Block diagonal and schur complement preconditioners for block-toeplitz systems with small size blocks

In this paper we consider the solution of Hermitian positive definite block-Toeplitz systems with small size blocks. We propose and study block diagonal and Schur complement preconditioners for such block-Toeplitz matrices. We show that for some block-Toeplitz matrices, the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers where this fixed number depends only on the size of the block. Hence, conjugate gradient type methods, when applied to solving these preconditioned block-Toeplitz systems with small size blocks, converge very fast. Recursive computation of such block diagonal and Schur complement preconditioners is considered by using the nice matrix representation of the inverse of a block-Toeplitz matrix. Applications to block-Toeplitz systems arising from least squares filtering problems and queueing networks are presented. Numerical examples are given to demonstrate the effectiveness of the proposed method. © 2007 Society for Industrial and Applied Mathematics.published_or_final_versio

How Ordinary Elimination Became Gaussian Elimination

Newton, in notes that he would rather not have seen published, described a process for solving simultaneous equations that later authors applied specifically to linear equations. This method that Euler did not recommend, that Legendre called "ordinary," and that Gauss called "common" - is now named after Gauss: "Gaussian" elimination. Gauss's name became associated with elimination through the adoption, by professional computers, of a specialized notation that Gauss devised for his own least squares calculations. The notation allowed elimination to be viewed as a sequence of arithmetic operations that were repeatedly optimized for hand computing and eventually were described by matrices.Comment: 56 pages, 21 figures, 1 tabl

Repetitive process control of additive manufacturing with application to laser metal deposition

Additive Manufacturing (AM) is a set of manufacturing processes which has promise in the production of complex, functional structures that cannot be fabricated with conventional manufacturing and the repair of high-value parts. However, a significant challenge to the adoption of additive manufacturing processes to these applications is proper process control. In order to enable closed-loop process control compact models suitable for control design and for describing the layer-by-layer material addition process are needed. This dissertation proposes a two-dimensional modeling and control framework, with an application to a specific metal-based AM process, whereby the deposition of the current layer is affected by both in-layer and layer-to-layer dynamics, both of which are driven by the state of the previous layer. The proposed modeling framework can be used to create two-dimensional dynamic models for the analysis of layer-to-layer stability and as a foundation for the design of layer-to-layer controllers for AM processes. In order to analyze the stability of this class of systems, linear repetitive process results are extended enabling the treatment of the process model as a two-dimensional analog of a discrete time system. For process control, the closed-loop repetitive process is again treated as a two-dimensional analog of a discrete time system for which controllers are designed. The proposed methodologies are applied to a metal-based AM process, Laser Metal Deposition (LMD), which is known to exhibit layer-to-layer unstable behavior and is also of significant interest to high-value manufacturing industries --Abstract, page iii

NASA Tech Briefs, July 1995

Topics include: mechanical components, electronic components and circuits, electronic systems, physical sciences, materials, computer programs, mechanics, machinery, manufacturing/fabrication, mathematics and information sciences, book and reports, and a special section of Federal laboratory computing Tech Briefs