Gradient WEB-spline finite element method for solving two-dimensional quasilinear elliptic problems

In this paper, a two-dimensional quasilinear elliptic problem of the form -divF(x,▽u)=g(x)$-\mathit{divF}(\mathbf{x}\text{,}▽u)=g(\mathbf{x})$ is considered. This problem is ill-conditioned and we therefore propose a modified iterative algorithm based on coupling of the Sobolev space gradient method and WEB-spline finite element method. Applying the preconditioned iterative method, which has been already provided by Farago and Karatson (2001) [1] reduces the our considered problem to a sequence of linear Poisson’s problems. Then the WEB-spline finite element method is applied to the approximate solution of these Poisson’s problems. In this sense, a convergence theorem is proved and the advantages of this technique than the gradient finite element method (GFEM) is also described. Finally, the presented method is tested on some examples and compared with GFEM. It is shown that the gradient WEB-spline finite element method gives better test results.     

Nonlinear conjugate gradient methods with structured secant condition for nonlinear least squares problems

In this paper, we deal with conjugate gradient methods for solving nonlinear least squares problems. Several Newton-like methods have been studied for solving nonlinear least squares problems, which include the Gauss-Newton method, the Levenberg-Marquardt method and the structured quasi-Newton methods. On the other hand, conjugate gradient methods are appealing for general large-scale nonlinear optimization problems. By combining the structured secant condition and the idea of Dai and Liao (2001) [20], the present paper proposes conjugate gradient methods that make use of the structure of the Hessian of the objective function of nonlinear least squares problems. The proposed methods are shown to be globally convergent under some assumptions. Finally, some numerical results are given.     

Sobolev方程的CN全离散化有限元格式

首先给出Sobolev方程关于时间二阶精度的Crank-Nicolson(CN)时间半离散格式,然后直接从时间二阶精度的CN时间半离散格式出发,构造CN全离散化的有限元格式,并给出这种时间二阶精度的CN全离散化有限元解的误差估计.本文研究方法使得理论证明变得更简便, 也是处理Sobolev方程的一种新的尝试.     

An improved parallel hybrid bi-conjugate gradient method suitable for distributed parallel computing

An improved parallel hybrid bi-conjugate gradient method (IBiCGSTAB(2) method, in brief) for solving large sparse linear systems with nonsymmetric coefficient matrices is proposed for distributed parallel environments. The method reduces five global synchronization points to two by reconstructing the BiCGSTAB(2) method in [G.L.G. Sleijpen, H.A. van der Vorst, Hybrid bi-conjugate gradient methods for CFD problems, in: M. Hafez, K. Oshima (Eds.), Computational Fluid Dynamics Review 1995, John Wiley & Sons Ltd, Chichester, 1995, pp. 457–476] and the communication time required for the inner product can be efficiently overlapped with useful computation. The cost is only slightly increased computation time, which can be ignored, compared with the reduction of communication time. Performance and isoefficiency analysis shows that the IBiCGSTAB(2) method has better parallelism and scalability than the BiCGSTAB(2) method. Numerical experiments show that the scalability can be improved by a factor greater than 2.5 and the improvement in parallel communication performance approaches 60%.     

A Levenberg–Marquardt method based on Sobolev gradients

We extend the theory of Sobolev gradients to include variable metric methods, such as Newton’s method and the Levenberg–Marquardt method, as gradient descent iterations associated with stepwise variable inner products. In particular, we obtain existence, uniqueness, and asymptotic convergence results for a gradient flow based on a variable inner product.     

Continuous interior penalty finite element methods for Sobolev equations with convection‐dominated term

We consider implicit and semi‐implicit time‐stepping methods for continuous interior penalty (CIP) finite element approximations of Sobolev equations with convection‐dominated term. Stability is obtained by adding an interior penalty term giving L² ‐control of the jump of the gradient over element faces. Several $\cal {A}$ ‐stable time‐stepping methods are analyzed and shown to be unconditionally stable and optimally convergent. We show that the contribution from the gradient jumps leading to an extended matrix pattern may be extrapolated from previous time steps, and hence handled explicitly without loss of stability and accuracy. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

Combined Hermite spectral-finite difference method for the Fokker-Planck equation

The convergence of a class of combined spectral-finite difference methods using Hermite basis, applied to the Fokker-Planck equation, is studied. It is shown that the Hermite based spectral methods are convergent with spectral accuracy in weighted Sobolev space. Numerical results indicating the spectral convergence rate are presented. A velocity scaling factor is used in the Hermite basis and is shown to improve the accuracy and effectiveness of the Hermite spectral approximation, with no increase in workload. Some basic analysis for the selection of the scaling factors is also presented.

     

A space–time discontinuous Galerkin method for linear convection-dominated Sobolev equations

This article presents a space–time discontinuous Galerkin (DG) finite element method for linear convection-dominated Sobolev equations. The finite element method has basis functions that are continuous in space and discontinuous in time, and variable spatial meshes and time steps are allowed. In the discrete intervals of time, using properties of the Radau quadrature rule, eliminates the restriction to space–time meshes of convectional space–time Galerkin methods. The existence and uniqueness of the approximate solution are proved. An optimal priori error estimate in L^∞(H¹) is derived. Numerical experiments are presented to confirm theoretical results.     

Application of the least squares method to solving linear differential-algebraic equations

We consider an application of the least squares method to numerical solving a linear system of ordinary differential equations (ODEs) with an identically singular or rectangular matrix multiplying the highest derivative of the desired vector-function. The behavior of gradient methods for minimizing the squared residual in Sobolev spaces and some other issues are discussed. Results of some numerical experiments are given.     

On the Gradient Method for Some Nonlinear Elliptic Boundary Value Problems

The infinite-dimensional gradient method is applied to the iterative solution of quasilinear elliptic boundary value problems. Earlier results on uniformly monotone problems are extended to a general case within the scope of Hilbert space well-posedness. Linear convergence is proved in Sobolev norm. This revised version was published online in June 2006 with corrections to the Cover Date.     

求解界约束优化的一种新的非单调谱投影梯度法

本文给出求解界约束优化问题的一种新的非单调谱投影梯度算法. 该算法是将谱投影梯度算法与Zhang and Hager [SIAM Journal on Optimization,2004,4（4）：1043-1056]提出的非单调线搜索结合得到的方法. 在合理的假设条件下,证明了算法的全局收敛性.数值实验结果表明,与已有的界约束优化问题的谱投影梯度法比较,利用本文给出的算法求解界约束优化问题是有竞争力的.     

Subcritical Hamilton–Jacobi fractional equation in

Solvability of Cauchy's problem in for fractional Hamilton–Jacobi equation (1.1) with subcritical nonlinearity is studied here both in the classical Sobolev spaces and in the locally uniform spaces. The first part of the paper is devoted to the global in time solvability of subcritical equation (1.1) in locally uniform phase space, a generalization of the standard Sobolev spaces. Subcritical growth of the nonlinear term with respect to the gradient is considered. We prove next the global in time solvability in classical Sobolev spaces, in Hilbert case. Regularization effect is used there to guarantee global in time extendibility of the local solution. Copyright © 2014 John Wiley & Sons, Ltd.     

A Newton linearized compact finite difference scheme for one class of Sobolev equations

In this article, a Newton linearized compact finite difference scheme is proposed to numerically solve a class of Sobolev equations. The unique solvability, convergence, and stability of the proposed scheme are proved. It is shown that the proposed method is of order 2 in temporal direction and order 4 in spatial direction. Moreover, compare to the classical extrapolated Crank‐Nicolson method or the second‐order multistep implicit–explicit methods, the proposed scheme is easier to be implemented as it only requires one starting value. Finally, numerical experiments on one and two‐dimensional problems are presented to illustrate our theoretical results.     

Global Convergence Properties of Nonlinear Conjugate Gradient Methods with Modified Secant Condition

Conjugate gradient methods are appealing for large scale nonlinear optimization problems. Recently, expecting the fast convergence of the methods, Dai and Liao (2001) used secant condition of quasi-Newton methods. In this paper, we make use of modified secant condition given by Zhang et al. (1999) and Zhang and Xu (2001) and propose a new conjugate gradient method following to Dai and Liao (2001). It is new features that this method takes both available gradient and function value information and achieves a high-order accuracy in approximating the second-order curvature of the objective function. The method is shown to be globally convergent under some assumptions. Numerical results are reported.     

Semi-Conjugate Direction Methods for Real Positive Definite Systems

In this preliminary work, left and right conjugate direction vectors are defined for nonsymmetric, nonsingular matrices A and some properties of these vectors are studied. A left conjugate direction (LCD) method for solving nonsymmetric systems of linear equations is proposed. The method has no breakdown for real positive definite systems. The method reduces to the usual conjugate gradient method when A is symmetric positive definite. A finite termination property of the semi-conjugate direction method is shown, providing a new simple proof of the finite termination property of conjugate gradient methods. The new method is well defined for all nonsingular M-matrices. Some techniques for overcoming breakdown are suggested for general nonsymmetric A. The connection between the semi-conjugate direction method and LU decomposition is established. The semi-conjugate direction method is successfully applied to solve some sample linear systems arising from linear partial differential equations, with attractive convergence rates. Some numerical experiments show the benefits of this method in comparison to well-known methods. This revised version was published online in July 2006 with corrections to the Cover Date.     

Determination of space‐dependent coefficients from temperature measurements using the conjugate gradient method

In this article, we consider coefficient identification problems in heat transfer concerned with the determination of the space‐dependent perfusion coefficient and/or thermal conductivity from interior temperature measurements using the conjugate gradient method (CGM). We establish the direct, sensitivity and adjoint problems and the iterative CGM algorithm which has to be stopped according to the discrepancy principle in order to reconstruct a stable solution for the inverse problem. The Sobolev gradient concept is introduced in the CGM iterative algorithm in order to improve the reconstructions. The numerical results illustrated for both exact and noisy data, in one‐ and two‐dimensions for single or double coefficient identifications show that the CGM is an efficient and stable method of inversion.     

攀登伪蒙特卡罗积分法在Sobolev和Korobov空间中的随机化误差

攀登伪蒙特卡罗积分法是由伪蒙特卡罗与蒙特卡罗方法混合而成的一种新方法，它体现了两者的优点．本文研究这种积分法在Sobolev空间和Korobov空间中的随机化误差．我们证明攀登(λ，t，m，s)-网积分法在这两个空间中的随机化误差的渐近阶为n^3/2[logn]^(s-1)/2。     

Accelerated conjugate gradient algorithm with finite difference Hessian/vector product approximation for unconstrained optimization

In this paper we propose a fundamentally different conjugate gradient method, in which the well-known parameter β_k is computed by an approximation of the Hessian/vector product through finite differences. For search direction computation, the method uses a forward difference approximation to the Hessian/vector product in combination with a careful choice of the finite difference interval. For the step length computation we suggest an acceleration scheme able to improve the efficiency of the algorithm. Under common assumptions, the method is proved to be globally convergent. It is shown that for uniformly convex functions the convergence of the accelerated algorithm is still linear, but the reduction in function values is significantly improved. Numerical comparisons with conjugate gradient algorithms including CONMIN by Shanno and Phua [D.F. Shanno, K.H. Phua, Algorithm 500, minimization of unconstrained multivariate functions, ACM Trans. Math. Softw. 2 (1976) 87–94], SCALCG by Andrei [N. Andrei, Scaled conjugate gradient algorithms for unconstrained optimization, Comput. Optim. Appl. 38 (2007) 401–416; N. Andrei, Scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, Optim. Methods Softw. 22 (2007) 561–571; N. Andrei, A scaled BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, Appl. Math. Lett. 20 (2007) 645–650], and new conjugacy condition and related new conjugate gradient by Li, Tang and Wei [G. Li, C. Tang, Z. Wei, New conjugacy condition and related new conjugate gradient methods for unconstrained optimization, J. Comput. Appl. Math. 202 (2007) 523–539] or truncated Newton TN by Nash [S.G. Nash, Preconditioning of truncated-Newton methods, SIAM J. on Scientific and Statistical Computing 6 (1985) 599–616] using a set of 750 unconstrained optimization test problems show that the suggested algorithm outperforms these conjugate gradient algorithms as well as TN.     

雷达波形设计的一个数值方法

考虑了雷达成像中的一个波形设计问题.基于匹配滤波,在Sobolev空间的框架下,使用Hermite多项式理论,并通过一种数值算法,实现了波形的优化设计.数值实验表明,该方法是有效的.相比于CW(continuous wave)脉冲和线性调频脉冲方法,该方法可以得到更高的分辨率.     

一类特殊空间上映射的梯度几何流柯西问题解的存在唯一性

通过几何分析方法与抛物型方程组解的逼近理论,研究特殊空间(一维球面S~1到二维球面S~2)上映射的梯度几何流柯西问题解的存在唯一性.利用能量法和空间本身特有的性质来解决能量守恒的问题,并利用适当的抛物型方程组逼近该梯度几何流,在适当的Sobolev空间中建立先验估计,找到其时间的一致正下界和抛物型方程组一列解的Sobo1ev范数的一致边界,借助于抛物型偏微分方程的理论,以此决定该柯西问题解的存在唯一性.