论文标题
Sylvester结构系统的快速解决方案，用于宇宙微波背景的空间源分离
Fast solution of Sylvester-structured systems for spatial source separation of the Cosmic Microwave Background
论文作者
论文摘要
用于大型多元数据集的许多统计方法的实现需要一个求解一个线性系统，该系统取决于方法，是观测值数量或每个单独的数据向量的维度。这通常是用数据大小和复杂性扩展方法的限制因素。在本文中，我们说明了Krylov子空间方法在宇宙学中源分离问题的统计解决方案中使用Krylov子空间方法，在该解决方案中，数据大小非常大，以直接解决所需系统的直接解决方案。描述了从文献中的技术改编的两种不同的方法：一种使用共轭梯度直接与Kronecker结构问题的方法，另一种使用sylvester矩阵方程进行了重新定义的方法。我们表明，两种方法都在可接受的计算时间内产生准确的解决方案，并且对当前可用的数据大小具有实际的内存要求。
Implementation of many statistical methods for large, multivariate data sets requires one to solve a linear system that, depending on the method, is of the dimension of the number of observations or each individual data vector. This is often the limiting factor in scaling the method with data size and complexity. In this paper we illustrate the use of Krylov subspace methods to address this issue in a statistical solution to a source separation problem in cosmology where the data size is prohibitively large for direct solution of the required system. Two distinct approaches, adapted from techniques in the literature, are described: one that uses the method of conjugate gradients directly to the Kronecker-structured problem and another that reformulates the system as a Sylvester matrix equation. We show that both approaches produce an accurate solution within an acceptable computation time and with practical memory requirements for the data size that is currently available.
