Fast solution of Sylvester-structured systems for spatial source separation of the Cosmic Microwave Background

TL;DR

This paper presents efficient Krylov subspace methods, including conjugate gradients and Sylvester matrix reformulation, to solve large Sylvester-structured systems in cosmological source separation, enabling scalable analysis of massive data sets.

Contribution

It introduces two novel Krylov-based approaches for solving Sylvester-structured systems in large-scale cosmological data analysis, improving scalability and efficiency.

Findings

Both methods achieve accurate solutions within acceptable computation time.

The approaches handle data sizes that are currently prohibitive for direct methods.

Memory requirements are practical for large cosmological data sets.

Abstract

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.