Syllabus
The course will cover primarily projection methods and, in particular, Krylov subspace methods in relation to the problem of moments and related issues. The emphasis will be on interconnections between the relevant topics from various disciplines, including the elements of numerical solution of partial differential equations, approximation theory and functional analysis. Tentative content:
1. Projection processes.
2. Krylov subspaces.
3. Basic methods.
4. Stieltjes moment problem.
5. Orthogonal polynomials, continued fractions, Gauss-Christoffel quadrature and model reduction .
6. Matrix representation and the method of conjugate gradients.
7. Vorobyev method of moments and non-symmetric generalizations.
8. Non-normality and spectral information.
Annotation
The course will deal with mathematical foundations of matrix iterative methods, in particular Krylov subspace methods, in connection with the areas of mathematics and computer science important for understanding basic principles and the state of the art. It will formulate open questions and explain existing misunderstandings going across the fields that prevent deeper understanding and the development of the theory as well as efficient use of the methods in applications.