基于自适应交叉近似的低秩分解算法研究

项目名称： 基于自适应交叉近似的低秩分解算法研究

项目编号： No.61501227

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 无线电电子学、电信技术

项目作者： 陈新蕾

作者单位： 南京航空航天大学

项目金额： 19万元

中文摘要： 基于自适应交叉近似(ACA)的低秩分解算法由于具有计算精度高、不依赖于积分核等优点近年来备受关注，但是这些算法均具有较高的渐近计算复杂度，这成为此类方法在(超)电大电磁目标数值仿真分析中的主要瓶颈。为了解决这一问题，本项目对基于ACA的高效低秩分解算法进行深入研究。首先，通过改进传统的ACA分解算法并借鉴“蝶形算法”的思想，研究出一种基于ACA的新型多层低秩分解技术，可以将迭代求解的计算复杂度降至理想的O(NlogN)或者O(Nlog2N)，这里N表示未知量的数目；然后，研究出基于该新型多层低秩分解技术的直接求解算法；在此基础上，与特征基函数法以及多层特征基函数法相结合，通过对缩减矩阵进行快速迭代和直接求解，发展出计算能力更强大的算法；最终将这些新算法应用到实际工程电磁问题中。这对于发展同时具有精度好、效率高、核独立等优点的电磁仿真算法具有重要的理论意义和应用价值。

中文关键词： 电磁计算；积分方程法；矩量法；低秩分解算法；自适应交叉近似

英文摘要： In recent years, the adaptive cross approximation (ACA)-based low-rank decomposition algorithms have attracted considerable attention because they are accurate and integral kernel-independent. However, these algorithms have relatively high asymptotic computational complexities, which is the major bottleneck of solving electrically (ultra-large) large targets. To address this problem, we focus on the more efficient ACA-based low-rank decomposition algorithms in this project. First, by improving the conventional ACA algorithm and referring to the butterfly algorithm, we propose a new ACA-based multilevel low-rank decomposition technique, which can reduce the computational complexity of the iterative solution to the ideal O(NlogN) or O(Nlog2N), where N is the number of unknowns. Then, we propose a more efficient direct algorithm base on the new multilevel low-rank decomposition technique. In addition, we develop more powerful methods by combining the new ACA-based algorithms with the characteristic basis function method (CBFM) and multilevel characteristic basis function method (MLCBFM). Finally, these new methods are applied to solve the electromagnetic practical engineering problems. The study has important theoretical and practical value for the development of the accurate, efficient and kernel-independent electromagnetic simulation algorithms.

英文关键词： Electromagnetic calculation;Integral equation method;Method of moments;Low-rank decomposition algorithm;Adaptive cross approximation