Residual information and regularization of discrete illposed problems 
 Abstract  The first part of this thesis presents a basic theoretical study of discrete illposed problems, where the objective is to find a useful parameterchoice method in the case of nonwhite 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
parameterchoice methods, which have turned out to provide new insight for the Lcurve  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 parameterchoice methods are proposed: the normalized cumulative periodogram, the crossspectral density and the KruskalWallis 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 parameterchoice method as it has performed very satisfactory for the problems considered in this work, both in the case of white and nonwhite noise. Furthermore, it tracks with the twonorm 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 KruskalWallis test is a statistical ranktest and it has proven efficient in determining a closetooptimal regularization parameter both in the case of white and nonwhite noise.
The second part of this thesis is a study of an acoustic illposed 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 parameterchoice 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 lowfrequent.  Keywords  discrete illposed problems, regularization, parameterchoicemethods, inverse acoustic problems  Type  Master's thesis [Industrial collaboration]  Year  2002  Publisher  Informatics and Mathematical Modelling, Technical University of Denmark, DTU  Address  Richard Petersens Plads, Building 321, DK2800 Kgs. Lyngby  Series  IMMEKS200221  Note  Supervised by Per Christian Hansen, IMM, in collaboration with Bruel & Kjaer, Denmark  Electronic version(s)  [zip]  BibTeX data  [bibtex]  IMM Group(s)  Scientific Computing 
