In this report we present some new numerical methods for unconstrained optimization. These methods apply update formulae that do not satisfy the quasi-Newton equation. We derive these new formulae by considering different techniques of approximating the objective function. Theoretical analyses are given to show the advantages of using non-quasi-Newton updates. Under mild conditions we prove that our new update formulae preserve global convergence properties. Numerical results are also presented.

In this paper we consider the $L^\infty$ convergence for quasi-conforming finite elements solving the boundary value problems of the biharmonic equation and give the nearly optimal order $L^\infty$ estimates.

In this paper, a finite difference scheme for the linear and nonlinear models of wheezes is given. The stability of the finite difference scheme for the linear model is obtained by using of von Neumann method. Moreover, the convergence and stability of the finite difference scheme for the nonlinear model are studied by the energy inequalities method. By some numerical computations, the relationships between angular frequency and wall position, fluid speed and amplitude are discussed. Finally, the author shows that the numerical results are coincided with Grotberg's theoretical results.

After surveying the theoretical aspects of Huber's $M$-estimator on underdetermined problems, two finite algorithms are presented. Both proceed in a constructive manner by moving from one partition to an adjacent one. One of the algorithm, which uses the tuning constant as a continuation parameter, also has the facility to simultaneously estimate the tuning constant and scaling factor. Stable and efficient implementation of the algorithms is presented together with numerical results. The $L_1$-norm problem is mentioned as a special case.

Fourier-Legendre spectral approximation for the unsteady Navier-Stokes equations is analyzed. The generalized stability and convergence are proved respectively.

The line search subproblem in unconstrained optimization is concerned with finding an acceptable steplength satisfying certain standard conditions. The conditions proposed in the early work of Armijo and Goldstein are sometimes replaced by those recommended by Wolfe because these latter conditions automatically allow positive definiteness of some popular quasi-Newton updates to be maintained. It is shown that a slightly modified form of quasi-Newton update allows positive definiteness to be maintained even if line searches based on the Armijo-Goldstein conditions are used.

Predictor-corrector algorithm for linear programming, proposed by Mizuno et al.$^{[1]}$, becomes the best well known in the interior point methods. The purpose of this paper is to extend these results in two directions. First, we modify the algorithm in order to solve convex quadratic programming with upper bounds. Second, we replace the corrector step with an iteration of Monteiro and Adler's algorithm$^{[2]}$. With these modifications, the duality gap is reduced by a constant factor after each corrector step for convex quadratic programming. It is shown that the new algorithm has a $O(\sqrt nL)$-iteration complexity.

In the paper a linear combination of both the standard mixed formulation and the displacement one of the Reissner-Mindlin plate theory is used to enhance stability of the former and to remove "locking" of the later. For this new stabilized formulation, a unified approach to convergence analysis is presented for a wide spectrum of finite element spaces. As long as the rotation space is appropriately enriched, the formulation is convergent for the finite element spaces of sufficiently high order. Optimal-order error estimates with constants independent of the plate thickness are proved for the various lower order methods of this kind.