Testing the calibration of GSH function coefficients using two different integration techniques: Monte-Carlo integration and midpoint rule integration in 3 dimensions.

1. Research Updates
January 18th 2016
Noah Paulson
Advisors: Surya R. Kalidindi (GT), David L.
McDowell (GT), Donald S. Shih (Boeing)
1

2. GSH Function Calibration Introduction
Goal: Determine optimum method to calibrate GSH coefficients for a known function
• Test function created as a linear combination of symmetrized (hexagonal) GSH basis
functions
• Note that the test function is real-valued
• The goal is to retrieve coefficients with different methods

3. GSH Function Calibration Methods
• Method 1: linear regression
• Samples on grid
• Random samples
• Method 2: orthonormal linear regression
• Samples on grid
• Random samples
• Method 3: Midpoint Integration
• Samples on grid
• Method 4: Monte-Carlo Integration
• Random samples
𝑋† 𝑋𝛽 = 𝑋† 𝑌
Covariance
Matrix
𝐶𝑙
𝑚𝑛
=
𝑛=1
𝑁
𝑇𝑙
𝑚𝑛
†
𝑔 𝑛 𝑓𝑡𝑒𝑠𝑡 𝑔 𝑛 sin Φ 𝑛
𝑛=1
𝑁
𝑇𝑙
𝑚𝑛
†
𝑔 𝑛 𝑇𝑙
𝑚𝑛
𝑔 𝑛 sin Φ 𝑛
Vector of
𝐶𝑙
𝑚𝑛
𝐶𝑙
𝑚𝑛
=
1
2𝑙 + 1 𝐹𝑍
𝑓𝑡𝑒𝑠𝑡 𝑔 𝑇𝑙
𝑚𝑛
†
𝑔 𝑑𝑔
𝑑𝑔 = sin Φ
3
2𝜋2
𝑑𝜙1 𝑑Φ𝑑𝜙2

4. GSH Function Calibration Accuracy
minimum function value -132.44
mean function value 68.49
maximum function value 238.57
standard dev of function values 93.30
0.01
0.1
1
10
100
1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07
MeanError
Number of Samples
Comparision of Accuracy in GSH Coefficient Determination
midpoint rule integration
Monte-Carlo integration