Diffractive multifocal lens analysis using complex Fourier series

2021 ◽  
Author(s):  
Jim Schwiegerling
Keyword(s):  
Fourier Series ◽  
Complex Fourier Series ◽  
Multifocal Lens

scholarly journals Estimating Ruin Probability in an Insurance Risk Model with Stochastic Premium Income Based on the CFS Method

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 982
Author(s):  
Yujuan Huang ◽  
Jing Li ◽  
Hengyu Liu ◽  
Wenguang Yu
Keyword(s):  
Fourier Series ◽  
Ruin Probability ◽  
Risk Model ◽  
Expansion Method ◽  
Nonparametric Estimator ◽  
Insurance Risk ◽  
Numerical Examples ◽  
Complex Fourier Series ◽  
Premium Income ◽  
Insurance Risk Model

This paper considers the estimation of ruin probability in an insurance risk model with stochastic premium income. We first show that the ruin probability can be approximated by the complex Fourier series (CFS) expansion method. Then, we construct a nonparametric estimator of the ruin probability and analyze its convergence. Numerical examples are also provided to show the efficiency of our method when the sample size is finite.

scholarly journals Optimal multistage relaxation with automatic parameter selection

2021 ◽  
Author(s):  
Alvin Wong
Keyword(s):  
Fourier Series ◽  
Relaxation Method ◽  
Accurate Solution ◽  
Convergence Time ◽  
Fourier Series Expansion ◽  
End User ◽  
Local Flow ◽  
Flow Problems ◽  
User Expertise ◽  
Complex Fourier Series

This research developed a numerical method that solves complicated fluid flow problems without requiring end-user expertise with the solver. This method is capable of obtaining a spatially accurate solution in the same time or better as a skilled user with a conventional solver. An explicit preconditioned multigrid solver was used in this research with a multistage relaxation method. The prosposed method utilizies a database with optimized relaxation method parameters for different local flow and mesh conditions. The parameters are optimized for the relaxation such that the error modes in a complex Fourier series expansion of the residual can be quickly reduced. The convergence time and iteration count of this method was compared against the same solver using default input values, as well as a pre-optimized solver, to simulate a skilled user for various geometries. Improvements in both comparisons were demonstrated.

scholarly journals Optimal multistage relaxation with automatic parameter selection

2021 ◽  
Author(s):  
Alvin Wong
Keyword(s):  
Fourier Series ◽  
Relaxation Method ◽  
Accurate Solution ◽  
Convergence Time ◽  
Fourier Series Expansion ◽  
End User ◽  
Local Flow ◽  
Flow Problems ◽  
User Expertise ◽  
Complex Fourier Series

This research developed a numerical method that solves complicated fluid flow problems without requiring end-user expertise with the solver. This method is capable of obtaining a spatially accurate solution in the same time or better as a skilled user with a conventional solver. An explicit preconditioned multigrid solver was used in this research with a multistage relaxation method. The prosposed method utilizies a database with optimized relaxation method parameters for different local flow and mesh conditions. The parameters are optimized for the relaxation such that the error modes in a complex Fourier series expansion of the residual can be quickly reduced. The convergence time and iteration count of this method was compared against the same solver using default input values, as well as a pre-optimized solver, to simulate a skilled user for various geometries. Improvements in both comparisons were demonstrated.

Generalized Heine’s identity for complex Fourier series of binomials

2010 ◽  
Vol 467 (2126) ◽  
pp. 333-345 ◽  
Author(s):  
Howard S. Cohl ◽  
Diego E. Dominici
Keyword(s):  
Fourier Series ◽  
Closed Form ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Complex Fourier Series

In his treatise, Heine ( Heine 1881 In Theorie und Anwendungen ) gave an identity for the Fourier series of the function , with , and z >1, in terms of associated Legendre functions of the second kind . In this paper, we generalize Heine’s identity for the function , with , and , in terms of . We also compute closed-form expressions for some  .

Sign in / Sign up

Export Citation Format

Share Document