Fast log-decon with a quasi-Newton solver |

I give the updating formulas of the Hessian as presented by Nocedal (1980). First, we define

, and

As described above, when , the iteration number, obeys , where is the storage limit, we have the BFGS update

(4) | |||

For we have the limited-memory update

(5) | |||

These equations show how the update of the Hessian is calculated.

Usually the L-BFGS method is implemented with a line search for the
step length
to ensure a
sufficient decrease of the misfit function.
Convergence properties of the L-BFGS method are guaranteed if
in equation (2) satisfies the *Wolfe
conditions* (Kelley, 1999):

and are constants to be chosen a priori and . For and we set and as proposed by Liu and Nocedal (1989). Equation (6) is a sufficient decrease condition that all line search algorithms must satisfy. Equation (7) is a curvature condition. The line search algorithm has to be carefully designed since it absorbs most of the computing time. I programmed a line search based on the More and Thuente (1994) method. Because the line search is time consuming, the step length is always tested first. This procedure saves a lot of computing time and is also recommended by Liu and Nocedal (1989). I now give the algorithm used to minimize any objective function involving nonlinear problems.

Fast log-decon with a quasi-Newton solver |

2012-10-29