Indo-French Spring School




Date
June 11 to 15, 2007.

Venue
University of Saint-Etienne
Faculté des Sciences et Techniques
Salle C205 et salle de séminaires C112
23 rue du Docteur Paul Michelon
42023 SAINT-ETIENNE CEDEX 02
Contact : Florence Vende
Florence Vende.

Addressed to
Master students,
Ph D Students,
Professors of mathematics,
Researchers on related domains.

Financially supported by
Programme ARCUS Rhône-Alpes/Inde,
The French Department of Foreign Affairs.

Subject
Spectral Approximation : theory, applications and numerics.

Chairman
Mrs Laurence Grammont, Maître de conférences in mathematics.
Member of LAMUSE (LAboratoire de Mathématiques de l'Université de Saint-Etienne)

Organizers
Laurence Grammont and Alain Largillier.

Executive secretary
Mrs Florence Vende.

Lecturers
Pr B.V. LIMAYE (Indian Institute of Technology, Bombay),
Pr.R.P. KULKARNI (Indian Institute of Technology, Bombay),
Pr.R. ALAM (Indian Institute of Technology, Guwahati),
Pr.M. SADKANE (Université de Brest),
Pr.M. AHUES (Université de Saint-Etienne).

The courses

Functional Analysis (Pr. B.V. Limaye)

  1. Normed Spaces
  2. Continuity of Linear Maps
  3. Banach Spaces
  4. Compact Linear Maps
  5. Zabreiko's Theorem
  6. Uniform Boudedness Principle
  7. Closed Graph and Open mapping Theorems


Approximate solutions of second kind integral equations (Pr. R.P. Kulkarni)

This lecture use the setting developped by B. Limaye and the results proved by M. Ahues such as various approximation methods.
Il will be proved error bounds and orers of convergence for specifi approximation spaces such as piecewise polynomial spaces. Results regarding interpolation will be proved there.
R. Kulkarni will present two grid methods and extrapolation techniques.

  1. Integral equations of the second kind.
  2. Polynomial ans spline interpolation.
  3. Collocation and Galerkin methods.
  4. Numerical quadrature.
  5. Nystrom method.
  6. Discrete Galerkin method.
  7. A new method.
  8. Implementation specifications.
  9. Two-grid method.


Pseudospectra of matrices and applications (Pr. R. Alam)

  1. Pseudo-spectrum of matrices.
  2. Pertubation estimation for eigenvalues.
  3. Stability of eigendecompositions.


Numerics and algorithms (Pr. M. Sadkane)

  1. GMRES (standard and block versions).
  2. Application to large Sylvester equations.
  3. Computation of eigenvalues and invariant subspaces by Arnoldi's method.
  4. Davidson type methods.
  5. Lyapunov stability and spectral dichotomy methods.


Spectral theory and approximation (Pr. M. Ahues)

  1. Invariant subspaces of a linear operator.
  2. Decomposition into invariant subspaces.
  3. Resolvent operator and Neumann expansions.
  4. Spectral projection.
  5. Spectral approximation.