A New Method for TSVD Regularization Truncated Parameter Selection

Zemin Wu,Caibing Xiang,Yude Tong,Shaofeng Bian

doi:10.1155/2013/161834

Abstract

The truncated singular value decomposition (TSVD) regularization applied in ill-posed problem is studied. Through mathematical analysis, a new method for truncated parameter selection which is applied in TSVD regularization is proposed. In the new method, all the local optimal truncated parameters are selected first by taking into account the interval estimation of the observation noises; then the optimal truncated parameter is selected from the local optimal ones. While comparing the new method with the traditional generalized cross-validation (GCV) andLcurve methods, a random ill-posed matrices simulation approach is developed in order to make the comparison as statistically meaningful as possible. Simulation experiments have shown that the solutions applied with the new method have the smallest mean square errors, and the computational cost of the new algorithm is the least.

Highlights

Ill-posed problem is widespread in the field of geophysical survey, such as GNSS rapid positioning, precise orbit solution of spacecraft, and downward continuation of airborne gravity [1,2,3,4,5,6,7]
Methods based on singular value decomposition (SVD) have drawn extensive attention, thanks to the numerical stability of their solutions [7]
The key point of local optimal truncated parameter selection is the estimation of the observation noises

Summary

Introduction

Ill-posed problem is widespread in the field of geophysical survey, such as GNSS rapid positioning, precise orbit solution of spacecraft, and downward continuation of airborne gravity [1,2,3,4,5,6,7]. Hanckowiak [12] and Lin et al [13] studied this problem Among those approaches, methods based on singular value decomposition (SVD) have drawn extensive attention, thanks to the numerical stability of their solutions [7]. Wiggins [14] and Xu [15] intensively studied the truncated singular value decomposition (TSVD) regularization method. In ill-posed problem, some singular values of the coefficient matrix approximate to 0, the least square estimation will enormously amplify the observation noises and degrade the precision. In TSVD regularization, items containing these small singular values are discarded to maintain the stability of the solution. The key point of the TSVD is how to select a proper truncated parameter to get the smallest mean square error of the solution.

Traditional Truncated Parameter Selecting Method

A New Truncated Parameter Selecting Method

Random Simulation Experiment

Conclusions