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A new regularization approach for numerical differentiation
Applied Mathematics in Science and Engineering ( IF 1.3 ) Pub Date : 2020-06-04 , DOI: 10.1080/17415977.2020.1763983
Abinash Nayak 1
Affiliation  

It is well known that the problem of numerical differentiation is an ill-posed problem and one requires regularization methods to approximate the solution. The commonly practiced regularization methods are (external) parameter-based like Tikhonov regularization, which has certain inherent difficulties associated with them. In such scenarios, iterative regularization methods serve as an attractive alternative. In this paper, we propose a novel iterative regularization method where the minimizing functional does not contain the noisy data directly, but rather a smoothed or integrated version of it. The advantage, in addition to circumventing the use of noisy data directly, is that the sequence of functions constructed during the descent process tends to avoid overfitting, and hence, does not corrupt the recovery significantly. To demonstrate the effectiveness of our method we compare the numerical results obtained from our method with the numerical results obtained from certain standard regularization methods such as Tikhonov regularization, Total-variation, etc.

中文翻译:

一种新的数值微分正则化方法

众所周知,数值微分问题是一个不适定问题,需要使用正则化方法来逼近解。常用的正则化方法是基于(外部)参数的,如 Tikhonov 正则化,它有一些与它们相关的固有困难。在这种情况下,迭代正则化方法是一种有吸引力的替代方法。在本文中,我们提出了一种新颖的迭代正则化方法,其中最小化函数不直接包含噪声数据,而是它的平滑或集成版本。除了直接避免使用噪声数据外,其优点是在下降过程中构建的函数序列倾向于避免过度拟合,因此不会显着破坏恢复。
更新日期:2020-06-04
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