Examination procedure
<p>Seminar talk</p>
Examination procedure notes
<p>The student will be asked to present the results of a project to be chosen together with the instructor.</p>
Prerequisites
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.
Syllabus
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
Bibliographical references
Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, 2003.
Additional references will be provided in class.