Krylov Subspace Methods and Preconditioning
Prerequisiti
Excellent knowledge of linear algebra and familiarity with the basics of numerical computing and scientific programming, for example in MATLAB, Python, or Julia.
The course is suitable for PhD students (I-II year) and for Master's students with previous exposure to numerical analysis. The course should be of interest also
to PhD student in Chemistry and Physics.
Programma
Introductory remarks
Krylov subspaces
Projection methods
The Lanczos and Arnoldi methods
The conjugate gradient method
Elements of approximation theory
Chebyshev and Faber polynomials
The Full Orthogonalization Method (FOM)
Minimum residual methods (MinRes, GMRES)
Hybrid methods (briefly)
Convergence analysis
The Faber-Manteuffel Theorem (statement only)
Preconditioning techniques (ILU, SPAI, AINV, AMG, etc.)
Rational Krylov methods
Krylov methods for computing eigenvalues and functions of matrices
Obiettivi formativi
To teach the mathematical foundations and computational aspects of Krylov methods (polynomial and rational) for the solution of linear systems and large eigenvalue problems, together with the related preconditioning techniques. At the end of the course, students will be able to use these methods for the solution of scientific problems, and to conduct research activities in this area of numerical analysis.
Riferimenti bibliografici
Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, 2003.
Additional references will be provided in class.