 # Computer experiments & Waves propagation

## 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

Waves

• Main models of wave propagation: acoustic waves, wave equation, Schrödinger equations.
• Boundary conditions: physical boundaries, artificial boundary (transparent, absorbing boundary conditions, perfectly matched layers).

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

Waves

• The principle of transparent and absorbing boundary conditions for wave propagation
• Two methods of uncertainty quantifications: chaos polynomial expansion and multi-level Monte Carlo methods

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

Waves

• Design theoretically and numerically transparent and absorbing boundary conditions.
• Perform numerically a UQ analysis based on intrusive methods (chaos polynomial methods) or non intrusive (multi-level Monte Carlo methods).

## Needed prerequisite

Gaussian vectors. Basics of PDE: methods of characteristics, finite difference methods for PDEs.

## Recommended prerequisite

Waves

Basics of PDE: methods of characteristics, finite difference methods for PDEs.

## 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...

## Bibliography

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

http://www.gaussianprocess.org/gpml/

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

http://journal-sfds.fr/article/view/53