Krylov Subspace Methods

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Prerequisites include a solid mathematical background and a first course on numerical linear algebra.  The course is aimed primarily at students in computational mathematics, physics, chemistry, and material sceince.. 


Brief review of stationary iterative methods for linear systems

Projection methods: Krylov subspaces

The Lanczos process and the conjugate gradient (CG) method

The method of minimal residuals (MINRES)

The Arnoldi process and the Full Orthogonalization Method (FOM)

The generalized minimal residual method (GMRES)

Other methods (BiCG, QMR, BiCGSTAB), briefly

Preconditioning techniques (ILU, AINV, SPAI, multilevel methods, etc.)

Application to eigenvalue problems and matrix functions

Rational Krylov methods

Educational aims

The main goal is to introduce students to the field of  preconditioned iterative methods for the solution of large linear systems (an ubiquitous problem in scientific computing). Techniques for eigenvalue problems and the computation of selected eigenvalues wll also be described. Attention will be devoted to the convergence theory of Krylov subspace methods.

Bibliographical references

Y. Saad, Iterative Methods for Sparse Linear Systems (2nd Ed.), Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. 

Other references will be provided during the course.