Regularized steepest descent

Note that this algorithm is equivalent to the regularized steepest descent method, which we will describe in Chapter 5. 

Thus, we can construct an iteration process for the regularized steepest descent as... 

This method uses the same ideas as the conventional conjugate gradient method. However, the iteration process is based on the calculation of the regularized steepest descent directions ... 

We can apply different types of minimization methods (steepest descent, Newton method, etc.) to the straightforward minimization of the corresponding misfit functional /(m). However, all these solutions have many limitations and are very sensitive to small variations of the observed data due to the principal instability of the inverse problem. To overcome this difficulty, we have to apply a regularizing method. 

The steepest descent method for nonlinear regularized least-squares inversion To solve the problem of minimization of the parametric functional using the steepest descent method, let us calculate the first variation of P (m,d), assuming that the operator A(m) is differentiable, so that... 

However, the regularized steepest descent dircc tions are now determined according to the formula for the least-squares method ... 

The first iteration of the steepest descent or any other gradient method is run usually with oq = 0. The initial value of the regularization parameter, Oi, is determined after the first iteration, mi, as a ratio... 

Based on expression (5.122) for the steepest ascent direction, we can derive all the formulae for the regularized steepest descent, Newton and conjugate gradi-( iit iiK thods for minimization of llu Tiklionot ])arametric functional with a yisendo-(.luadratic stabilizer. 

Beck and Berry - have performed a similar analysis on all of the clusters studied in the iV-dependence work of Beck et al. Steepest-descent quenches were performed at regular intervals along trajectories at three energies two... 

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