Robust canonical correlations: A comparative study

Summary  Several approaches for robust canonical correlation analysis will be presented and discussed. A first method is based on the definition of canonical correlation analysis as looking for linear combinations of two sets of variables having maximal (robust) correlation. A second method is based on alternating robust regressions. These methods are discussed in detail and compared with the more traditional approach to robust canonical correlation via covariance matrix estimates. A simulation study compares the performance of the different estimators under several kinds of sampling schemes. Robustness is studied as well by breakdown plots.     

Existence and global bounds for a fluid model of plasma display technology

This article considers the fluid model for the discharge of plasma particle species in display technology. The fluid equations are coupled with Poisson's equation, which describes the effect of the charged particles on the electric field. The diffusion and mobility coefficients for the positive ion particles depend on the electric field, while those for the electrons depend on the electron mean energy. The reaction rates are proportional to the products of the densities of the reacting particles involved in the particular ionization, conversion or recombination reactions. Moreover, the ionization coefficients are dependent on the electric field, which varies spatially and temporally. The main ionization and discharge reactions are described by an initial-boundary value problem for a system of coupled parabolic–elliptic partial differential equations. The system is first analyzed by upper–lower solution method. By means of the a priori bounds obtained for an arbitrary time, the existence of solution for the initial-boundary value problem is proved in an appropriate Hölder space.     

Banach spaces with projectional skeletons

A projectional skeleton in a Banach space is a σ-directed family of projections onto separable subspaces, covering the entire space. The class of Banach spaces with projectional skeletons is strictly larger than the class of Plichko spaces (i.e. Banach spaces with a countably norming Markushevich basis). We show that every space with a projectional skeleton has a projectional resolution of the identity and has a norming space with similar properties to Σ-spaces. We characterize the existence of a projectional skeleton in terms of elementary substructures, providing simple proofs of known results concerning weakly compactly generated spaces and Plichko spaces. We prove a preservation result for Plichko Banach spaces, involving transfinite sequences of projections. As a corollary, we show that a Banach space is Plichko if and only if it has a commutative projectional skeleton.     

解凸约束非线性单调方程组的无导数谱PRP投影算法

本文在著名PRP共轭梯度算法的基础上研究了一种无导数谱PRP投影算法,并证明了算法在求解带有凸约束条件的非线性单调方程组问题的全局收敛性.由于无导数和储存量小的特性,它更适应于求解大规模非光滑的非线性单调方程组问题.数值试验表明,新算法对给定的测试问题是有效的和稳定的.     

Linear Optimization in C (Ω) and Portfolio Insurance

Suppose that X is a subspace of C ( z ) generated by n linearly independent positive elements of C ( z ). In this article we study the problem of minimization of a positive linear functional p of X in X , under a finite number of linear inequalities. This problem does not have always a solution and if a solution exists we cannot determine it. In this article we show that if X is contained in a finite dimensional minimal lattice-subspace Y of C ( z ) (or equivalently, if X is contained in a finite dimensional minimal subspace Y of C ( z ) with a positive basis) and m = dim Y , then the minimization problem has a solution and we determine the solutions by solving an equivalent linear programming problem in . Finally note that this minimization problem has an important application in the portfolio insurance which was the motivation for the preparation of this article.     

Approximate Toeplitz Matrix Problem Using Semidefinite Programming

Given a data matrix, we find its nearest symmetric positive-semidefinite Toeplitz matrix. In this paper, we formulate the problem as an optimization problem with a quadratic objective function and semidefinite constraints. In particular, instead of solving the so-called normal equations, our algorithm eliminates the linear feasibility equations from the start to maintain exact primal and dual feasibility during the course of the algorithm. Subsequently, the search direction is found using an inexact Gauss-Newton method rather than a Newton method on a symmetrized system and is computed using a diagonal preconditioned conjugate-gradient-type method. Computational results illustrate the robustness of the algorithm.     

The Drazin inverses of products and differences of orthogonal projections

In this note, the Drazin inverses of products and differences of orthogonal projections on a Hilbert space are established.     

Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces

Let C be a closed convex subset of a real Hilbert space H and assume that T is a κ-strict pseudo-contraction on C with a fixed point, for some 0?κ<1. Given an initial guess x₀∈C and given also a real sequence {αn} in (0,1). The Mann's algorithm generates a sequence {xn} by the formula: xn+1=αnxn+(1−αn)Txn, n?0. It is proved that if the control sequence {αn} is chosen so that κ<αn<1 and , then {xn} converges weakly to a fixed point of T. However this convergence is in general not strong. We then modify Mann's algorithm by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strong convergent sequence. This result extends a recent result of Nakajo and Takahashi [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372-379] from nonexpansive mappings to strict pseudo-contractions.     

Projection 2 Goes Turbulent - and Fully Implicit

We describe and demonstrate both our newest projection method (fully implicit) for the incompressible Navier-Stokes equations via the finite element method and our newest turbulence model - a somewhat sophisticated nonlinear (cubic) eddy viscosity model that does not employ wall functions or normal distances to walls - the combination providing a method and a model that is each applicable in arbitrary geometry. We demonstrate both via a flow simulation past an automobile-like body at a Reynolds number of 4.3 million.     

An efficient iterative method for finding common fixed point and variational inequalities in Hilbert spaces

ABSTRACT

In this paper, we investigate the problem of finding a common solution to a fixed point problem involving demi-contractive operator and a variational inequality with monotone and Lipschitz continuous mapping in real Hilbert spaces. Inspired by the projection and contraction method and the hybrid descent approximation method, a new and efficient iterative method for solving the problem is introduced. Strong convergence theorem of the proposed method is established under standard and mild conditions. Our scheme generalizes and extends some of the existing results in the literature, and moreover, its computational effort is less per each iteration compared with related works.