Preconditioning for multilevel Toeplitz systems has long been a focal point of research in numerical linear algebra. In this work, we develop a novel preconditioning method for a class of nonsymmetric multilevel Toeplitz systems, which includes the all-at-once systems that arise from evolutionary partial differential equations. These systems have recently garnered considerable attention in the literature. To further illustrate our proposed preconditioning strategy, we specifically consider the application of solving a wide range of non-local, time-dependent partial differential equations in a parallel-in-time manner. For these equations, we propose a symmetric positive definite multilevel Tau preconditioner that is not only efficient to implement but can also be adapted as an optimal preconditioner. In this context, the proposed preconditioner is optimal in the sense that it enables mesh-independent convergence when using the preconditioned generalized minimal residual method. Numerical examples are provided to critically analyze the results and underscore the effectiveness of our preconditioning strategy.

翻译：多层Toeplitz系统的预条件技术长期以来一直是数值线性代数研究的焦点。本文针对一类非对称多层Toeplitz系统开发了一种新颖的预条件方法，该类系统包括由演化偏微分方程产生的全一次性（all-at-once）系统。近年来，这类系统在文献中受到了广泛关注。为进一步阐明所提出的预条件策略，我们特别考虑了在时间并行框架下求解各类非局部、时间依赖的偏微分方程的应用。针对这些方程，我们提出了一种对称正定的多层Tau预条件子，该预条件子不仅易于高效实现，还可被调整为最优预条件子。在此背景下，所提预条件子具有最优性，其含义在于：当采用预条件广义最小残差法时，它能实现与网格无关的收敛性。本文通过数值算例对结果进行了深入分析，并凸显了所提预条件策略的有效性。