Re weighting

The probability distributions P(cf>, ip) of

simulation sampled not only the states of Pp and C5 but also the states of or, ap, and ctp. 

We call this algorithm conjugate gradient re-weighted optimization because the weighting matrix is updated on every iteration (Portniaguine and Zhdanov,... 

The re-weighted regularized conjugate gradient (RCG) method in the space of weighted parameters... 

The regularized steepest ascent directions for the re-weighted RCG method are determined according to the formula... 

We call this algorithm the re-weighted RCG method, because the weighting matrix Wen is updated on each iteration. 

Therefore, on each iteration of the re-weighted RCG method we actually minimize the parametric functional with the different stabilizers, because the weighting matrix Wen is updated on each iteration. In order to insure the convergence of the misfit functional to the global minimum, we use adaptive regularization and decrease the ttn+i, if 7 > 1 ... 

Finally, we present an algorithm for the re-weighted RCG method of solving the linear inverse problem. It can be easily obtained from (5.151), assuming that all Frcchet derivative matrices F, are equal to the matrix of the weighted linear... 

Thus we can construct an iteration process of the re-weighted conjugate gradient method in a similar way as for the gravity problem, using recursions (7.26)-(7.27)... 

The re-weighted and regularized conjugate gradient method in the space of weighted model parameters, (5.152), discussed in Chapter 5 will be used. This method includes weighting of the model parameters, which forms a very important part of... 

Figure 7-7 Results of the focusing inversion obtained by the re-weighted regularized conjugate gradient method.

In numerical implementation of the re-weighted conjugate gradient scheme (5.152) we apply the following inverse weighting matrix according to (5.153) ... 

Figure 7-7 illustrates the focusing inversion result obtained by the re-weighted regularized conjugate gradient method. The plots of the misfit and parametric functionals are shown in the top panel of Figure 7-7. In this case the data fitting after 50 iterations is within 4% nevertheless the inverse image adec uately reconstructs the true model. We can clearly recognize two bodies in this image, and the densities correspond well to the true model.

The minimization problem (10.19) can be solved by the conjugate gradient or re-weighted conjugate gradient methods introduced in Chapter 5. 

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