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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...