Numerical differentiation by the polynomial-exponential basis
Numerical differentiation by the polynomial-exponential basis
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DOI：
10.48550/arxiv.2304.05909
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发表时间：
2023-04
期刊：
影响因子：
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通讯作者：
P. M. Nguyen;T. Le;L. Nguyen;M. Klibanov
中科院分区：
文献类型：
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作者：
P. M. Nguyen;T. Le;L. Nguyen;M. Klibanov
我们的目标是计算被噪声破坏的数据的导数。这是一项具有挑战性的任务，因为即使是少量的噪声也会导致计算中的重大错误。这主要是由于噪声的随机性，这可能导致高频波动。为了克服这一挑战，我们提出了一种方法，通过从给定数据的多项式指数基的傅里叶展开中消除高频项来近似数据。这种截断方法有助于正则化问题，而多项式-指数基的使用保证了计算的准确性。我们通过一维和二维的数值例子证明了我们方法的有效性。
Our objective is to calculate the derivatives of data corrupted by noise. This is a challenging task as even small amounts of noise can result in significant errors in the computation. This is mainly due to the randomness of the noise, which can result in high-frequency fluctuations. To overcome this challenge, we suggest an approach that involves approximating the data by eliminating high-frequency terms from the Fourier expansion of the given data with respect to the polynomial-exponential basis. This truncation method helps to regularize the issue, while the use of the polynomial-exponential basis ensures accuracy in the computation. We demonstrate the effectiveness of our approach through numerical examples in one and two dimensions.