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Center for Nonlinear Studies |
Regularization techniques have been widely used in many application domains for the solution of ill-posed inverse problems. Their major function is to penalize the oscillation in the resulting solution caused by the ill-posedness of the problems, while preserving enough fidelity. In this talk, I will go over two different types of regularization techniques including: Tikhonov regularization and Total Variation regularization. Problems of image restoration and image reconstruction will be utilized as testing problems. Three topics from different numerical aspects will be covered in this talk. Firstly, I will consider the parallelization of the least squares regularization problem by using multisplitting which is a method of domain decomposition. The algorithm is composed of global and local iterations. The global iteration requires that solutions are obtained in the local iteration to solve the sub problem system for multiple right-hand sides, where each right-hand side depends on the current global iterative solution. The algorithm is made more efficient by using updating of the initial local Krylov subspaces per problem with minimal restarts. We also study both the corresponding convergence and computational cost. The strength of our algorithm is illustrated by applying it to both image reconstruction and restoration problems. Secondly, I will analyze the choice of the regularization parameter, which plays a central role in the correct implementation of different types of regularization techniques, such as, Tikhonov or Total Variation. The Unbiased Predictive Risk Estimator (UPRE) method is first reviewed as applied for optimal parameter selection for Tikhonov Regularization. Then we discuss the difficulties for the Total Variation case: non-linearity and large scale computation. We propose a linearization scheme and a Krylov subspace approximation method to bypass these two difficulties for Total Variation regularization. Several image deblurring and denoising problems are presented to show the feasibility and accuracy of the algorithm. Thirdly, the emerging Graphical Processing Unit (GPU) has shown its enormous computability in high performance computation (HPC). Many efforts has been devoted to the efficient algorithm design of Krylov subspace solvers for large scale linear systems. In this part, I will introduce a well-known numerical method, Lanczos-Galerkin projection originated from the numerical analysis field to the GPU community. We shows that this method can fit well with the single precision accuracy of the current GPUs. One of the benefit gaining from this method is that, it overcomes the large amount of matrix vector (mat-vec) multiplications, which is usually the bottleneck for an efficient GPU based Krylov iterative algorithm. We discuss two famous solvers in this paper: Conjugate Gradient (CG) and LSQR. Numerical results running on GPU are provided to support our proposed algorithms.
Host: Brendt Wohlberg