Comparison of parameter choices in regularization algorithms in case of different information about noise level

A family of rules for the choice of the regularization parameter in the Lavrentiev method in the case of rough estimate of the noise level of the data

“…We made the computations on the sequence of parameters i D r i , i D 0; 1; : : : and r D 1:1. The parameter˛i is found as the minimizer of the corresponding functions N g k .˛/ (see (5.2)) in the interval OE˛; 1, where˛is the largest˛i for which the value of N g k .˛i / is C times larger than its value at its current minimum (see [8,10,24]). We used the value C D 1:2, in addition we found the first local minimum (corresponding to C D 1) and the global minimum (corresponding to C D 1).…”

“…Numerical examples in Section 7 show that the rules from the proposed family work well also in the case the noise level is under-or overestimated many times. Note that extensive numerical comparison of other parameter choice rules can be found in [3,[8][9][10]24]. …”

On parameter choice in the regularization of ill-posed problems with rough estimate of the noise level of the data

“…Investigation of the case 3 is motivated by the fact that classical rules for the choice of the regularization parameter that use the noise level (e.g. the discrepancy principle [18,26]) need an exact noise level: they fail in case of underestimated noise level and give large error in case of very moderate overestimation of the noise level (see [8]). On the other hand, for heuristic rules that do not use the noise level, the convergence of approximate solutions as y − y * → 0 can not be guaranteed (see [1]).…”

“…. ; q < 1, for example, q = 0.9; find α as the first α i for which From [8,10,11,20] it is known that this rule guarantees the convergence x α − x * → 0 as δ → 0, provided that lim y−y * /δ ≤ const, and the order optimal error estimate x α −x * ≤ constδ p/(p+1) holds for x * −x 0 ∈ R((A * A) p/2 ) for all p ≤ 2m, if y − y * ≤ δ and c 1 ≥ 0.24.…”

“…In some problems this rule chooses a too small α in the T-method and a too large n in the L-method. We propose to modify the functions to be minimized and to use the local minimizer instead of the global minimizer (see [6,8,20]). …”