunction with a discrete vortex model for the downstream wake. Denoting the filament non-dimensional rigidity (normalized by the tension force) by $\bar{\unicode[STIX]{x1D700}}$, it is first verified that the beam deflection and associated flow field converge to the membrane solution at $\bar{\unicode[STIX]{x1D700}}\rightarrow 0$. At low actuation frequencies, the differences between the membrane and beam motions are small, and both follow a nearly rigid-body motion parallel to the upstream-edge actuation. With increasing frequency, the differences between the beam and membrane become visible at increasingly lower values of $\bar{\unicode[STIX]{x1D700}}$, and the stabilizing effect of beam flexural rigidity, resulting in reduced flapping amplitudes, is apparent. Examining the beam motion near its edge points at non-small frequencies, semi-analytic approximations for the associated time-periodic displacements are obtained. Close to the actuated end, a layer of width $\bar{\unicode[STIX]{x1D700}}^{1/2}$ is found, where the flexural rigidity term in the equation of motion is balanced by the tension term. Here, the differences between the beam and membrane deflections are attributed to the additional zero-slope condition satisfied by the former. In the vicinity of the free end, a local Taylor expansion is carried out. A balance between the bending and inertia terms results in a layer of width $\propto \bar{\unicode[STIX]{x1D700}}^{1/4}/\bar{\unicode[STIX]{x1D714}}_{h}^{1/2}$, where $\bar{\unicode[STIX]{x1D714}}_{h}$ denotes the scaled heaving frequency. The layer is therefore thicker than the upstream layer for $\bar{\unicode[STIX]{x1D714}}_{h}\approx 1$, and becomes thinner with increasing $\bar{\unicode[STIX]{x1D714}}_{h}$. Within the layer, the beam deflects linearly with the distance from the edge, in marked difference from a membrane and in accordance with the free-end conditions satisfied by the former.

Solving DSGE Models - When Local Approximations Fail

This paper studies the effect of persistent growth risks on the solution accuracy of dynamic stochastic general equilibrium models. We find that whenever the economy is exposed to risks with long-lasting effects a local Taylor-expansion based solution method does not suffice. In particular, this affects asset pricing quantities and welfare implications. We compare perturbed solutions for various economies with solutions obtained with a projection algorithm. It is shown that besides slightly misstating macroeconomic moments a perturbed solution strongly understates the mean risk-free rate and the wealth consumption ratio. Further, we identify parameters driving the approximation error and compare different degrees of approximation. Finally, we show that a Chebyshev projection does a better job at approximating asset pricing and welfare quantities than a Taylor-expansion even for low order polynomials.

An accelerated stochastic vortex structure method for particle collision and agglomeration in homogeneous turbulence

Modeling the response of interacting particles, droplets, or bubbles to subgrid-scale fluctuations in turbulent flows is a long-standing challenge in multiphase flow simulations using the Reynolds-Averaged Navier-Stokes approach. The problem also arises for large-eddy simulation for sufficiently small values of the Kolmogorov-scale particle Stokes number. This paper expands on a recently proposed stochastic vortex structure (SVS) method for modeling of turbulence fluctuations for colliding or otherwise interacting particles. An accelerated version of the SVS method was developed using the fast multipole expansion and local Taylor expansion approach, which reduces computation speed by two orders of magnitude compared to the original SVS method. Detailed comparisons are presented showing close agreement of the energy spectrum and probability density functions of various fields between the SVS computational model, direct numerical simulation (DNS) results, and various theoretical and experimental results found in the literature. Results of the SVS method for particle collision rate and related measures of particle interaction exhibit excellent agreement with DNS predictions for homogeneous turbulent flows. The SVS method was also used with adhesive particles to simulate formation of particle agglomerates with different values of the particle Stokes and adhesion numbers, and various measures of the agglomerate structure are compared to the DNS results.

Asset Pricing in Production Economies When Capital Inputs are Heterogeneous

This paper studies asset pricing implications of heterogeneity across capital inputs. We build a general equilibrium model including two types of capital. The agents in our economy have Epstein and Zin (1989) preferences and technology is exposed to long-run risks in productivity growth. Following the literature, production inputs differ with respect to rates of depreciation and adjustment cost. We explain both, differences in returns and investment behaviour for two types of capital. Model implied asset pricing and relevant macroeconomic moments are in line with empirical data. To fully capture the effect of small but persistent shocks to the expected growth rate of productivity in a DSGE framework, the model is solved with a global non-linear solution algorithm. It is shown, when including long-run risks in productivity growth in conjunction with capital heterogeneity, using local Taylor-expansion based methods will not suffice, to obtain a satisfactory accuracy of approximation.

Experimental analysis and numerical simulation of pasta dough extrusion process

Pasta extrusion simulation still represent a powerful challenge from a computational point of view, for both the complexity of the rheological properties of semolina dough and the process itself, in which a polymerization phenomena, driven by a combination of pressure and temperature and strongly influenced by moisture content, takes place in the final part of the extruder barrel. In this work an integrated experimental-numerical approach is proposed for numerical simulation of pasta extrusion. An extensive set of rheological data in industrial range of moisture content (MC) and temperature obtained using a capillary rheometer is reported. To overcome the reduced accuracy of Arrhenius models for pasta dough viscosity published in literature, a numerical approach based on local Taylor expansion is proposed, matching exactly experimental data. The proposed model was then validated numerically comparing numerical results obtained in the framework of Computational Fluid Dynamics with experimental data obtained by using an experimental laboratory extruder (Sercom press, mod INRA). A moving mesh approach was adopted to model the screw dynamics inside the press barrel and a modified version of OpenFoam solver was developed. Rheological test and numerical simulations reciprocally validate each others: errors in rheological characterization would results in errors in numerical results, while wrong numerical modeling would not match experimental extrusion data. The proposed tool represent a validated basis for pasta production process optimization and reverse engineering of die design.

A Local Fractional Taylor Expansion and Its Computation for Insufficiently Smooth Functions

AbstractA general fractional Taylor formula and its computation for insufficiently smooth functions are discussed. The Aitken delta square method and epsilon algorithm are implemented to compute the critical orders of the local fractional derivatives, from which more critical orders are recovered by analysing the regular pattern of the fractional Taylor formula. The Richardson extrapolation method is used to calculate the local fractional derivatives with critical orders. Numerical examples are provided to verify the theoretical analysis and the effectiveness of our approach.

A theory of regularity structures

We introduce a new notion of “regularity structure” that provides an algebraic framework allowing to describe functions and/or distributions via a kind of “jet” or local Taylor expansion around each point. The main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose-built for the problem at hand. In particular, this allows to describe the local behaviour not only of functions but also of large classes of distributions. We then build a calculus allowing to perform the various operations (multiplication, composition with smooth functions, integration against singular kernels) necessary to formulate fixed point equations for a very large class of semilinear PDEs driven by some very singular (typically random) input. This allows, for the first time, to give a mathematically rigorous meaning to many interesting stochastic PDEs arising in physics. The theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems, possibly modified by the addition of diverging counterterms. These counterterms arise naturally through the action of a “renormalisation group” which is defined canonically in terms of the regularity structure associated to the given class of PDEs. Our theory also allows to easily recover many existing results on singular stochastic PDEs (KPZ equation, stochastic quantisation equations, Burgers-type equations) and to understand them as particular instances of a unified framework. One surprising insight is that in all of these instances local solutions are actually “smooth” in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions/distributions that play the role of “polynomials” in the theory. As an example of a novel application, we solve the long-standing problem of building a natural Markov process that is symmetric with respect to the (finite volume) measure describing the \(\Phi ^4_3\) Euclidean quantum field theory. It is natural to conjecture that the Markov process built in this way describes the Glauber dynamic of \(3\)-dimensional ferromagnets near their critical temperature.

Conditions for low-redshift positive apparent acceleration in smooth inhomogeneous models

It is known that a smooth LTB model cannot have a positive apparent central acceleration. Using a local Taylor expansion method we study the low-redshift conditions to obtain an apparent negative deceleration parameter $q^{app}(z)$ derived from the luminosity distance $D_L(z)$ for a central observer in a LTB space, confirming that central smoothness implies a positive central deceleration. Since observational data is only available at redshift greater than zero we find the critical values of the parameters defining a centrally smooth LTB model which give a positive apparent acceleration at $z>0$, providing a graphical representation of the conditions in the $q_0^{app},q_1^{app}$ plane, which are respectively the zero and first order terms of the central Taylor expansion of $q^{app}(z)$. We finally derive a coordinate independent expression for the apparent deceleration parameter based on the expansion of the relevant functions in red-shift rather than in the radial coordinate. We calculate $q^{app}(z)$ with two different methods to solve the null geodesic equations, one based on a local central expansion of the solution in terms of cosmic time and the other one using the exact analytical solution in terms of generalized conformal time. %The expansion of the solution in terms of cosmic time is quite useful also for other applications requiring foliation %of space-time in space-like hyper-surfaces, such as spatial averaging, which is much more difficult to study using the %analytical solution in terms of the generalized conformal time coordinate.

Meshless Finite Difference Method with Higher Order Approximation—Applications in Mechanics

This work is devoted to some recent developments in the Higher Order Approximation introduced to the Meshless Finite Difference Method (MFDM), and its application to the solution of boundary value problems in mechanics. In the MFDM, approximation of the sought function is described in terms of nodes rather than by means of any imposed structure like elements, regular meshes etc. Therefore, the MFDM, using arbitrarily irregular clouds of nodes using the Moving Weighted Least Squares (MWLS) approximation falls into the category of the Meshless Methods (MM). The MFDM, dating to early seventies, is one of the oldest and possibly the most developed one. In this paper considered are some techniques which lead to improvement of the MFDM solution�s quality. The main objective of this paper is the presentation and overview of new ideas and the development of the Higher Order solution approach in the MFDM provided by correction terms, preceded by a brief information about the current state-of-the art of this method. The main concept of the Higher Order Approximation (HOA) used here, is based on consideration of additional terms in the local Taylor expansion of the sought function. It shall be demonstrated that such a move may essentially improve, in many ways, efficiency and solution quality of the Higher Order MFDM. The Higher Order correction terms may be applied in many aspects of the MFDM solution approach. Among them one may distinguish the a-posteriori error estimation as well as adaptive solution process with multigrid strategy. Moreover, in the present work considered are: computational implementation of the Higher Order MFDM algorithms, examination of the above mentioned aspects using 1D and 2D benchmark tests, as well as an application of the Higher Order MFDM solution approach to selected boundary value problems in mechanics.

Corrections to the apparent value of the cosmological constant due to local inhomogeneities

Supernovae observations strongly support the presence of a cosmological constant, but its value, which we will call apparent, is normally determined assuming that the Universe can be accurately described by a homogeneous model. Even in the presence of a cosmological constant we cannot exclude nevertheless the presence of a small local inhomogeneity which could affect the apparent value of the cosmological constant. Neglecting the presence of the inhomogeneity can in fact introduce a systematic misinterpretation of cosmological data, leading to the distinction between an apparent and true value of the cosmological constant.We establish the theoretical framework to calculate the corrections to the apparent value of the cosmological constant by modeling the local inhomogeneity with a ΛLTB solution.Our assumption to be at the center of a spherically symmetric inhomogeneous matter distribution correspond to effectively calculate the monopole contribution of the large scale inhomogeneities surrounding us, which we expect to be the dominant one, because of other observations supporting a high level of isotropy of the Universe around us. By performing a local Taylor expansion we analyze the number of independent degrees of freedom which determine the local shape of the inhomogeneity, and consider the issue of central smoothness, showing how the same correction can correspond to different inhomogeneity profiles.Contrary to previous attempts to fit data using large void models our approach is quite general. The correction to the apparent value of the cosmological constant is in fact present for local inhomogeneities of any size, and should always be taken appropriately into account both theoretically and observationally.

Direct dynamics simulations using Hessian-based predictor-corrector integration algorithms

In previous research [J. Chem. Phys. 111, 3800 (1999)] a Hessian-based integration algorithm was derived for performing direct dynamics simulations. In the work presented here, improvements to this algorithm are described. The algorithm has a predictor step based on a local second-order Taylor expansion of the potential in Cartesian coordinates, within a trust radius, and a fifth-order correction to this predicted trajectory. The current algorithm determines the predicted trajectory in Cartesian coordinates, instead of the instantaneous normal mode coordinates used previously, to ensure angular momentum conservation. For the previous algorithm the corrected step was evaluated in rotated Cartesian coordinates. Since the local potential expanded in Cartesian coordinates is not invariant to rotation, the constants of motion are not necessarily conserved during the corrector step. An approximate correction to this shortcoming was made by projecting translation and rotation out of the rotated coordinates. For the current algorithm unrotated Cartesian coordinates are used for the corrected step to assure the constants of motion are conserved. An algorithm is proposed for updating the trust radius to enhance the accuracy and efficiency of the numerical integration. This modified Hessian-based integration algorithm, with its new components, has been implemented into the VENUS/NWChem software package and compared with the velocity-Verlet algorithm for the H(2)CO-->H(2)+CO, O(3)+C(3)H(6), and F(-)+CH(3)OOH chemical reactions.

Finite element modelling of the thermal outflow of three power plants in Huelva Estuary

A simple explicit, characteristic-based finite element method for the numerical simulation of the dispersion of the thermal outflow in Huelva estuary is considered in this paper. The derivation involves a local Taylor expansion of the convection–diffusion equation. The numerical model used was originally designed to simulate the dispersion of pollutants in the open sea and has now been reconfigured to simulate the dispersion of the thermal outflow of some power plants which may be constructed in Huelva estuary in the near future. Numerical results obtained are presented.

Nonparaxial model for the focusing of high-numerical-aperture photon sieves.

Recently, a paraxially individual far-field model was presented for the focusing and imaging analysis of pinhole photon sieves. By use of a local Taylor expansion of the integrated function of the Rayleigh-Sommerfeld diffraction formula, the small-size property of the individual pinholes, and the linear superposition principle, we extend this model to the nonparaxial case of high-numerical-aperture photon sieves. Some related problems, such as the validity range of this nonparaxial model and the selection conditions for the individual pinholes, are also discussed in detail.