Iterative Solution of Symmetric Quasi-Definite Linear Systems

Numerous applications, including computational optimization and fluid dynamics, give rise to block linear systems of equations said to have the quasi-definite structure. In practical situations, the size or density of those systems can preclude a factorization approach, leaving only iterative methods as the solution technique. Known iterative methods, however, are not specifically designed to take advantage of the quasi-definite structure.

This book discusses the connection between quasi-definite systems and linear least-squares problems, the most common and best understood problems in applied mathematics, and explains how quasi-definite systems can be solved using tailored iterative methods for linear least squares (with half as much work!). To encourage researchers and students to use the software, it is provided in MATLAB, Python, and Julia.

The authors provide a concise account of the most well-known methods for symmetric systems and least-squares problems, research-level advances in the solution of problems with specific illustrations in optimization and fluid dynamics, and a website that hosts software in three languages.

Audience: This book is intended for researchers and advanced graduate students in computational optimization, computational fluid dynamics, computational linear algebra, data assimilation, and virtually any computational field in which saddle-point systems occur. The software should appeal to all practitioners, even those not technically inclined.

Contents: List of Algorithms; List of Theorems; Preface; Chapter 1: Introduction; Chapter 2: Preliminaries; Chapter 3: Overview of Existing Direct and Iterative Methods; Chapter 4: Fundamental Processes; Chapter 5: Iterative Methods Based on Reduced Equations; Chapter 6: Full-Space Iterative Methods; Chapter 7: Software and Numerical Experiments; Chapter 8: Discussion and Open Questions; Bibliography; Index.

Dominique Orban is associate professor of computational mathematics at École Polytechnique in Montréal and a member of the GERAD research center for decision analysis. He has a background in continuous optimization and has a keen interest in the linear algebra problems occurring at the core of methods for optimization. He believes one does not fully understand a numerical method until it has been implemented in a programming language. He has authored over 40 papers and several software packages on optimization and linear algebra and is constantly looking for better implementations of crucial methods in those fields. Dominique is a member of SIAM, the Mathematical Optimization Society, and the Association for Computing Machinery.

Mario Arioli is adjunct professor of at the mathematics and computer science department of Emory University. After he retired from Rutherford Appleton Laboratory, UK, he has been visiting scientist and professor at several universities in Germany (TU Berlin and Bergische University of Wuppertal) and in France (Institut de Mathématiques de Toulouse, Université Paul Sabatier, Toulouse, and INP-ENSEEIHT, Toulouse). He has a background in numerical analysis and his interests range from numerical linear algebra and round-off error analysis of algorithms to graph theory and the numerical solution of PDEs. He has authored more than 60 papers and contributed to the HSL software library. Mario is a member of SIAM.