Interior hybrid proximal extragradient methods for the linear monotone complementarity problem

We present new infeasible path-following methods for linear monotone complementarity problems based on Auslender, Teboulle and Ben-Tiba’s log-quadratic barrier functions. The central paths associated with these barriers are always well defined and, for those problems which have a solution, convergent to a pair of complementary solutions. Starting points in these paths are easy to compute. The theoretical iteration-complexity of these new path-following methods is derived and improved by a strategy which uses relaxed hybrid proximal-extragradient steps to control the quadratic term. Encouraging preliminary numerical experiments are presented.     

Approximation of the Quadratic Numerical Range of Block Operator Matrices

In this paper we establish an approximation of the quadratic numerical range of bounded and unbounded block operator matrices by variational methods. Applications to Hain?CLüst operators are given.     

Delay-dependent stability of linear multistep methods for DAEs with multiple delays

This paper aims to investigate the asymptotic stability of linear multistep (LM) methods for linear differential-algebraic equations (DAEs) with multiple delays. Based on the argument principle, we first establish the delay-dependent stability criteria of analytic solutions; then, we propose some practically checkable conditions for weak delay-dependent stability of numerical solutions derived by implicit LM methods. Lagrange interpolations are used to compute the delayed terms. Several numerical examples are given to illustrate the theoretical results.     

Banach Spaces of Universal Taylor Series in the Disc Algebra

We consider the problem of how to compute eigenvalues of a self-adjoint operator when a direct application of the Galerkin (finite-section) method is unreliable. The last two decades have seen the development of the so-called quadratic methods for addressing this problem. Recently a new perturbation approach has emerged, the idea being to perturb eigenvalues off the real line and, consequently, away from regions where the Galerkin method fails. We propose a simplified perturbation method which requires no á priori information and for which we provide a rigorous convergence analysis. The latter shows that, in general, our approach will significantly outperform the quadratic methods. We also present a new spectral enclosure for operators of the form A + iB where A is self-adjoint, B is self-adjoint and bounded. This enables us to control, very precisely, how eigenvalues are perturbed from the real line. The main results are demonstrated with examples including magnetohydrodynamics, Schrödinger and Dirac operators.     

Minimax Methods for Open-Loop Equilibra in $N$-Person Differential Games Part III: Duality and Penalty Finite Element Methods

The equilibrium strategy for $N$-person differential games can be obtained from a min-max problem subject to differential constraints. The differential constraints can be treated by the duality and penalty methods and then an unconstrained problem can be obtained. In this paper we develop methods applying the finite element methods to compute solutions of linear-quadratic $N$-person games using duality and penalty formulations. The calculations are efficient and accurate. When a (4,1)-system of Hermite cubic splines are used, our numerical results agree well with the theoretical predicted rate of convergence for the Lagrangian. Graphs and numerical data are included for illustration.     

On Regularization Methods for Inverse Problems of Dynamic Type

In this paper, we consider new regularization methods for linear inverse problems of dynamic type. These methods are based on dynamic programming techniques for linear quadratic optimal control problems. Two different approaches are followed: a continuous and a discrete one. We prove regularization properties and also obtain rates of convergence for the methods derived from both approaches. A numerical example concerning the dynamic EIT problem is used to illustrate the theoretical results.     

Non‐stationary iterated Tikhonov–Morozov method and third‐order differential equations for the evaluation of unbounded operators

In this paper we analyse the non‐stationary iterative Tikhonov–Morozov method analytically and numerically for the stable evaluation of differential operators and for denoizing images. A relationship between non‐stationary iterative Tikhonov–Morozov regularization and a filtering technique based on a differential equation of third order is established and both methods are shown to be effective for denoizing images and for the stable evaluation of differential operators. The theoretical results are verified numerically on model problems in ultrasound imaging and numerical differentiation. Copyright © 2000 John Wiley & Sons, Ltd.     

Fundamental solutions for hermite and subelliptic operators

In this article, we introduce a geometric method based on multipliers to compute heat kernels for operators with potentials. Using the heat kernel, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point on Euclidean space and on Heisenberg groups. As a consequence, we obtain the fundamental solutions for the sub-laplacian □ _J in a family of quadratic submanifolds. The research is partially supported by a William Fulbright Reserch Grant and a Competitive Research Grant at Georgetown University.     

On the Asymptotic and Numerical Analyses of Exponentially III-Conditioned Singularly Perturbed Boundary Value Problems

Asymptotic and numerical methods are used to study several classes of singularly perturbed boundary value problems for which the underlying homogeneous operators have exponentially small eigenvalues. Examples considered include the familiar boundary layer resonance problems and some extensions and certain linearized equations associated with metastable internal layer motion. For the boundary layer resonance problems, a systematic projection method, motivated by the work of De Groen [1], is used to analytically calculate high-order asymptotic solutions. This method justifies and extends some previous results obtained from the variational method of Grasman and Matkowsky [2]. A numerical approach, based on an integral equation formulation, is used to accurately compute boundary layer resonance solutions and their associated exponentially small eigenvalues. For various examples, the numerical results are shown to compare very favorably with two-term asymptotic results. Finally, some Sturm-Liouville operators with exponentially small spectral gap widths are studied. One such problem is applied to analyzing metastable internal layer motion for a certain forced Burgers equation.     

Some iterative methods for the largest positive definite solution to a class of nonlinear matrix equation

In this paper, we propose some inversion-free iteration methods for finding the largest positive definite solution of a class of nonlinear matrix equation. Then, we consider the properties of the solution for this nonlinear matrix equation. Also, we establish Newton’s iteration method for finding the largest positive definite solution and prove its quadratic convergence. Furthermore, we derive the semi-local convergence of the Newton’s iteration method. Finally, some numerical examples are presented to illustrate the effectiveness of the theoretical results and the behavior of the considered methods.     

Delay-Dependent Stability of Linear Multistep Methods for Neutral Systems with Distributed Delays

This paper considers the asymptotic stability of linear multistep (LM) methods for neutral systems with distributed delays. In particular, several sufficient conditions for delay-dependent stability of numerical solutions are obtained based on the argument principle. Compound quadrature formulae are used to compute the integrals. An algorithm is proposed to examine the delay-dependent stability of numerical solutions. Several numerical examples are performed to verify the theoretical results.     

Superconvergence analysis of fully discrete finite element methods for semilinear parabolic optimal control problems

We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization is based on difference methods, whereas the space discretization is done using finite element methods. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. First, we define a fully discrete finite element approximation scheme for the semilinear parabolic control problem. Second, we derive the superconvergence properties for the control, the state and the adjoint state. Finally, we do some numerical experiments for illustrating our theoretical results.     

求解非线性方程重根的平方收敛迭代法

研究了具有重根的非线性方程的迭代方法,对基于动力系统的新牛顿类方法作了修改,改进方法仍保持了牛顿方法的二阶收敛性.数值实验结果验证了方法的有效性.     

无穷维Hamilton算子的二次数值域

研究了一类无界无穷维Hamilton算子的二次数值域的性质,进而,应用二次数值域来刻画了无穷维Hamilton算子谱的分布范围,并给出了二次数值域的闭包包含谱集的结论.     

Numerical Accuracy of Ladder Schemes for Parallel Transport on Manifolds

Parallel transport is a fundamental tool to perform statistics on Riemannian manifolds. Since closed formulae do not exist in general, practitioners often have to resort to numerical schemes. Ladder methods are a popular class of algorithms that rely on iterative constructions of geodesic parallelograms. And yet, the literature lacks a clear analysis of their convergence performance. In this work, we give Taylor approximations of the elementary constructions of Schild’s ladder and the pole ladder with respect to the Riemann curvature of the underlying space. We then prove that these methods can be iterated to converge with quadratic speed, even when geodesics are approximated by numerical schemes. We also contribute a new link between Schild’s ladder and the Fanning scheme which explains why the latter naturally converges only linearly. The extra computational cost of ladder methods is thus easily compensated by a drastic reduction of the number of steps needed to achieve the requested accuracy. Illustrations on the 2-sphere, the space of symmetric positive definite matrices and the special Euclidean group show that the theoretical errors we have established are measured with a high accuracy in practice. The special Euclidean group with an anisotropic left-invariant metric is of particular interest as it is a tractable example of a non-symmetric space in general, which reduces to a Riemannian symmetric space in a particular case. As a secondary contribution, we compute the covariant derivative of the curvature in this space.

     

Application of LQR techniques to the adaptive control of quasilinear parabolic PDEs

LQR problems for linear parabolic PDEs have been studied in detail in the literature for the past 3 to 4 decades. The solvability of feedback control problems for a large class of problems is well understood. In recent years numerical methods for the approximation of the corresponding Riccati operators have been developed. These methods are able to calculate the feedback operator directly and thus can compute the solutions to linear problems efficiently. Here we study the applicability of such techniques to the control of quasilinear equations via local linearization in an adaptive control setting. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximation Properties For Modified q-Bernstein-Kantorovich Operators

The purpose of this article is to give a generalization of q-Bernstein-Kantorovich operators. We present some approximation theorems. We compute the rate of convergence and error estimation of these operators by means of the modulus of continuity. Furthermore, we give some numerical examples to show comparisons in illustrative graphics for the convergence of these operators to various functions.     

二次算子族及非负无穷维Hamilton算子的谱分布

本文讨论了一类在弦和梁的微小振动中出现的二次算子族L(λ)=λ~2MλK-A的谱分布问题,进而将所得结论与无穷维Hamilton算子联系起来,利用无穷维Hamilton算子的特殊结构,得到了一类非负无穷维Hamilton算子的谱分布,这为无穷维Hamilton算子的半群方法提供了理论保证.     

Spectral Inclusion Properties of Unbounded Hamiltonian Operators

In this paper, the authors investigate the spectral inclusion properties of the quadratic numerical range for unbounded Hamiltonian operators. Moreover, some examples are presented to illustrate the main results.