The use of Global Terrain Method elements (OCFE)

Basic concept of Global Terrain Method (Lucia and Feng,2002)

A method to find all physically meaningful solutions and singular points for a given (non) linear system of equations (F=0)

Based on intelligent movement along the valleys and ridges of the least-squares function of the system (FTF)

The task : tracing out lines that ‘connect’ the stationary points of FTF.

Mathematical background

Valleys and ridges in the terrain of FTF could be represented as the solutions (V) to:

Applying KKT conditions to the above optimization problem we get the following optimization problem

Thus solutions or stationary points are obtained as solutions to an eigen-value problem

Initial movement

It can be calculated from M or H using Lanzcos or some other eigenvalue-eigenvector technique (Sridhar and Lucia)

Direction

Downhill: Eigendirection of negative Eigenvalue

Uphill: Eigendirection of positive Eigenvalue

V = opt gTg such that FTF = L, for all L єL

F: a vector function, g = 2JTF, J: Jacobian matrix, L: the level-set of all contours

Hi : The Hessian for the i th function