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:

$${\mathbf{x}}_{\mathbf{1}}\mathbf{=}\mathbf{A}{\mathbf{x}}_{\mathbf{0}}$$

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

$${\lambda }_0=\frac{{\mathbf{x}}^{\mathrm{\top }}\mathbf{A}\mathbf{x}}{{\mathbf{x}}^{\mathrm{\top }}\mathbf{x}}$$

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

$${\beta }_0=\gamma \frac{{\mathbf{x}}^{\mathrm{\top }}{\mathbf{J}}^{\mathrm{\top }}\mathbf{J}\mathbf{x}}{{\mathbf{x}}^{\mathrm{\top }}{\mathbf{W}}^{\mathrm{\top }}\mathbf{W}\mathbf{x}}$$

Return type:

float

Returns:

beta0