Toeplitz系统的超快速Tikhonov正则化
Superfast Tikhonov Regularization of Toeplitz Systems
AI总结：
针对Toeplitz系统的Tikhonov正则化问题，提出一种基于扩展-变换技术的超快速算法，将其转化为切向插值问题，实现O(n (log n)^2)的计算复杂度。
摘要：
Toeplitz结构的线性系统经常出现在实际工程问题中。相应地，已经开发了许多利用Toeplitz结构来提高求解这些系统计算效率的算法。最早的Toeplitz系统“快速”算法需要O(n^2)次操作，而最近的“超快速”算法将成本降低到O(n (log n)^2)或更低。在这项工作中，我们提出了一种用于Toeplitz系统Tikhonov正则化的超快速算法。使用“扩展-变换”技术，我们的算法将Tikhonov正则化的Toeplitz系统转化为一种称为切向插值的专门多项式问题。在这种表述下，我们可以在仅O(n (log n)^2)次操作内计算出解。我们使用数值模拟来展示算法的复杂度，并验证其返回稳定解的能力。
英文摘要：
Toeplitz-structured linear systems arise often in practical engineering problems. Correspondingly, a number of algorithms have been developed that exploit Toeplitz structure to gain computational efficiency when solving these systems. The earliest "fast" algorithms for Toeplitz systems required O(n^2) operations, while more recent "superfast" algorithms reduce the cost to O(n (log n)^2) or below. In this work, we present a superfast algorithm for Tikhonov regularization of Toeplitz systems. Using an "extension-and-transformation" technique, our algorithm translates a Tikhonov-regularized Toeplitz system into a type of specialized polynomial problem known as tangential interpolation. Under this formulation, we can compute the solution in only O(n (log n)^2) operations. We use numerical simulations to demonstrate our algorithm's complexity and verify that it returns stable solutions.
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