The iterated Golub-Kahan-Tikhonov method

Abstract

The Golub-Kahan-Tikhonov method is a popular solution technique for large linear discrete ill-posed problems. This method first applies partial Golub-Kahan bidiagonalization to reduce the size of the given problem and then uses Tikhonov regularization to compute a meaningful approximate solution of the reduced problem. It is well known that iterated variants of this method often yield approximate solutions of higher quality than the standard non-iterated method. Moreover, it produces more accurate computed solutions than the Arnoldi method when the matrix that defines the linear discrete ill-posed problem is far from symmetric. This paper starts with an ill-posed operator equation in infinite-dimensional Hilbert space, discretizes the equation, and then applies the iterated Golub-Kahan-Tikhonov method to the solution of the latter problem. An error analysis that addresses all discretization and approximation errors is provided. Additionally, a new approach for choosing the regularization parameter is described. This solution scheme produces more accurate approximate solutions than the standard (non-iterated) Golub-Kahan-Tikhonov method and the iterated Arnoldi-Tikhonov method.

Figures (5)

• Figure 1: Example \ref{['ex4']} - Exact solution $x^{\dagger}_{n}$ (upper left) and observed image $y_n^{\delta}$ (upper right). Approximate solutions $x^{\delta,\ell}_{\alpha,n,i}$ computed by the iGKT method with $\ell=80$ and $i=2000$ with $\alpha$ determined using \ref{['alpha_iGKT']} (lower left) and $x^{\delta,\ell}_{\alpha,n,i}$ computed by the iGKT method with $\ell=40$ and $i=2000$ with $\alpha$ determined using \ref{['alpha_iGKTimp']} (lower right). Here $n=256$ and $\xi= 1\%$.
• Figure 2: Example \ref{['ex3']} - Exact solution $x^{\dagger}_{n}$ (Up-Left) and observed image $y_n^{\delta}$ (upper right). Approximate solutions $x^{\delta,\ell}_{\alpha,n,i}$ computed by the iGKT method with $\ell=60$, $i=500$, and $\alpha$ determined by solving \ref{['alpha_iGKT']} (lower left) and $x^{\delta,\ell}_{\alpha,n,i}$ computed by the iGKT method with $\ell=30$, $i=500$, and $\alpha$ determined using \ref{['alpha_iGKTimp']} (lower right). Here $n=256$ and $\xi= 1\%$.
• Figure 3: Example \ref{['ex1']} - (Left) Values of $\frac{\|(\tilde{\mathcal{R}}_{\ell}T_n-\tilde{T}_n^{(\ell)})x_n^{\dagger}\|_2}{\|\tilde{\mathcal{R}}_{\ell}T_nx_n^{\dagger}\|_2}$ (continuous line) and $\frac{\|(\mathcal{R}_{\ell}T_n-T_n^{(\ell)})x_n^{\dagger}\|_2}{\|\mathcal{R}_{\ell}T_nx_n^{\dagger}\|_2}$ (dashed line) in logarithmic scale. (Right) Values of $\frac{\tilde{h}_{\ell}}{\|T_n\|_2}$ (continuous line) and $\frac{h_{\ell}}{\|T_n\|_2}$ (dashed line). Here $\ell=1,\ldots,30$.
• Figure 4: Example \ref{['ex1']} - True image $x^{\dagger}_{n}$ (upper left) and observed image $y_n^{\delta}$ (upper right). Approximate solutions $x^{\delta,\ell}_{\alpha,n,i}$ computed by the iGKT method with $i=200$ and $\alpha$ determined using \ref{['alpha_iGKTimp']} (lower left) and $x^{\delta,\ell}_{\alpha,n,i}$ computed by the iAT method with $i=100$ and $\tilde{\alpha}$ determined using \ref{['alpha_iATimp']} (lower right) with $\ell=20$. Here $n=256$ and $\xi= 2\%$.
• Figure 5: Example \ref{['ex5']} - Exact solution $x^{\dagger}_{n}$ (Up-Left) and observed data (sinogram) $y_n^{\delta}$ (upper right). Approximate solutions $x^{\delta,\ell}_{\alpha,n,i}$ computed by the iGKT method with $\alpha$ determined by \ref{['alpha_iGKT']} (lower left) and $x^{\delta,\ell}_{\alpha,n,i}$ computed by the iGKT method with $\alpha$ determined by \ref{['alpha_iGKTimp']} (lower right) with $\ell=12$ and $i=50$. Here $n=256$ and $\xi= 1\%$.

Theorems & Definitions (24)

• Proposition 1
• Proposition 2
• Corollary 3
• Proposition 4
• Proposition 5
• Corollary 6
• Proposition 7
• proof
• Remark 1
• Example 1