Analytical Methods for Structured Matrix Computations

The design of fast algorithms is not only about achieving faster speeds but also about retaining the ability to control the error and numerical stability. This is crucial to the reliability of computed numerical solutions. This dissertation studies topics related to structured matrix computations with an emphasis on their numerical analysis aspects and algorithms. The methods discussed here are all based on rich analytical results that are mathematically justified. In chapter 2, we present a series of comprehensive error analyses to an analytical matrix compression method and it serves as a theoretical explanation of the proxy point method. These results are also important instructions on optimizing the performance. In chapter 3, we propose a non-Hermitian eigensolver by combining HSS matrix techniques with a contour-integral based method. Moreover, probabilistic analysis enables further acceleration of the method in addition to manipulating the HSS representation algebraically. An application of the HSS matrix is discussed in chapter 4 where we design a structured preconditioner for linear systems generated by AIIM. We improve the numerical stability for the matrix-free HSS construction process and make some additional modifications tailored to this particular problem

Model-based X-ray CT Image and Light Field Reconstruction Using Variable Splitting Methods.

Model-based image reconstruction (MBIR) is a powerful technique for solving ill-posed inverse problems. Compared with direct methods, it can provide better estimates from noisy measurements and from incomplete data, at the cost of much longer computation time. In this work, we focus on accelerating and applying MBIR for solving reconstruction problems, including X-ray computed tomography (CT) image reconstruction and light field reconstruction, using variable splitting based on the augmented Lagrangian (AL) methods. For X-ray CT image reconstruction, we combine the AL method and ordered subsets (OS), a well-known technique in the medical imaging literature for accelerating tomographic reconstruction, by considering a linearized variant of the AL method and propose a fast splitting-based ordered-subset algorithm, OS-LALM, for solving X-ray CT image reconstruction problems with penalized weighted least-squares (PWLS) criterion. Practical issues such as the non-trivial parameter selection of AL methods and remarkable memory overhead when considering the finite difference image variable splitting are carefully studied, and several variants of the proposed algorithm are investigated for solving practical model-based X-ray CT image reconstruction problems. Experimental results show that the proposed algorithm significantly accelerates the convergence of X-ray CT image reconstruction with negligible overhead and greatly reduces the noise-like OS artifacts in the reconstructed image when using many subsets for OS acceleration. For light field reconstruction, considering decomposing the camera imaging process into a linear convolution and a non-linear slicing operations for faster forward projection, we propose to reconstruct light field from a sequence of photos taken with different focus settings, i.e., a focal stack, using an alternating direction method of multipliers (ADMM). To improve the quality of the reconstructed light field, we also propose a signal-independent sparsifying transform by considering the elongated structure of light fields. Flatland simulation results show that our proposed sparse light field prior produces high resolution light field with fine details compared with other existing sparse priors for natural images.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/108981/1/hungnien_1.pd

Development of scalable linear solvers for engineering applications

The numerical simulation of modern engineering problems can easily incorporate millions or even billions of unknowns. In several applications, particularly those with diffusive character, sparse linear systems with symmetric positive definite (SPD) matrices need to be solved, and multilevel methods represent common choices for the role of iterative solvers or preconditioners. The weak scalability showed by those techniques is one of the main reasons for their popularity, since it allows the solution of linear systems with growing size without requiring a substantial increase in the computational time and number of iterations. On the other hand, single-level preconditioners such as the adaptive Factorized Sparse Approximate Inverse (aFSAI) might be attractive for reaching strong scalability due to their simpler setup. In this thesis, we propose four multilevel preconditioners based on aFSAI targeting the efficient solution of ill-conditioned SPD systems through parallel computing. The first two novel methods, namely Block Tridiagonal FSAI (BTFSAI) and Domain Decomposition FSAI (DDFSAI), rely on graph reordering techniques and approximate block factorizations carried out by aFSAI. Then, we introduce an extension of the previous techniques called the Multilevel Factorization with Low-Rank corrections (MFLR) that ensures positive definiteness of the Schur complements as well as improves their approximation with the aid of tall-and-skinny correction matrices. Lastly, we present the adaptive Smoothing and Prolongation Algebraic MultiGrid (aSPAMG) preconditioner belonging to the adaptive AMG family that introduces the use of aFSAI as a flexible smoother; three strategies for uncovering the near-null space of the system matrix and two new approaches to dynamically compute the prolongation operator. We assess the performance of the proposed preconditioners through the solution of a set of model problems along with real-world engineering test cases. Moreover, we perform comparisons to other approaches such as aFSAI, ILU (ILUPACK), and BoomerAMG (HYPRE), showing that our new methods prove comparable, if not superior, in many test cases