Determining Atmospheric Boundary Layer Height with the Numerical Differentiation Method Using Bending Angle Data from COSMIC

This paper presents a new method to estimate the height of the atmospheric boundary layer(ABL) by using COSMIC radio occultation bending angle(BA) data. Using the numerical differentiation method combined with the regularization technique, the first derivative of BA profiles is retrieved, and the height at which the first derivative of BA has the global minimum is defined to be the ABL height. To reflect the reliability of estimated ABL heights, the sharpness parameter is introduced, according to the relative minimum of the BA derivative. Then, it is applied to four months of COSMIC BA data(January, April, July, and October in 2008), and the ABL heights estimated are compared with two kinds of ABL heights from COSMIC products and with the heights determined by the finite difference method upon the refractivity data. For sharp ABL tops(large sharpness parameters), there is little difference between the ABL heights determined by different methods, i.e.,the uncertainties are small; whereas, for non-sharp ABL tops(small sharpness parameters), big differences exist in the ABL heights obtained by different methods, which means large uncertainties for different methods. In addition, the new method can detect thin ABLs and provide a reference ABL height in the cases eliminated by other methods. Thus, the application of the numerical differentiation method combined with the regularization technique to COSMIC BA data is an appropriate choice and has further application value.

Hermite interpolation and its numerical differentiation formula involving symmetric functions

By utilizing symmetric functions,this paper presents explicit representations for Hermite interpolation and its numerical differentiation formula.And the corresponding error estimates are also provided.

Two Notes on Numerical DifFerentiation Formulae

Some new conclusions on asymptotic properties and inverse problems of numerical differentiation formulae have been drawn in this paper.In the first place,several asymptotic properties of intermediate points of numerical differentiation formulae are presented by using Taylor's formula.And then,based on the ideas of algebraic accuracy,several inverse problems of numerical differentiation formulae are given.

Numerical differentiation of noisy data with local optimum by data segmentation

A new numerical differentiation method with local opti- mum by data segmentation is proposed. The segmentation of data is based on the second derivatives computed by a Fourier devel- opment method. A filtering process is used to achieve acceptable segmentation. Numerical results are presented by using the data segmentation method, compared with the regularization method. For further investigation, the proposed algorithm is applied to the resistance capacitance （RC） networks identification problem, and improvements of the result are obtained by using this algorithm.

Explicit Representations for Local Lagrangian Numerical Differentiation

It is shown that in Lagrangian numerical differentiation formulas, the coefficients are explicitly expressed by means of cycle indicator polynomials of symmetric group. Moreover, asymptotic expansions of the remainders are also explicitly represented as a fixed number of interpolation nodes approaching infinitely to the point at which the derivative is evaluated. This implies that complete explicit formulas for local Lagrangian numerical differentiation can be obtained.

ADAPTIVE CHOICE OF THE REGULARIZATION PARAMETER IN NUMERICAL DIFFERENTIATION

We investigate a novel adaptive choice rule of the Tikhonov regularization parameter in numerical differentiation which is a classic ill-posed problem. By assuming a general unknown Holder type error estimate derived for numerical differentiation, we choose a regularization parameter in a geometric set providing a nearly optimal convergence rate with very limited a-priori information. Numerical simulation in image edge detection verifies reliability and efficiency of the new adaptive approach.

NUMERICAL SOLUTION OF QUASILINEAR SINGULARLY PERTURBED ORDINARY DIFFERENTIAL EQUATION WITHOUT TURNING POINTS

In this paper we consider a quasilinear second order ordinary diferential equation with a small parameter Firstly an approximate problem is constructed. Then an iterative procedure is developed. Finally we give an algorithm whose accuracy is good for arbitrary e>0 .

A Radial Basis Function Method with Improved Accuracy for Fourth Order Boundary Value Problems

Accurately approximating higher order derivatives is an inherently difficult problem. It is shown that a random variable shape parameter strategy can improve the accuracy of approximating higher order derivatives with Radial Basis Function methods. The method is used to solve fourth order boundary value problems. The use and location of ghost points are examined in order to enforce the extra boundary conditions that are necessary to make a fourth-order problem well posed. The use of ghost points versus solving an overdetermined linear system via least squares is studied. For a general fourth-order boundary value problem, the recommended approach is to either use one of two novel sets of ghost centers introduced here or else to use a least squares approach. When using either ghost centers or least squares, the random variable shape parameter strategy results in significantly better accuracy than when a constant shape parameter is used.

Nonlinear feedback synchronisation control between fractional-order and integer-order chaotic systems

This paper focuses on the synchronisation between fractional-order and integer-order chaotic systems. Based on Lyapunov stability theory and numerical differentiation, a nonlinear feedback controller is obtained to achieve the synchronisation between fractional-order and integer-order chaotic systems. Numerical simulation results are presented to illustrate the effectiveness of this method.

Adaptive fault-tolerant control of heavy lift launch vehicle via differential algebraic observer

A novel adaptive fault-tolerant control scheme in the differential algebraic framework was proposed for attitude control of a heavy lift launch vehicle （HLLV）. By using purely mathematical transformations, the decoupled input-output representations of HLLV were derived, rendering three decoupled second-order systems, i.e., pitch, yaw and roll channels. Based on a new type of numerical differentiator, a differential algebraic observer （DAO） was proposed for estimating the system states and the generalized disturbances, including various disturbances and additive fault torques. Driven by DAOs, three improved proportional-integral- differential （PID） controllers with disturbance compensation were designed for pitch, yaw and roll control. All signals in the closed-loop system were guaranteed to be ultimately uniformly bounded by utilization of Lyapunov＇s indirect method. The convincing numerical simulations indicate that the proposed control scheme is successful in achieving high performance in the presence of parametric perturbations, external disturbances, noisy corruptions, and actuator faults.

Generator,multiquadric generator,quasi-interpolation and multiquadric quasi-interpolation

The aim of this survey paper is to propose a new concept ＂generator＂. In fact, generator is a single function that can generate the basis as well as the whole function space. It is a more fundamental concept than basis. Various properties of generator are also discussed. Moreover, a special generator named multiquadric function is introduced. Based on the multiquadric generator, the multiquadric quasi-interpolation scheme is constructed, and furthermore, the properties of this kind of quasi-interpolation are discussed to show its better capacity and stability in approximating the high order derivatives.

Reconstruction of high order derivatives by new mollification methods

In this paper, the problem of reconstructing numerical derivatives from noisy data is considered. A new framework of mollification methods based on the L generalized solution regularization methods is proposed. A specific algorithm for the first three derivatives is presented in the paper, in which a modification of TSVD, termed cTSVD is chosen as the regularization technique. Numerical examples given in the paper verify the theoretical results and show efficiency of the new method.

An algorithm for frequency estimation of signals composed of multiple single-tones

The high-accuracy, wide-range frequency estimation algorithm for multi-component signals presented in this paper, is based on a numerical differentiation and central Lagrange interpolation. With the sample sequences, which need at most 7 points and are sampled at a sample frequency of 25600 Hz, and computation sequences, using employed a formulation proposed in this paper, the frequencies of each component of the signal are all estimated at an accuracy of 0.001% over 1 Hz to 800 kHz with the amplitudes of each component of the signal varying from 1 V to 200 V and the phase angle of each component of the signal varying from 0° to 360°. The proposed algorithm needs at most a half cycle for the frequencies of each component of the signal under noisy or non-noisy conditions. A testing example is given to illustrate the proposed algorithm in Matlab environment.

A new algorithm for reconstructing the three-dimensional flow field of the oceanic mesoscale eddy

A new algorithm for reconstructing the three-dimensional flow field of the oceanic mesoscale eddies is proposed in this paper,based on variational method.Firstly,with the numerical differentiation Tikhonov regularizer,we reconstruct the continuous horizontal flow field on discrete grid points at each layer in the oceanic region,in terms of the horizontal flow field observations.Secondly,benefitting from the variational optimization analysis and its improvement,we reconstruct a three-dimensional flow field under the constraint of the horizontal flow and the vertical flow.The results of simulation experiments validate that the relative error of the new algorithm is lower than that of the finite difference method in the case of high grid resolution,which still holds in the case of unknown observational errors or in the absence of vertical velocity boundary conditions.Finally,using the reanalysis horizontal data sourcing from SODA and the proposed algorithm,we reconstruct three-dimensional flow field structure for the real oceanic mesoscale eddy.

Monte-Carlo simulation of a stochastic differential equation

For solving higher dimensional diffusion equations with an inhomogeneous diffusion coefficient,Monte Carlo（MC） techniques are considered to be more effective than other algorithms, such as finite element method or finite difference method. The inhomogeneity of diffusion coefficient strongly limits the use of different numerical techniques. For better convergence, methods with higher orders have been kept forward to allow MC codes with large step size. The main focus of this work is to look for operators that can produce converging results for large step sizes. As a first step, our comparative analysis has been applied to a general stochastic problem.Subsequently, our formulization is applied to the problem of pitch angle scattering resulting from Coulomb collisions of charge particles in the toroidal devices.

Within-Cluster Variability Exponent for Identifying Coherent Structures in Dynamical Systems

We propose a clustering-based approach for identifying coherent flow structuresin continuous dynamical systems. We first treat a particle trajectory over a finitetime interval as a high-dimensional data point and then cluster these data from differentinitial locations into groups. The method then uses the normalized standarddeviation or mean absolute deviation to quantify the deformation. Unlike the usualfinite-time Lyapunov exponent (FTLE), the proposed algorithm considers the completetraveling history of the particles. We also suggest two extensions of the method. To improvethe computational efficiency, we develop an adaptive approach that constructsdifferent subsamples of the whole particle trajectory based on a finite time interval. Tostart the computation in parallel to the flow trajectory data collection, we also developan on-the-fly approach to improve the solution as we continue to provide more measurementsfor the algorithm. The method can efficiently compute the WCVE over adifferent time interval by modifying the available data points.

A robust model-free controller for a three-phase grid-connected photovoltaic system based on ultra-local model

In this paper, a robust model-free controller for a grid-connected photovoltaic (PV) system is designed. The system consists of a PV generator connected to a three-phase grid by a DC/AC converter. The control objectives of the overall system are to extract maximum power from the PV source, to control reactive power exchange and to improve the quality of the current injected into the grid. The model-free control technique is based on the use of an ultra-local model instead of the dynamic model of the overall system. The local model is continuously updated based on a numerical differentiator using only the input–output behavior of the controlled system. The model-free controller consists of a classical feedback controller and a compensator for the effects of internal parameter changes and external disturbances. Simulation results illustrate the efficiency of the controller for grid-connected PV systems.