Finite difference spectral approximation for the time‐space fractional telegraph equation and its parameter estimation

The object of this paper is to present the numerical solution of the time‐space fractional telegraph equation. The proposed method is based on the finite difference scheme in temporal direction and Fourier spectral method in spatial direction. The fast Fourier transform (FFT) technique is applied to practical computation. The stability and convergence analysis are strictly proven, which shows that this method is stable and convergent with (2?α) order accuracy in time and spectral accuracy in space. Moreover, the Levenberg‐Marquardt (L‐M) iterative method is employed for the parameter estimation. Finally, some numerical examples are given to confirm the theoretical analysis.     

A special complementarity function revisited

ABSTRACT

Recently, a local framework of Newton-type methods for constrained systems of equations has been developed. Applied to the solution of Karush–Kuhn–Tucker (KKT) systems, the framework enables local quadratic convergence under conditions that allow nonisolated and degenerate KKT points. This result is based on a reformulation of the KKT conditions as a constrained piecewise smooth system of equations. It is an open question whether a comparable result can be achieved for other (not piecewise smooth) reformulations. It will be shown that this is possible if the KKT system is reformulated by means of the Fischer–Burmeister complementarity function under conditions that allow degenerate KKT points and nonisolated Lagrange multipliers. To this end, novel constrained Levenberg–Marquardt subproblems are introduced. They allow significantly longer steps for updating the multipliers. Based on this, a convergence rate of at least 1.5 is shown.     

Nonmonotone Levenberg–Marquardt Algorithms and Their Convergence Analysis

In this paper, two nonmonotone Levenberg–Marquardt algorithms for unconstrained nonlinear least-square problems with zero or small residual are presented. These algorithms allow the sequence of objective function values to be nonmonotone, which accelerates the iteration progress, especially in the case where the objective function is ill-conditioned. Some global convergence properties of the proposed algorithms are proved under mild conditions which exclude the requirement for the positive definiteness of the approximate Hessian T(x). Some stronger global convergence properties and the local superlinear convergence of the first algorithm are also proved. Finally, a set of numerical results is reported which shows that the proposed algorithms are promising and superior to the monotone Levenberg–Marquardt algorithm according to the numbers of gradient and function evaluations.     

Logistic回归系数极大似然估计的计算

本文介绍了Logstic回归系数的极大似然估计（maximum likelihood）的计算,并引入Levenberg—Marquardt算法,通过这代给出了Logistic回归系数的极大似然估计。     

THREE-DIMENSIONAL EXTRUDATE SWELL: FORMULATION WITH THE STREAM TUBE METHOD AND NUMERICAL RESULTS FOR A NEWTONIAN FLUID

Stream tube analysis, already applied to two-dimensional extrudate swell problems involving rate and integral constitutive equations for incompressible fluids, is now considered in the problem of free surface determination in a three-dimensional flow situation. The method allows computation of the unknown free surface by considering only a ‘peripheral stream tube’ limited by the wall and the jet surface and an inner stream surface. Those boundary surfaces are determined by considering the conservation equations together with boundary condition equations, solved by the Levenberg/Marquatdt optimization algorithm. The method leads to a considerable reduction in the number of degrees of freedom and the storage area. As in a previous study in the two-dimensional case, singularity problems in the vicinity of the junction points between the wall and the free surface are avoided. However, the numerical method still allows evaluati on of stress peaks due to the singularity at the exit, as may be observed for results obtained with a Newtonian fluid in a duct of square cross-section.     

A Levenberg–Marquardt algorithm for unconstrained multicriteria optimization

To compute one of the nonisolated Pareto-critical points of an unconstrained multicriteria optimization problem a Levenberg–Marquardt algorithm is applied. Sufficient conditions for an error bound are provided to prove its fast local convergence. A globalized version is shown to converge to a Pareto-optimal point under convexity assumptions.     

A smoothing Levenberg–Marquardt method for the extended linear complementarity problem

We consider the extended linear complementarity problem (XLCP), of which the linear and horizontal linear complementarity problems are two special cases. We reformulate the XLCP to a smooth equation by using some smoothing functions and propose a Levenberg–Marquardt method to solve the system of smooth equation. The global convergence and local superlinear convergence rate are established under certain conditions. Numerical tests show the effectiveness of the proposed algorithm.     

Tools and strategies for processing diffusion-ordered 2D NMR spectroscopy (DOSY) of a broad,featureless resonance: an application to methylaluminoxane (MAO)

DOSY has been extremely successful in many studies of molecular weight distributions, especially when the components are separable along the chemical shift axis. However, an unresolved NMR resonance yields the familiar problem of overlapping exponential decays. In a study of methylaluminoxane (MAO), a set of data processing and simulation tools were developed: read Bruker data files (Matlab); preliminary non-linear least-squares fit with f-test (Matlab); movie generation of the fits (Matlab); conversion of diffusion coefficients to molecular masses through molecular volumes (Gaussian-98); and simulation of DOSY data sets for various molecular mass distributions (Mathematica). These tools are presented here and briefly compared with other DOSY analysis methods.Electronic Supplementary Material Supplementary material is available in the online version of this article at . The following Matlab and Mathematica files are made available: Plot_Raw_DOSY.m, Fit_Two_Component.m (which calls One_Gaussian_LEASTSQ.m, Two_Gaussian_LEASTSQ.m, and pFTest.m), and DOSY_theory.nb. Note: Matlab v5.2 optimization toolbox, as supplied, lacked a confidence interval subroutine; upon our request, confint.m was provided to the authors by Mathworks, Inc. Newer versions of Matlab have a similar program already included in the optimization