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Preconditioning Techniques for Large Linear Systems
De 30 de Julho a 03 de Agosto de 2012
Preconditioning Techniques for Large Linear Systems
Prof. Michele Benzi, Emory University, Atlanta, GA
Resumo
An overview of modern techniques for solving large sparse systems of li- near equations using preconditioned Krylov subspace methods. The course will emphasize the general mathematical principles underlying the design of efficient and robust solution techniques for large linear systems.
Descrição After a review of basic linear algebra notions, we recall some standard solution methodsforlinearsystems(LUandCholeskyfactorization, classicaliterativemethods, stationary iterations, Krylov subspace methods). Next, we discuss a variety of general- purpose, algebraic methods (including incomplete factorization and sparse approxi- mate inverses). A few special-purpose solvers for large linear systems in saddle point form arising from specific applications, mostly in Computational Fluid Mechanics, are also presented. Additional topics may include solvers for complex matrices and singular linear systems.
Tópicos
1. Preliminaries (2 hours) (a) Review of matrix theory (b) Direct methods (LU and Cholesky factorizations) (c) Theclassicaliterations(Jacobi, Gauss-Seidel, SOR,SSOR,Cimmino, Kacz- marz; block variants)
2. Krylov subspace methods (2 hours) (a) Krylov subspaces (b) "Optimal"methodsforsymmetricmatrices(CG,MINRES)andfornonsym- metric ones (GMRES); Faber-Manteuffel Theorem (c) Other methods for nonsymmetric systems (QMR, BiCGSTAB) (d) Remarks on the convergence properties of Krylov methods (e) Flexible and inexact variants (f) Stopping criteria
3. Algebraic preconditioners for general sparse matrices (4 hours) (a) Generalities on preconditioning (b) Incomplete Cholesky factorizations (c) Incomplete LU factorizations (ILU(0), ILU(k), ILUT) (d) Existence and stability issues (e) Reordering techniques (Red/Black, RCM, MD, GND...) (f) Approximate inverse preconditioners: SPAI and AINV (g) Robust incomplete factorizations: RIF, BIF
4. Preconditioners for saddle point problems (2 hours) (a) Examples and properties of saddle point systems (b) Solution algorithms for saddle point problems (c) Augmented Lagrangian (AL) formulations (d) AL-based preconditioners for incompressible flow problems
5. Additional topics (2 hours) (a) Preconditioners for complex linear systems (b) Restricted Additive Schwarz preconditioning (c) Singular linear systems, Markov chains
Informações Complementares
Local: sala C-116, bloco C, Centro de Tecnologia, UFRJ, Ilha do Fundão. Datas: 30 de julho a 3 de agosto de 2012.
Horário: diariamente, de 10h às 12h. (Apenas em 30/08, haverá uma segunda aula de 14h às 16h.)
Organização: Programa de Pós-Graduação em Matemática Aplicada, Instituto de Ma- temática, UFRJ, Departamento de Matemática Aplicada, Instituto de Matemática e Estatística, UERJ e Programa de Pós-Graduação em Engenharia Mecânica, Faculdade de Engenharia, UERJ .
