Parameter selection methods for axisymmetric flame tomography through Tikhonov regularization

Abstract

Deconvolution of optically collected axisymmetric flame data is equivalent to solving an ill-posed problem subject to severe error amplification. Tikhonov regularization has recently been shown to be well suited for stabilizing this deconvolution, although the success of this method hinges on choosing a suitable regularization parameter. Incorporating a parameter selection scheme transforms this technique into a reliable automatic algorithm that outperforms unregularized deconvolution of a smoothed data set, which is currently the most popular way to analyze axisymmetric data. We review the discrepancy principle, L-curve curvature, and generalized cross-validation parameter selection schemes and conclude that the L-curve curvature algorithm is best suited to this problem.

Comparingparameter choice methods for regularization of ill-posed problems

“…For several (but not all) problems it has been observed to give a reasonably good and robust parameter choice, and it can cope with correlated errors [1,3,29,67,71,73,75]. However, it is known theoretically that the L-curve method (from [75]) has serious limitations [41,146].…”

“…18, GCV mostly performs well for both Tikhonov and spectral cut-off regularization with white noise. It does not perform so well for Tikhonov regularization in the cases where (µ, ν) equals (1, 3), (1,5) and (3,5). These are the cases affected by saturation (since ν > µ + 1/2), for which the minimizer of the prediction risk is not so close to the minimizer of the X -norm risk.…”

“…Considering the number of parameter choice methods, there are relatively few comparative studies of them [1,3,9,29,45,60,63,67,89,101,113,125,131,141], [71, chap. 7].…”

“…A c c e p t e d M a n u s c r i p t which is of the same form as (3). This approach extends to the representation x = c i φ i with respect to an arbitrary orthonormal basis {φ i } by assuming that c ∼ N (0, Γ).…”

Simultaneous soot temperature and volume fraction measurements in axis-symmetric flames by a two-dimensional modulated absorption/emission technique

“…Daun [29], a Tikhonov regularization can be used to stabilize the deconvolution process. Following their procedure, the value of the regularization parameter Λ is selected so that it is located in the corner of the L-curve obtained when plotting…”