非线性最优化的投影型算法

以作者的研究成果为主，以利用梯度投影的各种技巧为主线，对非线性最优化的有关带转轴运算的、广义投影的、线性系统的、超线性收敛的以及统一算法模型的各种算法进行系统、简要的综述，最后提出若干展望．     

SEMI-LINEAR SYSTEMS OF BACKWARD STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS IN $R^n$

This paper explores the diffeomorphism of a backward stochastic ordinary differential equation (BSDE) to a system of semi-linear backward stochastic partial differential equations (BSPDEs), under the inverse of a stochastic flow generated by an ordinary stochastic differential equation (SDE). The author develops a new approach to BSPDEs and also provides some new results. The adapted solution of BSPDEs in terms of those of SDEs and BSDEs is constructed. This brings a new insight on BSPDEs, and leads to a probabilistic approach. As a consequence, the existence, uniqueness, and regularity results are obtained for the (classical, Sobolev, and distributional) solution of BSPDEs.The dimension of the space variable x is allowed to be arbitrary n, and BSPDEs are allowed to be nonlinear in both unknown variables, which implies that the BSPDEs may be nonlinear in the gradient. Due to the limitation of space, however, this paper concerns only classical solution of BSPDEs under some more restricted assumptions.     

THE LARGEST SOLUTION OF LINEAR EQUATION OVER THE COMPLETE HEYTING ALGEBRA

Let (L, ≤, ∨, ∧) be a complete Heyting algebra. In this article, the linear system Ax = b over a complete Heyting algebra, where classical addition and multiplication operations are replaced by ∨ and ∧ respectively, is studied. We obtain: (i) the necessary and suffcient conditions for S(A, b)≠■l; (ii) the necessary conditions for |S(A, b)| = 1. We also obtain the vector  ∈ Ln and prove that it is the largest element of S(A, b) if S(A, b)≠■.     

线性系统标准分解的解耦问题

本文解决了线性系统标准分解在其分解的四个子空间上解耦为四个独立子系统的问题。     

线性DEDS的机械化求解方法

利用Gauss消元法原理，提出了线性DEDS研究中方程求解、传递矩阵求取、矩阵伪逆运算的机械化方法，为DEDS的建模、分析、计算提供了新的途径．     

广义线性系统Ex=Ax解的存在唯一性

研究了广义线性系统Ex=AX的初值问题解的存在唯一性的充要条件和求解公式．     

二阶变系数线性系统的稳定性

本文研究变系数二阶线性微分方程组零解的稳定性。引入了时变方向场的概念，用定性方法建立了稳定及不稳定的定理。     

预紧弹性约束对系统主共振的影响

用平均法研究了含预紧弹性约束的单自由度系统在简谐激励下的主共振及其稳定性。发现预紧使主共振明显复杂化：一是存在四种持久的主共振，其运动与约束一擦边即可能失稳；二是共存的周期运动会从三种增加到五种，其中二种不稳定     

关于线性系统稳定性的一个比较判别法

本文建立了一个关于齐次线性系统的比较定理,借以将对系统 x=A(t)x 的渐近稳定性判定归结于对某个系统 z=C(t)z 的考察.适当地选定 C(t),得到某些简便有效的渐近稳定性判别法。     

THE RESTRICTIVELY PRECONDITIONED CONJUGATE GRADIENT METHODS ON NORMAL RESIDUAL FOR BLOCK TWO-BY-TWO LINEAR SYSTEMS

The restrictively preconditioned conjugate gradient （RPCG） method is further developed to solve large sparse system of linear equations of a block two-by-two structure. The basic idea of this new approach is that we apply the RPCG method to the normal-residual equation of the block two-by-two linear system and construct each required approximate matrix by making use of the incomplete orthogonal factorization of the involved matrix blocks. Numerical experiments show that the new method, called the restrictively preconditioned conjugate gradient on normal residual （RPCGNR）, is more robust and effective than either the known RPCG method or the standard conjugate gradient on normal residual （CGNR） method when being used for solving the large sparse saddle point problems.