Research UpdatesJanuary 18th 2016

Noah Paulson

Advisors: Surya R. Kalidindi (GT), David L.

McDowell (GT), Donald S. Shih (Boeing)

1

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

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

𝐶𝑙𝑚𝑛 =

𝑛=1𝑁 𝑇𝑙

𝑚𝑛†𝑔𝑛 𝑓𝑡𝑒𝑠𝑡 𝑔𝑛 sinΦ𝑛

𝑛=1𝑁 𝑇𝑙

𝑚𝑛†𝑔𝑛

𝑇𝑙𝑚𝑛 𝑔𝑛 sinΦ𝑛

Vector of 𝐶𝑙𝑚𝑛

𝐶𝑙𝑚𝑛 =

1

2𝑙 + 1 𝐹𝑍

𝑓𝑡𝑒𝑠𝑡 𝑔 𝑇𝑙𝑚𝑛

†𝑔 𝑑𝑔

𝑑𝑔 = sinΦ3

2𝜋2𝑑𝜙1𝑑Φ𝑑𝜙2

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

Mea

n E

rro

rNumber of Samples

Comparision of Accuracy in GSH Coefficient Determination

midpoint rule integration

Monte-Carlo integration