Convergence Estimates for Some Regularization Methods to Solve a Cauchy Problem of the Laplace Equation

In this paper,we give a general proof on convergence estimates for some regularization methods to solve a Cauchy problem for the Laplace equation in a rectangular domain.The regularization methods we considered are:a non-local boundary value problem method,a boundary Tikhonov regularization method and a generalized method.Based on the conditional stability estimates,the convergence estimates for various regularization methods are easily obtained under the simple verifications of some conditions.Numerical results for one example show that the proposed numerical methods are effective and stable.

An Indirect Finite Element Method for Variable-Coefficient Space-Fractional Diffusion Equations and Its Optimal-Order Error Estimates

We study an indirect finite element approximation for two-sided space-fractional diffusion equations in one space dimension.By the representation formula of the solutions u(x)to the proposed variable coefficient models in terms of v(x),the solutions to the constant coefficient analogues,we apply finite element methods for the constant coefficient fractional diffusion equations to solve for the approximations vh(x)to v(x)and then obtain the approximations uh(x)of u(x)by plugging vh(x)into the representation of u(x).Optimal-order convergence estimates of u(x)−uh(x)are proved in both L2 and Hα∕2 norms.Several numerical experiments are presented to demonstrate the sharpness of the derived error estimates.

Some theoretical problems on variational data assimilation

Theoretical aspects of variational data assimilation （VDA） for a simple model with both global and local observational data are discussed. For the VDA problems with global observational data, the initial conditions and parameters for the model are revisited and the model itself is modified. The estimates of both error and convergence rate are theoretically made and the vahdity of the method is proved. For VDA problem with local observation data, the conventional VDA method are out of use due to the ill-posedness of the problem. In order to overcome the difficulties caused by the ill-posedness, the initial conditions and parameters of the model are modified by using the improved VDA method, and the estimates of both error and convergence rate are also made. Finally, the validity of the improved VDA method is proved through theoretical analysis and illustrated with an example, and a theoretical criterion of the regularization parameters is proposed.

Relative efficiency appraisal of discrete choice modeling algorithms using small-scale maximum likelihood estimator through empirically tailored computing environment

Discrete choice models are widely used in multiple sectors such as transportation, health, energy, and marketing, etc., where the model estimation is usually carried out by using commercial software. Nonetheless, tailored computer codes offer modellers greater flexibility and control of unique modelling situation. Aligned with empirically tailored computing environment, this research discusses the relative performance of six different algorithms of a discrete choice model using three key performance measures： convergence time, number of iterations, and iteration time. The computer codes are developed by using Visual Basic Application （VBA）. Maximum likelihood function （MLF） is formulated and the mathematical relationships of gradient and Hessian matrix are analytically derived to carry out the estimation process. The estimated parameter values clearly suggest that convergence criterion and initial guessing of parameters are the two critical factors in determining the overall estimation performance of a custom-built discrete choice model.

THE OPTIMAL CONVERGENCE ORDER OF THE DISCONTINUOUS FINITE ELEMENT METHODS FOR FIRST ORDER HYPERBOLIC SYSTEMS

In this paper, a discontinuous finite element method for the positive and symmetric, first-order hyperbolic systems （steady and nonsteady state） is constructed and analyzed by using linear triangle elements, and the O（h^2）-order optimal error estimates are derived under the assumption of strongly regular triangulation and the Ha-regularity for the exact solutions. The convergence analysis is based on some superclose estimates of the interpolation approximation. Finally, we discuss the Maxwell equations in a two-dimensional domain, and numerical experiments are given to validate the theoretical results.

CONVERGENCE AND OPTIMALITY OF ADAPTIVE MIXED METHODS FOR POISSON’S EQUATION IN THE FEEC FRAMEWORK

Finite Element Exterior Calculus (FEEC) was developed by Arnold, Falk, Winther andothers over the last decade to exploit the observation that mixed variational problems canbe posed on a Hilbert complex, and Galerkin-type mixed methods can then be obtained bysolving finite-dimensional subcomplex problems. Chen, Holst, and Xu (Math. Comp. 78(2009) 35–53) established convergence and optimality of an adaptive mixed finite elementmethod using Raviart–Thomas or Brezzi–Douglas–Marini elements for Poisson’s equationon contractible domains in R^2, which can be viewed as a boundary problem on the deRham complex. Recently Demlow and Hirani (Found. Math. Comput. 14 (2014) 1337–1371) developed fundamental tools for a posteriori analysis on the de Rham complex.In this paper, we use tools in FEEC to construct convergence and complexity resultson domains with general topology and spatial dimension. In particular, we construct areliable and efficient error estimator and a sharper quasi-orthogonality result using a noveltechnique. Without marking for data oscillation, our adaptive method is a contractionwith respect to a total error incorporating the error estimator and data oscillation.

EXPONENTIAL TIME DIFFERENCING-PADE FINITE ELEMENT METHOD FOR NONLINEAR CONVECTION-DIFFUSION-REACTION EQUATIONS WITH TIME CONSTANT DELAY

In this paper,ETD3-Padéand ETD4-PadéGalerkin finite element methods are proposed and analyzed for nonlinear delayed convection-diffusion-reaction equations with Dirichlet boundary conditions.An ETD-based RK is used for time integration of the corresponding equation.To overcome a well-known difficulty of numerical instability associated with the computation of the exponential operator,the Padéapproach is used for such an exponential operator approximation,which in turn leads to the corresponding ETD-Padéschemes.An unconditional L^(2) numerical stability is proved for the proposed numerical schemes,under a global Lipshitz continuity assumption.In addition,optimal rate error estimates are provided,which gives the convergence order of O(k^(3)+h^(r))(ETD3-Padé)or O(k^(4)+h^(r))(ETD4-Padé)in the L^(2)norm,respectively.Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes.

Regularization of the Inverse Problem for Time Fractional Pseudo-parabolic Equation with Non-local in Time Conditions

This paper is devoted to identifying an unknown source for a time-fractional diffusion equation in a general bounded domain.First,we prove the problem is non-well posed and the stability of the source function.Second,by using the Modified Fractional Landweber method,we present regularization solutions and show the convergence rate between regularization solutions and sought solution are given under a priori and a posteriori choice rules of the regularization parameter,respectively.Finally,we present an illustrative numerical example to test the results of our theory.

A Second Order Nonconforming Rectangular Finite Element Method for Approximating Maxwell's Equations

Abstract The main objective of this paper is to present a new rectangular nonconforming finite element scheme with the second order convergence behavior for approximation of Maxwell＇s equations. Then the corresponding optimal error estimates are derived. The difficulty in construction of this finite element scheme is how to choose a compatible pair of degrees of freedom and shape function space so as to make the consistency error due to the nonconformity of the element being of order O（h^3）, properly one order higher than that of its interpolation error O（h^2） in the broken energy norm, where h is the subdivision parameter tending to zero.

A Modified Crank-Nicolson Numerical Scheme for the Flory-Huggins Cahn-Hilliard Model

In this paper we propose and analyze a second order accurate numericalscheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential. A modified Crank-Nicolson approximation is applied to the logarithmic nonlinear term, while the expansive term is updated by an explicit second order AdamsBashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. A nonlinear artificial regularization term is added in the numerical scheme,which ensures the positivity-preserving property, i.e., the numerical value of the phasevariable is always between -1 and 1 at a point-wise level. Furthermore, an unconditional energy stability of the numerical scheme is derived, leveraging the special formof the logarithmic approximation term. In addition, an optimal rate convergence estimate is provided for the proposed numerical scheme, with the help of linearizedstability analysis. A few numerical results, including both the constant-mobility andsolution-dependent mobility flows, are presented to validate the robustness of the proposed numerical scheme.

SOME RESULTS ON ESTIMATION OF THE TAIL INDEX OF A DISTRIBUTION

The author obtains the rate of strong convergence,mean squared error and optimal choice of the“smoothing parameter”(the sample fraction)of a tail index estimator which was proposed by the author from Pickands’estimator,and called modified Pickands’estimator.The similar results about Hill’s estimator are also obtained,which generalize the corresponding results in.Besides,some comparisons between Hill’s estimator and the modified Pickands’estimator are given.