Let X and Y be Banach spaces. For A ∈ L ( X ), B ∈ L ( Y ), C ∈ L ( Y, X ), let M C be the operator matrix defined on X ⊕ Y by M_C = ( [ A C; 0 B ]) ∈ L(X ⊕ Y) . In this paper we investigate the decomposability for M C . We consider Bishop’s property ( β ), decomposition property ( δ ) and Dunford’s property ( C ) and obtain the relationship of these properties between M C and its entries. We explore how σ * ( M C ) shrinks from σ * ( A ) ∪ σ * ( B ), where σ * denotes σ β , σ δ , σ C , σ dec . In particular, we develop some sufficient conditions for equality σ * ( M C ) = σ * ( A ) ∪ σ * ( B ). Besides, we consider the perturbation of these properties for M C and show that in perturbing with certain operators C the properties for M C keeps with A, B . Some examples are given to illustrate our results. Furthermore, we study the decomposability for ( [ 0 A; B 0 ]) . Finally, we give applications of decomposability for operator matrices.
To solve the tensor completion problem,a cyclic algorithm for low rank tensor completion is proposed.Based on the alternating direction multiplier method,the sub-problem is circularly updated,which effectively reduces the cost of tensor expansion,matrix folding and singular value decomposition in the iterative process.At the same time,the convergence analysis of the algorithm is given under reasonable assumptions.Finally,the numerical experiments show that the proposed algorithm is more efficient than other algorithm.
The analysis theory on the time scales not only effectively unifies the theory of continuous analysis and discrete analysis, but also becomes an indispensable mathematics tool in such domains of natural science and social science. Moreover, the oscillation and non-oscillation theories are the key issues of qualitative study of the neutral dynamic equations, which have attracted people’s attention. Therefore, the dynamical properties of certain second-order delay nonlinear noncanonical dynamic equations on time scales are discussed. By using the calculus theory on the time scales and the generalized Riccati transformation and some necessary analytic techniques, two new dynamical properties for the equations under certain conditions are established. The results extend and improve the part of the results established in previous literatures. Finally, the effectiveness of the theoretical results are verified by two examples.
针对机器学习中一类有限光滑凸函数和的最小化问题,提出一种新的随机方差约简梯度下降算法.新算法的特点是将随机方差约简梯度算法和一种谱梯度BB步长方法有机结合,从而可以充分发挥两种方法的优势.另外,初始步长可以任意选取,且步长在算法运行中可以自适应地计算更新.此外,新算法使用了重要样本抽样方法,可以大大减少计算工作量.最后,在通常的假设条件下证明了新算法具有R-线性收敛速度,并给出了复杂度分析.数值实验验证了新算法是可行有效的.
提出了一种求解不规则边界上有Robin边界条件的椭圆方程的Cartesian网格方法.该椭圆方程经重写后转化为定义在矩形区域上的椭圆界面问题,进而采用水平集浸入界面方法(IIM)对其进行求解.特别地,Robin边界条件采用单边三次插值离散.随后,利用该方法求解定义在不规则区域上的Navier-Stokes程.Navier-Stokes方程的解法器由求解速度方程的虚拟流体方法(GFM)和辅助变量方程的IIM耦合而成.数值测试表明,椭圆方程的解法器能够产生二阶精度的数值解和梯度,而且能够快速收敛,Navier-Stokes方程的解法器产生了二阶精度的速度及一阶精度的压力.圆柱绕流的仿真验证了 Navier-Stokes方程解法器的鲁棒性.
