论文标题
用Tikhonov正则化估算解决方案的平滑度和数据噪声
Estimating solution smoothness and data noise with Tikhonov regularization
论文作者
论文摘要
经典tikhonov正则化的主要缺点是，在实践中，使用理论结果的参数通常是所需的参数，例如，寻求的解决方案和噪声水平的平滑度是未知的。在本文中，我们在新的细节中调查了Tikhonov正则化中的残差视为正则化参数的函数。我们表明，残留物具有一些限制，即有关未知解决方案和噪声水平的信息。通过计算大量正则化参数的近似解决方案，我们可以从剩余的一组噪声数据和正向操作员中提取两个参数。残差的平滑度允许重新审视参数选择规则，并以一种新颖的方式将A-Priori，A-Posteriori和启发式规则相关联，该方式模糊了参数选择规则的经典划分之间的界限。所有结果均伴有数值实验。
A main drawback of classical Tikhonov regularization is that often the parameters required to apply theoretical results, e.g., the smoothness of the sought-after solution and the noise level, are unknown in practice. In this paper we investigate in new detail the residuals in Tikhonov regularization viewed as functions of the regularization parameter. We show that the residual carries, with some restrictions, the information on both the unknown solution and the noise level. By calculating approximate solutions for a large range of regularization parameters, we can extract both parameters from the residual given only one set of noisy data and the forward operator. The smoothness in the residual allows to revisit parameter choice rules and relate a-priori, a-posteriori, and heuristic rules in a novel way that blurs the lines between the classical division of the parameter choice rules. All results are accompanied by numerical experiments.
