Optimization of Shape Parameters for Radial Basis Functions
Investigates optimization of shape parameters for radial basis functions in a research project conducted by Salome Kakhaia and Mariam Razmadze under the supervision of Ramaz Botchorishvili and Tinatin Davitashvili at Tbilisi State University on August 24, 2018.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
Optimization of Shape Parameters for Radial Basis Functions Salome Kakhaia, Mariam Razmadze Supervisors -Ramaz Botchorishvili Tinatin Davitashvili Department of Mathematics Tbilisi State University August 24, 2018 1
Aim: Define method for shape parameter Identification Improve accuracy of radial basis function Interpolation Applications for SmartLab measurements 2
RBF - Radial Basis Function ? = ?(?) ? = ? ?? - Euclidian distance ?? - centers Gaussian RBF - ? = ? ???? ?? - shape parameters RBF Interpolant, given by a linear combination : ?2=0.01 ?2=0.001 ? Figure 1. ? ? = ??? ? ??,?? ?=? 3
Importance of the Shape Parameter Figure 1 Figure 1 : Choosing Basis Functions and Shape Parameters for Radial Basis Function Methods Michael Mongillo October 25, 2011 4
Optimization of Parameters 1D Individual Shape parameter for each basis Problem and Data orientated Best approximation rate in mid points between nodes More accurate Interpolation process 5
Optimization of Parameters 1D ? ? = ? ? ? ? Error function ? ? - exact ?0 ?1 ?2 ? ? - gaussian RBF ?1/2 ?3/2 RBF Centers : ??, ??, ?? Taylor series expansion around x1/2 in terms of ?/2: ? ? ? ??/? = ?? ? ??/? + ? ??* ? . . 2 ?1 2= ?( ?) assumed : ?? . ? ??/? = ?? ?"(??/?) ??(??/?)for ?( ?3) ?? + ?( ?)) + ?( ??) ?= ?(?"(??/?) + ??? ?? 6
Optimized Shape Parameters 1D ?"(??/?) ??(??/?) ?= ?? ?= ?( ?) ? ?? ?? ?"(??/?) ??(??/?) ?= ?? ? ??/? ~? ??/?~?( ??) ?+ + ?? ? ? ?? ?= ?? 7
Function Approximation 1D ? ? = ??? ??+? ? Mean Squared Error : RBF (optimized parameters) - 0.0004 Polynomial - 0.0128 RBF (random parameters) - 0.0128 8
SmartLab Measurements No2 April 13, 2018 ? ? ? ? Real measured value in node (7) - 57.92 Predicted by RBF Interpolation - 57.23 Predicted by Polynomial - 61.22 10
SmartLab Measurements CO April 13, 2018 ? ? ? ? Real measured value in node (7) - 0.78 Predicted by RBF Interpolation - 0.89 Predicted by Polynomial - 0.91 11
SmartLab Measurements Elements Mean squared Error CO NO2 CH4 Real value 0.78 57.92 2.43 RBF(optimized parameters) 0.89 57.23 2.19 Polynomial 0.91 61.22 2.17 RBF (random parameter ?2=0.0001) 0.91 61.25 2.17 12
Optimization of Parameter 2D (?3,?3) ? ?,? = ? ?,? ? ?,? Taylor series expansion around x0,y0 2+? ?0,?0 ?? ? 2) + O( ?4) + O( ?4) ? ?0,?0 = (? ?0,?0 ?? ? 4 4 ?/2 (?1,?1) (?2,?2) (?0,?0) ?/2 ? = 1 4(??? ?2+??? ?2+ 2?2f ( ?2+ ?2)) + O( ?4) + O( ?4)+ O ?2 ?2 Optimization condition : (?4,?4) (??? ?2+??? ?2+ 2?2f ( ?2+ ?2)) = 0 13
Optimization of Parameter 2D (?3,?3) ??? ?2+??? ?2 2? ( ?2+ ?2) ?2= (?0,?0) ?/2 For higher accuracy: add a new data location (?0,?0) (?1,?1) (?2,?2) (?0,?0) ?/2 Add a zero degree polynomial term to RBF interpolant 4 ? ? = ??? ? ??,?? + ?? (?4,?4) ?=1 14
Optimised Shape Parameter 2D Obtained : 2( ?(?1,?1) + ?(?3,?3) + ?(?2,?2) + ?(?4,?4) 4?(?0,?0)) ( ?2+ ?2) ?(?0,?0) ?2 ? ??,??= ?( ??) + ?( ??) - error in ??,?? given by RBF with optimised shape parameters While: ? ??,??= ?( ??) + ?( ??) - error in ??,??given by Polynomial Interpolation. Achieved high order of convergence around the location ?0,?0 15
Function Approximation 2D ? ?? ? ?,? = (?+(??)?)+ ?+? Optimized Shape Parameter = -0.06 Mean Squared error over the area : RBF Polynomial - 7.697 - 0.286 Exact RBF Polynomial 16
Function Approximation 2D ? ? ?,? = ??? ?? ?+???(?) Optimized Shape Parameter = 0.246 Mean Squared Error over the area : RBF - 0.002 Polynomial - 0.033 Exact RBF Polynomial 17
SmartLab Measurements 2D CO April 4, 2018 18
SmartLab Measurements 2D ?? ? ? ? ? Assumption : surface is flat and extra factors do not influence the results 19
Results in unmeasured locations CO - April 4th, 19:20 Prediction of CO value in A location: RBF Interpolation 0.472 Polynomial Interpolation 2.222 Polynomial Rbf A 20
Multiple Variable Shape Parameters What if we add more parameters? While, Gaussian RBF has 1 shape parameter Multiple variable shape parameters provide adaptive basis Adaptive basis can achieve higher order of accuracy based on optimizing parameters. 21
RBF transformation Multiple variable shape parameters Transformation : Circle ? ? = ? new shape : ? = ? ? ? ? ? ? has multiple shape parameters 22
RBF transformation Multiple variable shape parameters results: 23
References A RBF-WENO finite volume method for hyperbolic conservation laws with the monotone polynomial interpolation method Jingyang Guo, Jae-Hun Jung Radial basis function ENO and WENO finite difference methods based on the optimization of shape parameters Jingyang Guo, Jae-Hun Jung Inventing the Circle Johan Gielis Geniaal bvba , 2003. 24
