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Computer experiments & Stochastic calculus


Program (detailed content) :


Computer Experiments

  • Introduction: computer experiments and metamodelling. Examples of applications
  • Two famous metamodels : chaos polynomials and Gaussian process regression (Kriging)
  • Simulation of unconditional / conditional Gaussian processes
  • Accounting for external knowledge and covariance kernel customization
  • Metamodel-based optimisation (Bayesian optimisation)
  • Design of computer experiments: focus on space-filling design
  • Global sensitivity analysis: focus on ANOVA decomposition (Sobol decomposition)
  • Industrial application: uncertainty quantification


Stochastic calculus

  • Continuous-time stochastic processes and martingales. Introduction to stopping-times.
  • Construction of the Brownian movement and the stochastic integral then derivation of Itô's formula. Solving a Dirichlet problem using the Brownian motion.
  • Introduction to stochastic differential equations (SDE) then derivation of the Fokker-Planck equations. Solving a parabolic equation using the solution of a SDE.



Organization :

  • Course, exercises, computer lab with R software.


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


Computer Experiment

  • Metamodelling for optimization / uncertainty quantification of a computer code
  • At least the two main families of metamodels : chaos polynomials and Gaussian processes
  • Kernel customization to account for external knowledge
  • Design of computer experiments
  • Global sensivity analysis


Stochastic calculus

  • The brownian motion as well as the Wiener integral and Itô’s formula
  • The relationship between a stochastic differential equation and its Fokker-Planck equation.
  • The rewriting of a parabolic or elliptical problem using a well-chosen stochastic process.



The student should be able :


Computer Experiments

  • At a theoretical level, to do computations for:
    • covariance kernels and Gaussian process
    • ANOVA decomposition, Sobol indices
  • At a practical level, to perform the complete methodology for analyzing a computer code
    • design of experiments
    • metamodel construction / evaluation
    • application to optimization / uncertainty quantification of a computer code

Stochastic calculus

  • Derive simple models on noise filtration and stochastic control.
  • Numerically implement the resolution of a parabolic or elliptic equation using a particle-based probabilistic method.

Needed prerequisite

Gaussian vectors. Probability. ODE. Basics of 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...


Gaussian process for machine learning, C. E. Rasmussen and C. K. I. Williams, The MIT Press, 2006.


B. Iooss. Revue sur l'analyse de sensibilité globale de modèles numériques. Journal de la Société Française de Statistique, 152:1-23, 2011


Øksendal, B. (2003). Stochastic differential equations. In Stochastic differential equations (pp. 65-84). Springer, Berlin, Heidelberg.