用信息场和共轭梯度技术求解线性方程——应用于宇宙微波背景数据分析
Solving linear equations with messenger-field and conjugate gradients techniques - an application to CMB data analysis
AI总结：
本文比较了信息场方法与预条件共轭梯度方法在求解线性方程中的性能，指出预条件共轭梯度方法在迭代次数和求解时间上通常表现更优，并通过宇宙微波背景数据中的例子验证了这一结论，同时指出预条件共轭梯度方法的性能可能依赖于初始向量。
摘要：
我们讨论了涉及信息场的线性系统求解器，并将其与（预条件的）共轭梯度方法进行比较。我们展示信息场技术对应于适当预条件的初始线性方程组的固定点迭代。然后我们论证，将共轭梯度求解器应用于相同的预条件系统，或者等价地使用相同的预条件器对原始系统应用预条件共轭梯度求解器，在迭代次数和求解时间方面通常至少能保证可比甚至更好的性能。我们通过两个来自宇宙微波背景数据分析的常见例子来说明我们的结论：维纳滤波和制图。此外，与CMB领域中的标准观点相反，我们展示了预条件共轭梯度求解器的性能可能在很大程度上依赖于初始向量。这一观察在高信噪比天空制图的情况下尤为重要，因此应为下一代CMB实验的相关性。
英文摘要：
We discuss linear system solvers invoking a messenger-field and compare them with (preconditioned) conjugate gradients approaches. We show that the messenger-field techniques correspond to fixed point iterations of an appropriately preconditioned initial system of linear equations. We then argue that a conjugate gradient solver applied to the same preconditioned system, or equivalently a preconditioned conjugate gradient solver using the same preconditioner and applied to the original system, will in general ensure at least a comparable and typically better performance in terms of the number of iterations to convergence and time-to-solution. We illustrate our conclusions on two common examples drawn from the Cosmic Microwave Background data analysis: Wiener filtering and map-making. In addition, and contrary to the standard lore in the CMB field, we show that the performance of the preconditioned conjugate gradient solver can depend importantly on the starting vector. This observation seems of particular importance in the cases of map-making of high signal-to-noise sky maps and therefore should be of relevance for the next generation of CMB experiments.
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