# Modelling

## Objectives

At the end of this module, the student will have understood and be able to explain (main concepts) :
The modelling of a concrete problem (on simple examples coming from physics, or from finance), by a mathematical model expressed in terms of algebraic equations (least square approximation, splines, parametric optimization) or partial differential equations (PDE).
The student should have understood the following notions :
Functional optimisation, piecewise functions and C^k continuity ; natural cubic splines and their local and global representations. Approximated solution of an overdetermined linear system, singular values of matrices, numerical optimization by iterations.
Fourier method (series of orthogonal functions) for the resolution of linear PDE.
Fundamentals of the Finite Difference method (order of a scheme, stability, convergence).
Formal definition of the Brownian motion and principles of the Monte-Carlo method for parabolic PDE.
Robustness and Stability of a numerical model or method.
The student will be able to :
- Model a simple problem (spline approximation, parametric optimization, PDE).
- Determine and compute the interpolating spline and the smoothing spline of n given points.
- Check the homogeneity of a mathematical formula.
- Solve an overdetermined linear system, choosing between four different methods (normal equations, Gram-Schmidt, modified Gram-Schmidt, Householder).
- Choose an appropriate optimization method and solve an optimization problem by this method.
- Solve a linear PDE with the Fourier method.
- Analyse stability and consistency of a finite difference scheme.
- Program with MATLAB a finite difference scheme or the Monte-Carlo method for solving a linear parabolic PDE.

## Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...