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) and
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 [
The standard form of the observation equation is (nonstandard form can be transformed to a standard one) as follows:
The condition number of normal matrix (
According to application experiences, if
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. Setting a small positive number
Normally, the mean square error (MSE) is used to evaluate the quality of ill-posed problem 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. Many scholars have extensively studied this problem. Golub et al. [
Based on statistical point of view, Golub et al. [
The disadvantage of GCV method is that the GCV function converges very slowly in some cases. It may lead to the GCV function minimization while
Hansen [
The original discreet point which lies closest to the maximum curvature point from the left side is selected. Its truncated serial number
Further descriptions of the
In this section, we develop a new approach to selecting the truncated parameter. We focus on the MSE of the solution, making some proper estimation, and finally derive the conditions that the optimal truncated parameter should meet. In this section and the subsequent sections,
The MSE of TSVD solution is
Taking the singular values decomposition into consideration, the least square solution of the observation equation can be written as
According to (
Inserting (
As mentioned above, if
Denoting
As
Inserting (
Equation (
Then, we estimate
We have to treat
Scheme of
In order to overcome this shortcoming, the interval estimation of
To get a proper interval estimation of
The estimation of the observation noises is
Thus, according to the definition of
Considering the
Selecting
Equation (
There is an undetermined parameter in inequalities (
If there is only one local optimal value, we regard it as the optimal value. If not, assuming that there are
The mean square error function can be transformed as
Assuming that
According to inequalities (
If we regard
As
According to (
In Sections Begin else, skip this step. else, make pairwise comparisons of the local optimal truncated parameters according to ( End
Although we can use a particular coefficient matrix
In order to make the comparison of the three methods as persuasive as possible, we have to simulate the coefficient matrices by taking into account the three issues mentioned above. According to (
Theoretically, the dimensions
Utilizing the Gram-Schmidt method, unitary matrix
To make the simulation meaningful, we sample 100 random generated coefficient matrices for our simulation experiment. The dimensions and condition numbers of these matrices are shown in Figures
Dimensions and condition numbers of the 100 coefficient matrices randomly generated, respectively.
In this section, we will compare the performances of TSVD solutions applied with the GCV,
Results comparison between GCV and the new methods in two observation noise levels.
Seen in Figures
Results comparison between
As seen in Figures
(a) time consuming comparisons of the three methods; (b) time consuming of the new algorithm in two noise levels.
From Figures
A new truncated parameter selecting method is proposed in this paper. Focusing on how to decrease the MSE of the solution, we divide the truncated parameter selection into two parts: local optimal and optimal truncated parameter selection. Detailed analyses are made and finally the equations to select the local optimal and the optimal truncated parameter are obtained. Although the derivation of the new algorithm is somehow trivial, its execution is quite simple and efficient.
The key point of local optimal truncated parameter selection is the estimation of the observation noises. To overcome the inaccuracy of point estimation, we apply the interval estimation with the confidence level
The selection of optimal truncated parameter is based on the local optimal truncated parameters. We regard the parameter vector as a random variant vector, making use of the stability of its regularization solutions, and finally derive the equations of optimal truncated parameter selection.
A random ill-posed matrices simulation approach is developed and a great deal of experiments is made in Section
This work was financially supported by the National Key Basic Research and Development Program (2012CB719902) and the National Natural Science Foundation of China (nos. 41274013, 41374082, and 61203193).