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Computer experiments & Stochastic calculus

Presentation

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.

Objectives

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...