Inexact Generalized Golub-Kahan Methods for Large-Scale Bayesian Inverse Problems

TL;DR

This paper introduces an efficient approach by developing an inexact generalized Golub-Kahan decomposition that can incorporate varying degrees of inexactness in the forward model to solve large-scale generalized Tikhonov regularized problems.

Abstract

Solving large-scale Bayesian inverse problems presents significant challenges, particularly when the exact (discretized) forward operator is unavailable. These challenges often arise in image processing tasks due to unknown defects in the forward process that may result in varying degrees of inexactness in the forward model. Moreover, for many large-scale problems, computing the square root or inverse of the prior covariance matrix is infeasible such as when the covariance kernel is defined on irregular grids or is accessible only through matrix-vector products. This paper introduces an efficient approach by developing an inexact generalized Golub-Kahan decomposition that can incorporate varying degrees of inexactness in the forward model to solve large-scale generalized Tikhonov regularized problems. Further, a hybrid iterative projection scheme is developed to automatically select Tikhonov regularization parameters. Numerical experiments on simulated tomography reconstructions demonstrate the stability and effectiveness of this novel hybrid approach.

Figures (6)

• Figure 1: Set up of parallel-beam X-ray CT where the detector collects data from a finite number of projection angles $\boldsymbol{\theta}$ and the data from a set of $\boldsymbol{\theta}$ constitute the sinogram.
• Figure 2: Both (a) and (b) present comparisons of standard, generalized, inexact, and inexact generalized iterative methods for image reconstruction. The legend LSQR corresponds to the standard method, genLSQR corresponds to the generalized method, iLSQR cooresponds to the inexact method, and igenLSQR correponds to the inexact generalized method. In (a) we present the image reconstructions at the end of $50$ iterations, and in (b) we provide the relative errors norms along the iterations.
• Figure 3: Comparisons of standard, generalized, inexact, and inexact generalized hybrid approach with optimal regularization parameter. HyBR corresponds to the hybrid method based on GK, genHyBR corresponds to the generalized hybrid method based on genGK, iHyBR corresponds to the inexact hybrid method based on iGK, and igenHyBR corresponds to the inexact generalized hybrid method based on igenGK. In (a) we present the image reconstructions at the end of $50$ iterations, (b) provides the relative errors norms along the iterations.
• Figure 4: Comparison of optimal, DP, and WGCV methods in choosing the Tikhonov regularization parameter $\lambda$. The relative error norms are computed along iterations where each regularization method is incorporated into igenHyBR in computing solutions.
• Figure 5: Comparisons of reconstruction results for sinograms acquired at inexact angles, solved without regularization. (a) starts from smaller degree of inexactness ($\alpha_1 = 10^{-1}$), (b) starts from larger degree of inexactness ($\alpha_1 = 1$).