simpeg.directives.PairedBetaEstimate_ByEig.initialize#

PairedBetaEstimate_ByEig.initialize()[source]#

The initial beta is calculated by comparing the estimated eigenvalues of \(J^T J\) and \(W^T W\). To estimate the eigenvector of A, we will use one iteration of the Power Method:

\[\mathbf{x_1 = A x_0}\]

Given this (very course) approximation of the eigenvector, we can use the Rayleigh quotient to approximate the largest eigenvalue.

\[\lambda_0 = \frac{\mathbf{x^\top A x}}{\mathbf{x^\top x}}\]

We will approximate the largest eigenvalue for both JtJ and WtW, and use some ratio of the quotient to estimate beta0.

\[\beta_0 = \gamma \frac{\mathbf{x^\top J^\top J x}}{\mathbf{x^\top W^\top W x}}\]
Return type:

float

Returns:

beta0