dorsal/arxiv
View SchemaNumerical differentiation: local versus global methods
| Authors | Karsten Ahnert, Markus Abel |
|---|---|
| Categories | |
| ArXiv ID | physics/0510176 |
| URL | https://arxiv.org/abs/physics/0510176 |
Abstract
In the context of the analysis of measured data, one is often faced with the task to differentiate data numerically. Typically, this occurs when measured data are concerned or data are evaluated numerically during the evolution of partial or ordinary differential equations. Usually, one does not take care for accuracy of the resulting estimates of derivatives because modern computers are assumed to be accurate to many digits. But measurements yield intrinsic errors, which are often much less accurate than the limit of the machine used, and there exists the effect of ``loss of significance'', well known in numerical mathematics and computational physics. The problem occurs primarily in numerical subtraction, and clearly, the estimation of derivatives involves the approximation of differences. In this article, we discuss several techniques for the estimation of derivatives. As a novel aspect, we divide into local and global methods, and explain the respective shortcomings. We have developed a general scheme for global methods, and illustrate our ideas by spline smoothing and spectral smoothing. The results from these less known techniques are confronted with the ones from local methods. As typical for the latter, we chose Savitzky-Golay filtering and finite differences. Two basic quantities are used for characterization of results: The variance of the difference of the true derivative and its estimate, and as important new characteristic, the smoothness of the estimate. We apply the different techniques to numerically produced data and demonstrate the application to data from an aeroacoustic experiment. As a result, we find that global methods are generally preferable if a smooth process is considered. For rough estimates local methods work acceptably well.
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"abstract": "In the context of the analysis of measured data, one is often faced with the\ntask to differentiate data numerically. Typically, this occurs when measured\ndata are concerned or data are evaluated numerically during the evolution of\npartial or ordinary differential equations. Usually, one does not take care for\naccuracy of the resulting estimates of derivatives because modern computers are\nassumed to be accurate to many digits. But measurements yield intrinsic errors,\nwhich are often much less accurate than the limit of the machine used, and\nthere exists the effect of ``loss of significance\u0027\u0027, well known in numerical\nmathematics and computational physics. The problem occurs primarily in\nnumerical subtraction, and clearly, the estimation of derivatives involves the\napproximation of differences. In this article, we discuss several techniques\nfor the estimation of derivatives. As a novel aspect, we divide into local and\nglobal methods, and explain the respective shortcomings. We have developed a\ngeneral scheme for global methods, and illustrate our ideas by spline smoothing\nand spectral smoothing. The results from these less known techniques are\nconfronted with the ones from local methods. As typical for the latter, we\nchose Savitzky-Golay filtering and finite differences. Two basic quantities are\nused for characterization of results: The variance of the difference of the\ntrue derivative and its estimate, and as important new characteristic, the\nsmoothness of the estimate. We apply the different techniques to numerically\nproduced data and demonstrate the application to data from an aeroacoustic\nexperiment. As a result, we find that global methods are generally preferable\nif a smooth process is considered. For rough estimates local methods work\nacceptably well.",
"arxiv_id": "physics/0510176",
"authors": [
"Karsten Ahnert",
"Markus Abel"
],
"categories": [
"physics.comp-ph",
"physics.data-an"
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"title": "Numerical differentiation: local versus global methods",
"url": "https://arxiv.org/abs/physics/0510176"
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