This paper is an extension of [1]. In this paper the descent and ascent segments are introduced to replace respectively the descent and ascent directions in [1] and are used to extend the concepts of S-basin and basin of a minimizer of a function. Lemmas and theorems similar to those in [1] are proved for the filled function $$P(x,r,p)= \frac{1}{r+F(x)}exp(-|x-x^*_1|^2/\rho^2),$$ which is the same as that in [1], where $x^*_1$ is a constrained local minimizer of the problem (0.3) below and $$F(x)=f(x)+\sum^{m'}_{i=1}\mu_i|c_i(x)|+ \sum^m_{i=m'+1}\mu_i max(0, -c_i(x))$$ is the exact penalty function for the constrained minimization problem

$\mathop{\rm min}\limits_x f(x)$,

subject to $$c_i(x) = 0 , i = 1, 2, \cdots, m',$$ $$c_i(x) \ge 0 , i = m'+1, \cdots, m,$$ where $μ_i ＞ 0 \ (i=1, 2, \cdots, m)$ are sufficiently large. When $x^*_1$ has been located, a saddle point or a minimizer $\hat{x}$ of $P(x,r,\rho)$ can be located by using the nonsmooth minimization method with some special termination principles. The $\hat{x}$ is proved to be in a basin of a lower minimizer $x^*_2$ of $F(x)$, provided that the ratio $\rho^2/[r+F(x^*_1)]$ is appropriately small. Thus, starting with $\hat{x}$ to minimize $F(x)$, one can locate $x^*_2$. In this way a constrained global minimizer of (0.3) can finally be found and termination will happen.

A modified Calahan method for parabolic equations with time-dependent coefficients is presented. It is shown that the convergence order is $O(h^{r+1}+k^3)$ while the convergence order obtained in [1] for a standard Calahan method is only $O(h^{r+1}+k^2)$.

This paper presents a new coupling of the finite element method and the boundary element method to solve the two-dimensional exterior Helmholtz problems by using the asymptotic radiation conditions in [1], in which the coupling relations are the same as C. Johnson and J. C. Nedelec's. The error estimates are derived and results of numerical calculation in comparison with analytic solution verify the theoretical estimates.

We consider in this note how the principal angles between column spaces R(A) and R(B) change when the elements in A and B are subject to perturbations. The basic idea in the proof of our results is that the non-zero cosine values of the principal angles between R(A) and R(B) coincide with the non-zero singular values of $P_AP_B$, the product of two orthogonal projections, and consequently we can apply a perturbation theorem of orthogonal projections proved by the author[4].

An explanatory model class belonging to the family of Space-Time Transfer Function-Noise (STTFN) process is presented. The paper develops a three-stage iterative procedure for building STTFN models of the rainfall-runoff process. Four precipitation and runoff stations located in a watershed in southern Ontario, Canada, sampled at 15-day intervals are used for the numerical analysis. Three STTF models are identified. The model parameters are estimated by the polytope technique, a nonlinear optimization algorithm. Two of the developed space-time models proved adequate in describing the spatio-temporal characteristics of precipitation and runoff time series.

Some main results in nondifferentiable optimization are reviewed. In section 2, we discuss subgradient methods. Section 3 is about the cutting plane method and the bundle methods are studied in section4. Trust region methods for composite nonsmooth optimization are discussed in section 5.

This paper gives a thorough analysis of the local refinement method on plane polygonal domains with special attention to the treatment of reentrant corner. Convergence rates of the finite element method under various norms are derived via a systematic treatment of the interpolation theory in weighted Sobolev spaces. It is proved that by refining the mesh suitably, the finite element approximations for problems with singularities achieve the same convergence rates as those for smooth solutions.