Comparing Krylov Subspace Methods for Solving Linear Systems

Algebraic iterative reconstruction methods are based on the idea of constructing a sequence of approximations to the solution of a linear inverse problem using algebraic operations. These methods have been shown to be highly effective in improving the quality of CT images, but they can be computationally intensive and require careful parameter tuning for optimal performance.
Block Kaczmarz Methods
One popular algebraic iterative reconstruction method is the block Kaczmarz method. This method works by dividing the image into smaller blocks and solving a linear system for each block using a randomized iteration. The key insight behind this method is that small blocks can be solved more efficiently than a single large matrix, allowing for faster convergence to the solution.
Analysis of Block Kaczmarz Methods
To understand how block Kaczmarz methods work, it is helpful to consider their geometric interpretation. Each block Kaczmarz step can be viewed as a transformation that maps the current estimate towards the true solution. This transformation is based on the inverse of the small submatrix formed by the current iterate and the measurement vector for that block. By applying this transformation multiple times, we can iteratively improve our estimate of the solution.
Convergence Analysis
The convergence analysis of block Kaczmarz methods involves studying the behavior of the iterates as the number of iterations increases. Key to this analysis is understanding the effect of the randomization on the convergence properties of the method. In general, block Kaczmarz methods are shown to converge quadratically with respect to the number of iterations, which means that the error decreases at a slower rate than the number of iterations. This can lead to instability in some cases, particularly when the iterates are close to the exact solution.
Heuristic Stopping Criterion
To address this stability issue, we propose a heuristic stopping criterion based on the magnitude of the iterates. Specifically, we stop the algorithm after kmax steps if the magnitude of the iterates is below a certain threshold. This allows us to terminate the algorithm early when the iterates are close to the exact solution, reducing the risk of instability.
Conclusion
In conclusion, algebraic iterative reconstruction methods offer a powerful tool for solving linear inverse problems in computerized tomography. By understanding the geometric interpretation and convergence properties of block Kaczmarz methods, we can develop effective heuristics for improving their performance and stability. With these insights, we can demystify these techniques and make them more accessible to a wider range of practitioners.