Radial Basis Functions Salome Kakhaia Mariam Razmadze Supervisors Ramaz Botchorishvili Tinatin Davitashvili Department of Mathematics Tbilisi State University 1 August 24 2018 ID: 809451
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Slide1
Optimization ofShape Parameters forRadial Basis Functions
Salome Kakhaia, Mariam Razmadze Supervisors - Ramaz Botchorishvili Tinatin DavitashviliDepartment of MathematicsTbilisi State University
1
August 24, 2018
Slide22Aim:
Define method for shape parameter Identification Improve accuracy of radial basis function InterpolationApplications for SmartLab measurements
Slide3RBF - Radial Basis Function
- Euclidian distance
- centers
Gaussian RBF -
- shape parameters
RBF Interpolant, given by a linear combination :
=0.01
=0.001
3
Figure 1.
Slide4Importance of the Shape Parameter
4Figure 1 :
Choosing Basis Functions and Shape Parameters for Radial Basis Function Methods Michael Mongillo October 25, 2011
Figure 1
Slide55 Individual Shape parameter for each basis
Problem and Data orientated Best approximation rate in mid points between nodesMore accurate Interpolation process
Optimization of Parameters 1D
Slide6Error function
- exact
- gaussian RBF
RBF Centers
:
,
,
Taylor series expansion around
in terms of
:
*.. assumed :
.
for
6
Optimization of Parameters 1D
Slide77
Optimized Shape Parameters 1D
Slide8Function Approximation 1D8
Mean Squared Error :
RBF (optimized parameters) - 0.0004
Polynomial - 0.0128
RBF (random parameters) - 0.0128
Slide99
SmartLab Measurements
Slide1010
SmartLab Measurements
Real measured value in node
(7)
- 57.92
Predicted by RBF Interpolation - 57.23
Predicted by Polynomial - 61.22
No
2
April 13, 2018
Slide1111
SmartLab Measurements
Real measured value in node
(7)
- 0.78
Predicted by RBF Interpolation - 0.89
Predicted by Polynomial - 0.91
CO
April 13, 2018
Slide1212
ElementsMean squared Error
CONO2CH4
Real value
0.78
57.92
2.43
RBF
(optimized parameters)
0.8957.232.19Polynomial 0.9161.222.17RBF (random parameter0.0001) 0.9161.252.17 ElementsMean squared ErrorCONO2CH4Real value0.7857.922.43RBF (optimized parameters)
0.8957.232.19
Polynomial 0.91
61.222.170.9161.252.17
SmartLab Measurements
Slide13Taylor series expansion around
+
+
+
Optimization
condition :
+
13
Optimization of Parameter 2D
Slide14For higher accuracy: add a new data location
Add a zero degree polynomial term to RBF interpolant
14
Optimization of Parameter 2D
Slide15Optimised Shape Parameter 2D
Obtained :
=
-
error in
While:
=
-
error in
given by Polynomial Interpolation.
Achieved high order of convergence around the location
15
Slide16Exact
RBF
Polynomial
16
Optimized Shape Parameter = -0.06
Mean Squared error over the area :
RBF - 0.286
Polynomial - 7.697
Function Approximation 2D
Slide17Exact
RBF Polynomial
17
Function Approximation 2D
Optimized Shape Parameter = 0.246
Mean Squared Error over the area :
RBF - 0.002
Polynomial - 0.033
Slide1818
CO
April 4, 2018
SmartLab Measurements 2D
Slide1919
Assumption :
surface is flat and extra factors do not influence the results
SmartLab Measurements 2D
Slide2020
Results in unmeasured locations
A
Polynomial
CO - April 4
th
, 19:20
Prediction of CO value in
A
location:
RBF Interpolation – 0.472Polynomial Interpolation – 2.222Rbf
Slide21What if we add more parameters?While, Gaussian RBF has 1 shape parameter
Multiple variable shape parameters provide adaptive basisAdaptive basis can achieve higher order of accuracy based on optimizing parameters.
21
Multiple Variable Shape Parameters
Slide22RBF transformationMultiple variable shape parameters
22Transformation :Circle
new shape :
has multiple shape parameters
Slide2323
RBF transformation
Multiple variable shape parameters
results:
Slide24References
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
Slide25Thank you !