dorsal/arxiv
View SchemaA new method for numerical inversion of the Laplace transform
| Authors | Bruno Huepper, Eli Pollak |
|---|---|
| Categories | |
| ArXiv ID | physics/9807051 |
| URL | https://arxiv.org/abs/physics/9807051 |
| DOI | 10.1063/1.479059 |
Abstract
A formula of Doetsch ({\em Math. Zeitschr.} {\bf 42}, 263 (1937)) is generalized and used to numerically invert the one-sided Laplace transform ${\hat C}(\beta)$. The necessary input is only the values of ${\hat C}(\beta)$ on the positive real axis. The method is applicable provided that the functions $\hat{C}(\beta)$ belong to the function space $L^2_\alpha$ defined by the condition that $ G(x) = e^{x\alpha}\hat{C}(e^x),~ \alpha > 0$ has to be square integrable. This space includes sums of exponential decays ${\hat C}(\beta)=\sum_n^{\infty}a_n e^{-\beta E_n}$, e.g. partition functions with $a_n = 1$. In practice, the inversion algorithm consists of two subsequent fast Fourier transforms. High accuracy inverted data can be obtained, provided that the signal is also highly accurate. The method is demonstrated for a harmonic partition function and resonant transmission through a barrier. We find accurately inverted functions even in the presence of noise.
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"abstract": "A formula of Doetsch ({\\em Math. Zeitschr.} {\\bf 42}, 263 (1937)) is\ngeneralized and used to numerically invert the one-sided Laplace transform\n${\\hat C}(\\beta)$. The necessary input is only the values of ${\\hat C}(\\beta)$\non the positive real axis. The method is applicable provided that the functions\n$\\hat{C}(\\beta)$ belong to the function space $L^2_\\alpha$ defined by the\ncondition that $ G(x) = e^{x\\alpha}\\hat{C}(e^x),~ \\alpha \u003e 0$ has to be square\nintegrable. This space includes sums of exponential decays ${\\hat\nC}(\\beta)=\\sum_n^{\\infty}a_n e^{-\\beta E_n}$, e.g. partition functions with\n$a_n = 1$. In practice, the inversion algorithm consists of two subsequent fast\nFourier transforms. High accuracy inverted data can be obtained, provided that\nthe signal is also highly accurate. The method is demonstrated for a harmonic\npartition function and resonant transmission through a barrier. We find\naccurately inverted functions even in the presence of noise.",
"arxiv_id": "physics/9807051",
"authors": [
"Bruno Huepper",
"Eli Pollak"
],
"categories": [
"physics.data-an",
"physics.comp-ph"
],
"doi": "10.1063/1.479059",
"title": "A new method for numerical inversion of the Laplace transform",
"url": "https://arxiv.org/abs/physics/9807051"
},
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