纳米设备中非线性矩阵方程数值解研究

In the modeling of nanoscale devices, nano scientists usually make use of the non-equilibrium Green's function to describe the local density of quantum state and are of great interests to obtain such a function numerically. Although the Green's function has been shown in theory to be a solution of a class of nonlinear matrix equations, the research on how to compute it from numerics is still a fresh research subject. This proposal focuses on the computation of the non-equilibrium Green's function by using the iterative methods and the direct methods. The main research contents are comprised of the following topics: design of the efficient fixed-point iterative schemes via minimizing the spectral radius of the Fréchet derivative in a class of weighted fixed-point methods; construction of the cyclic reduction algorithm from the infinite block tridiagonal matrix corresponded to the Green's function; design of the Newton-type method that can preserve positive definite imaginary part in iterative matrix sequence; analysis of the numerical stability of the structure-preserving direct algorithm by the representation theory on the invariant subspace. The derived research results are expected not only to contribute to the completeness of the computation of the Green's function both in theory and algorithm aspects, but also to propel further developments of related scientific areas.

在纳米设备建模理论中，纳米科学家经常用非均衡格林函数来对量子状态的局部密度进行描述并对如何从数值上求出这样的格林函数抱有极大的兴趣。从理论上来说，这一格林函数已被证明为某一类非线性矩阵方程的解，然而从数值上对如何求解这一类非线性方程的研究才刚刚起步。本项目将从迭代方法和直接方法两个方面对格林函数的数值计算进行深入研究。研究内容包括：从最小化加权系数不动点方法Fréchet导数的谱半径出发设计有效的不动点迭代格式；直接从格林函数对应的无限块三对角矩阵入手设计循环约化型算法；通过纳米设备中非线性方程特殊结构将迭代矩阵的实部与虚部分开，采用相关的正定化技巧设计能够保证迭代矩阵具有正定虚部的牛顿型迭代格式；结合不变子空间表示理论分析保结构直接算法的稳定性。预期获得的理论和算法成果不但有助于纳米设备建模中格林函数数值计算方法的完善，还将对其他相关交叉学科的发展起到极大的推动作用。

本项目对一类纳米设备建模中产生的非线性矩阵方程的数值求解方法进行了研究，研究内容包含了具有实际意义的解的存在性，新算法的设计、收敛性分析、复杂度分析以及误差分析等等。尤其在大规模方程数值解方面取得了较好的研究成果。获得的成果既丰富了纳米设备建模中非线性矩阵方程数值解算法又对其他相关领域中非线性矩阵方程的数值求解方法发展起到了推动作用。