Comparison among Several Numerical Integration Methods for Kramers-Kronig Transformation

1988 ◽  
Vol 42 (6) ◽  
pp. 952-957 ◽  
Author(s):  
Koji Ohta ◽  
Hatsuo Ishida
Keyword(s):  
Fourier Transform ◽  
Numerical Integration ◽  
Spectral Data ◽  
Analytical Formula ◽  
Computation Time ◽  
The Other ◽  
Integration Methods ◽  
Fourier Transform Methods ◽  
Double Fourier Transform ◽  
Numerical Integration Methods

Several numerical integration methods are compared in order to search out the most effective method for the Kramers-Kronig transformation, using the analytical formula of the Kramers-Kronig transformation of a Lorentzian function as a reference. The methods to be compared involve the use of (1) Maclaurin's formula, (2) trapezium formula, (3) Simpson's formula, and (4) successive double Fourier transform methods. It is found that Maclaurin's formula, in which no special approximation is necessary for the pole part of the integration, gives the most accurate results, and also that its computation time is short. Successive Fourier transform is less accurate than the other methods, but it takes the least time when used without zero-filling. These results have important relevance for programs used to obtain optical constant spectra and to analyze spectral data.

scholarly journals Benchmarking of numerical integration methods for ODE models of biological systems

2020 ◽  
Author(s):  
Philipp Städter ◽  
Yannik Schälte ◽  
Leonard Schmiester ◽  
Jan Hasenauer ◽  
Paul L. Stapor
Keyword(s):  
Systems Biology ◽  
Numerical Integration ◽  
Computation Time ◽  
Biological Processes ◽  
Unknown Parameters ◽  
Integration Methods ◽  
Ode Models ◽  
Stability And Bifurcation Analysis ◽  
Numerical Integration Methods ◽  
The Impact

AbstractOrdinary differential equation (ODE) models are a key tool to understand complex mechanisms in systems biology. These models are studied using various approaches, including stability and bifurcation analysis, but most frequently by numerical simulations. The number of required simulations is often large, e.g., when unknown parameters need to be inferred. This renders efficient and reliable numerical integration methods essential. However, these methods depend on various hyperparameters, which strongly impact the ODE solution. Despite this, and although hundreds of published ODE models are freely available in public databases, a thorough study that quantifies the impact of hyperparameters on the ODE solver in terms of accuracy and computation time is still missing. In this manuscript, we investigate which choices of algorithms and hyperparameters are generally favorable when dealing with ODE models arising from biological processes. To ensure a representative evaluation, we considered 167 published models. Our study provides evidence that most ODEs in computational biology are stiff, and we give guidelines for the choice of algorithms and hyperparameters. We anticipate that our results will help researchers in systems biology to choose appropriate numerical methods when dealing with ODE models.

scholarly journals Benchmarking of numerical integration methods for ODE models of biological systems

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Philipp Städter ◽  
Yannik Schälte ◽  
Leonard Schmiester ◽  
Jan Hasenauer ◽  
Paul L. Stapor
Keyword(s):  
Systems Biology ◽  
Numerical Integration ◽  
Computation Time ◽  
Biological Processes ◽  
Unknown Parameters ◽  
Integration Methods ◽  
Ode Models ◽  
Stability And Bifurcation Analysis ◽  
Numerical Integration Methods ◽  
The Impact

AbstractOrdinary differential equation (ODE) models are a key tool to understand complex mechanisms in systems biology. These models are studied using various approaches, including stability and bifurcation analysis, but most frequently by numerical simulations. The number of required simulations is often large, e.g., when unknown parameters need to be inferred. This renders efficient and reliable numerical integration methods essential. However, these methods depend on various hyperparameters, which strongly impact the ODE solution. Despite this, and although hundreds of published ODE models are freely available in public databases, a thorough study that quantifies the impact of hyperparameters on the ODE solver in terms of accuracy and computation time is still missing. In this manuscript, we investigate which choices of algorithms and hyperparameters are generally favorable when dealing with ODE models arising from biological processes. To ensure a representative evaluation, we considered 142 published models. Our study provides evidence that most ODEs in computational biology are stiff, and we give guidelines for the choice of algorithms and hyperparameters. We anticipate that our results will help researchers in systems biology to choose appropriate numerical methods when dealing with ODE models.

scholarly journals On the relationship between long-period comets and large trans-Neptunian planetary bodies

2016 ◽  
Vol 12 (S325) ◽  
pp. 263-265
Author(s):  
Rustam Guliyev ◽  
Ayyub Guliyev
Keyword(s):  
Solar System ◽  
Numerical Integration ◽  
Dynamical Evolution ◽  
Integration Methods ◽  
Long Period ◽  
Planetary Bodies ◽  
Numerical Integration Methods ◽  
Relationship Of ◽  
The Relationship

AbstractIn the present work we investigate the possible relationship of long-period comets with five large and distant trans-Neptunian bodies (Sedna, Eris, 2007 OR10, 2012 VP113and 2008 ST291) in order to determine the probability of the transfer of a part of these kind of comets to the inner of the Solar System. To identify such relationships, we studied the relative positions of the comet orbits and listed TNOs. Using numerical integration methods, we examined dynamical evolution of the comets and have found one encounter of comet C/1861J1 and Eris.

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