# 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...