 # Partial Derivative Equations  and Monte Carlo methods

## Presentation

Linear PDE  (analysis, numerics):

•  PDE models : examples ; the 4 fundamental linear PDE. Classification.
• Principle of the Finite Difference method: consistancy, stability, convergence.
• Laplace-Poisson equation (elliptic): explicit solution (by separation of variables), maximum principle, FD schemes.
• Heat equation (parabolic): explicit solution (by Fourier transform), FD schemes  (explicit, implicit, splitting). A non linear case.
• Transport equation (hyperbolic) : explicit solution -  characteristics, schemes, equivalent equation.
• Waves equation (hyperbolic) : explicit solution - characteristics, schemes.
• Practical works: stability, accuracy ; modeling.

Monte-Carlo:

Generation of random numbers, simulation by inversion of the distribution function, by the reject method and by some specific methods, Monte-Carlo Methods (convergence, rate of convergence, variance reduction by using different methods).

## Objectives

At the end of this module, the student will have understood and be able to explain (main concepts):

PDE:

• The four fundamentals PDE models, with their solution behaviors
• The Finite Difference discretization method

Monte-Carlo:

The fundamental principles of simulating random variables and Monte-Carlo methods.

The student will be able to:

PDE:

• To model basic fundamental phenomena by employing PDE
• To derive a Finite Difference scheme (consistent, stable, convergent).

Monte-Carlo:

Simulate a random variable by different methods, use probabilistic,  choose appropriate techniques for variance reduction and error estimation.

## Needed prerequisite

PDE:

Differential calculus, analysis, ODE

Basic numerical methods

Monte-Carlo:

A basic course on probabilities.

## 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...