Residual information and regularization of discrete ill-posed problems

AbstractThe first part of this thesis presents a basic theoretical study of discrete ill-posed problems, where the objective is to find a useful parameter-choice method in the case of non-white noise influencing the problem. Complex white noise is defined, and the relation between the Fourier and SVD bases is studied. It is verified that the first SVD components correspond to low frequency Fourier components, and as the index of the SVD components increases the frequency in the Fourier domain grows.

After these introductory studies follows a description of two approaches to find new
parameter-choice methods, which have turned out to provide new insight for the L-curve - however they are not new methods. These two approaches are: monitoring the normalized inner product and tracking the diagonal of the covariance matrix.

Three new parameter-choice methods are proposed: the normalized cumulative periodogram, the cross-spectral density and the Kruskal-Wallis test. The normalized cumulative periodogram is based on the Fourier transform of the residual and performs best in the case of white noise in the problem. It has not proved very efficient in the case of other types of noise though it does provide information on the spectral behavior of the residual. The cross spectral density is a promising parameter-choice method as it has performed very satisfactory for the problems considered in this work, both in the case of white and non-white noise. Furthermore, it tracks with the two-norm difference between the exact solution and the regularized solution as a function of the regularization parameter. This method is also based on the Fourier transform but the behavior can be explained by use of the SVD. The Kruskal-Wallis test is a statistical rank-test and it has proven efficient in determining a close-to-optimal regularization parameter both in the case of white and non-white noise.

The second part of this thesis is a study of an acoustic ill-posed problem; a vibrating speaker in a box. Two types of noise relevant for the field of acoustics are introduced; sensor mismatch errors and misalignment errors and the proposed parameter-choice methods are tested on the acoustic problem with the different types of noise. Sensor mismatch errors cause amplitude and phase errors to influence the problem and both amplitude and phase errors resemble white noise. Misalignment errors are divided into random and systematic misalignment errors where the random misalignment errors appear to be slightly dominated by ''middle'' SVD components whereas systematic misalignment errors are predominantly low-frequent.
Keywordsdiscrete ill-posed problems, regularization, parameter-choicemethods, inverse acoustic problems
TypeMaster's thesis [Industrial collaboration]
PublisherInformatics and Mathematical Modelling, Technical University of Denmark, DTU
AddressRichard Petersens Plads, Building 321, DK-2800 Kgs. Lyngby
NoteSupervised by Per Christian Hansen, IMM, in collaboration with Bruel & Kjaer, Denmark
Electronic version(s)[zip]
BibTeX data [bibtex]
IMM Group(s)Scientific Computing