On the convergence of Ritz pairs and refined Ritz vectors for quadratic eigenvalue problems

For a given subspace, the Rayleigh-Ritz method projects the large quadratic eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar to the Rayleigh-Ritz method for the linear eigenvalue problem, the Rayleigh-Ritz method defines the Ritz values and the Ritz vectors of the QEP with respect to the projection subspace. We analyze the convergence of the method when the angle between the subspace and the desired eigenvector converges to zero. We prove that there is a Ritz value that converges to the desired eigenvalue unconditionally but the Ritz vector converges conditionally and may fail to converge. To remedy the drawback of possible non-convergence of the Ritz vector, we propose a refined Ritz vector that is mathematically different from the Ritz vector and is proved to converge unconditionally. We construct examples to illustrate our theory.     

The regularizing effect of the Golub-Kahan iterative bidiagonalization and revealing the noise level in the data

Regularization techniques based on the Golub-Kahan iterative bidiagonalization belong among popular approaches for solving large ill-posed problems. First, the original problem is projected onto a lower dimensional subspace using the bidiagonalization algorithm, which by itself represents a form of regularization by projection. The projected problem, however, inherits a part of the ill-posedness of the original problem, and therefore some form of inner regularization must be applied. Stopping criteria for the whole process are then based on the regularization of the projected (small) problem. In this paper we consider an ill-posed problem with a noisy right-hand side (observation vector), where the noise level is unknown. We show how the information from the Golub-Kahan iterative bidiagonalization can be used for estimating the noise level. Such information can be useful for constructing efficient stopping criteria in solving ill-posed problems.     

Minimal residual methods for large scale Lyapunov equations

Projection methods have emerged as competitive techniques for solving large scale matrix Lyapunov equations. We explore the numerical solution of this class of linear matrix equations when a Minimal Residual (MR) condition is used during the projection step. We derive both a new direct method, and a preconditioned operator-oriented iterative solver based on CGLS, for solving the projected reduced least squares problem. Numerical experiments with benchmark problems show the effectiveness of an MR approach over a Galerkin procedure using the same approximation space.     

A computational method for large‐scale differential symmetric Stein equation

We propose a numerical method for solving large‐scale differential symmetric Stein equations having low‐rank right constant term. Our approach is based on projection the given problem onto a Krylov subspace then solving the low dimensional matrix problem by using an integration method, and the original problem solution is built by using obtained low‐rank approximate solution. Using the extended block Arnoldi process and backward differentiation formula (BDF), we give statements of the approximate solution and corresponding residual. Some numerical results are given to show the efficiency of the proposed method.     

A block Arnoldi-type contour integral spectral projection method for solving generalized eigenvalue problems

For generalized eigenvalue problems, we consider computing all eigenvalues located in a certain region and their corresponding eigenvectors. Recently, contour integral spectral projection methods have been proposed for solving such problems. In this study, from the analysis of the relationship between the contour integral spectral projection and the Krylov subspace, we conclude that the Rayleigh–Ritz-type of the contour integral spectral projection method is mathematically equivalent to the Arnoldi method with the projected vectors obtained from the contour integration. By this Arnoldi-based interpretation, we then propose a block Arnoldi-type contour integral spectral projection method for solving the eigenvalue problem.     

Convergence Properties of Projection and Contraction Methods for Variational Inequality Problems

In this paper we develop the convergence theory of a general class of projection and contraction algorithms (PC method), where an extended stepsize rule is used, for solving variational inequality (VI) problems. It is shown that, by defining a scaled projection residue, the PC method forces the sequence of the residues to zero. It is also shown that, by defining a projected function, the PC method forces the sequence of projected functions to zero. A consequence of this result is that if the PC method converges to a nondegenerate solution of the VI problem, then after a finite number of iterations, the optimal face is identified. Finally, we study local convergence behavior of the extragradient algorithm for solving the KKT system of the inequality constrained VI problem. \keywords{Variational inequality, Projection and contraction method, Predictor-corrector stepsize, Convergence property.} \amsclass{90C30, 90C33, 65K05.} Accepted 5 September 2000. Online publication 16 January 2001.     

General fractional variational problem depending on indefinite integrals

In this report, we consider two kind of general fractional variational problem depending on indefinite integrals include unconstrained problem and isoperimetric problem. These problems can have multiple dependent variables, multiorder fractional derivatives, multiorder integral derivatives and boundary conditions. For both problems, we obtain the Euler-Lagrange type necessary conditions which must be satisfied for the given functional to be extremum. Also, we apply the Rayleigh-Ritz method for solving the unconstrained general fractional variational problem depending on indefinite integrals. By this method, the given problem is reduced to the problem for solving a system of algebraic equations using shifted Legendre polynomials basis functions. An approximate solution for this problem is obtained by solving the system. We discuss the analytic convergence of this method and finally by some examples will be showing the accurately and applicability for this technique.     

有界约束半光滑方程组的非单调投影信赖域方法

投影信赖域策略结合非单调线搜索算法解有界约束非线性半光滑方程组.基于简单有界约束的非线性优化问题构建信赖域子问题,半光滑类牛顿步在可行域投影得到投影牛顿的试探步,获得新的搜索方向,结合非单调线搜索技术得到回代步,获得新的步长.在合理的条件下,证明算法不仅具有整体收敛性且保持超线性收敛速率.引入非单调技术能克服高度非线性的病态问题,加速收敛性进程,得到超线性收敛速率.     

Solving optimal control problems for the unsteady Burgers equation in COMSOL Multiphysics

The optimal control of unsteady Burgers equation without constraints and with control constraints are solved using the high-level modelling and simulation package COMSOL Multiphysics. Using the first-order optimality conditions, projection and semi-smooth Newton methods are applied for solving the optimality system. The optimality system is solved numerically using the classical iterative approach by integrating the state equation forward in time and the adjoint equation backward in time using the gradient method and considering the optimality system in the space-time cylinder as an elliptic equation and solving it adaptively. The equivalence of the optimality system to the elliptic partial differential equation (PDE) is shown by transforming the Burgers equation by the Cole-Hopf transformation to a linear diffusion type equation. Numerical results obtained with adaptive and nonadaptive elliptic solvers of COMSOL Multiphysics are presented both for the unconstrained and the control constrained case.     

Convergence of a projected gradient method variant for quasiconvex objectives

We present a version of the projected gradient method for solving constrained minimization problems with a competitive search strategy: an appropriate step size rule through an Armijo search along the feasible direction, thereby obtaining global convergence properties when the objective function is quasiconvex or pseudoconvex. In contrast to other similar step size rules, this one requires only one projection onto the feasible set per iteration, rather than one projection for each tentative step during the search for the step size, which represents a considerable saving when the projections are computationally expensive.