Numerical Analysis

Numerical Analysis courses at SMA

(last edited 13.06.2016)

Contact person: Professor Marco Picasso

Bachelor (3rd semester) :

Bachelor (5th and 6th semesters) :

Master (not always given):

Description :

Computational Science is a multidisciplinary field that uses computers to solve complex problems. Computational science includes mathematical modeling, numerical analysis, scientific computing, algorithms, computer science. Applications of computational science are numerical simulations (reconstructions/prediction of past/future events), model fitting, data analysis, optimization of known scenarios. Computational Science has strong mathematical links with analysis, probability and statistics.

Bachelor students interested in the Numerical Analysis/Scientific Computing/Computational Science track may choose:

- the Master of Mathematics with the Applied Mathematics orientation (90 credits),
- the Master of Applied Mathematics (ingéniérie mathématique in French, industrial internship, 120 credits),
- the Master of Computational Science and Engineering (120 credits).

Bachelor courses :

Numerical analysis. Stability, consistance, convergence of numerical methods, interpolation, numerical integration and differentiation, direct and iterative methods for the resolution of large linear systems, numerical resolution of non linear (system of) equations, numerical methods for ordinary differential equations.

Numerical approximation of PDEs 1. Partial Differential Equations, Finite element method, Galerkin approximation, convergence analysis.

Advanced numerical analysis. Runge-Kutta methods. Constrainted and unconstrainted numerical optimization.

Master courses :

Computational linear algebra. Krylov subspace methods. Singular value problems. Direct sparse factorizations. Efficient implementation on modern computer architectures. Matrix functions. Low-rank matrix and tensor approximation.

Numerical approximation of PDEs 2.  A priori and a posteriori error estimates, for elliptic, parabolic and hyperbolic pde's, adaptive algorithms.

Numerical integration of dynamical systems. Numerical integration of multi-scale or stiff  differential equations. Numerical methods preserving geometric structures of dynamical systems (Hamiltonian systems, reversible systems, systems with first integrals, etc.).

Numerical integration of stochastic differential equations. Introduction to stochastic processes. Ito calculus and stochastic differential equations. Numerical methods for stochastic differential equations (strong and weak convergence, stability, etc.). Stochastic simulations and multi-level Monte-Carlo methods.

Numerical methods for conservation laws. Introduction to the development, analysis, and application of computational methods for solving conservation laws with an emphasis on finite volume, high-order essentially non-oscillatory schemes, and discontinuous Galerkin methods.

Useful/related courses :



Related Minors :