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Partial Derivative Equations  and Monte Carlo methods


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.



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).


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


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



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


The student will be able to:


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



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

Needed prerequisite


Differential calculus, analysis, ODE

Basic numerical methods



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...