[C1]

Eulerian and Lagrangian models of continuum mechanics

Notification: April 04, 2013 by Prof. P. Fischer.

Prof. A. Iollo   (Dpt. Mathematics. U. Bordeaux 1, France)

We will derive the hyperbolic conservation laws that can model for example either gas flows or large deformations of a compressible elastic material in either a physical (Eulerian) or reference (Lagrangian) domain. The constitutive law characterizing each different (gas, liquid, elastic) material will be studied under the hypothesis of material indifference and thermodynamic consistency. Then, we will investigate the PDE system obtained in terms of characteristic pattern and hyperbolicity conditions. Finally, we will put at work this knowledge to devise consistent and stable schemes to integrate these models thanks to finite-volume schemes.

[C2]

Reduced order modeling applications

Notification: April 05, 2013 by Prof. Filippo Terragni

Prof. Filippo Terragni   (Dpt. of Materials Science and Engineering and Chemical Engineering. Modelling and Numerical Simulation Group. U. Carlos III de Madrid, Spain)

Modeling is an essential ingredient of scientific research and many industrial tasks, embracing different fields of application. There exists a large variety of mathematical models, which are shaped according to the nature of the problem to solve and the involved tools, all of them aiming at providing insights on and simplifying some (physical) scenario. The feasibility of their practical implementation (nowadays largely performed by means of computers) is a key point to remember in real applications. Reduced order models (ROMs) represent a wide class of models that make use of a small number of degrees of freedom, while preserving a satisfactory accuracy. In other words, these models are able to reduce the complexity of the problem, as they allow for a good description and approximation of the solutions in terms of few essential features (modes), thus reducing the necessary computational resources (storage and CPU time). The course is dedicated to the introduction of ROMs based on singular value decomposition (SVD), high-order singular value decomposition (HOSVD), and proper orthogonal decomposition (POD). Various applications of practical interest are presented and discussed, like compression of aerodynamic databases, reconstruction of images and gappy data, fast simulation in fluid dynamics, and construction of bifurcation diagrams.

[C3]

Time Series Analysis: methods and some applications in finance

Notification: April 15, 2013 by Prof. Cláudia Nunes  

Prof. Cláudia Nunes Philippart   (Dpt. Mathematics, U. of Lisbon, Portugal)

The subject of financial time series analysis has attracted substantial attention in recent years, especially with the 2003 Nobel awards to Professors Robert Engle and Clive Granger. Nowadays there are modern developments concerning the analysis of financial data, regarding, for example, high-frequency analysis and stochastic volatility. Also there is the need to understand and use software. In this course we will present several models used in the analysis of financial data. We will start by the basic ones (like the ARIMA models, which are linear models), and then we will present the main characteristics of models where the volatility is itself a stochastic process, leading to the heteroscedastic models, as the GARCH model. The concepts will be illustrated with the use of real data. Finally we will mention briefly other approaches and tools, as the Kalman-filter and the Markov-chain Monte-Carlo simulation.

[C4]

SIMULATION OF GEOPHYSICAL FLOWS

Notification: May 27, 2013 by Prof. Carlos Parés

Prof. Carlos Parés   (University of Málaga, Spain)

Shallow water equations are widely used in ocean and hydraulic engineering to model flows in rivers, reservoir or coastal areas, among some other applications such as floods, dam breaks, tsunami propagation in the ocean, etc. In the form to be considered in this course, these equations constitute a hyperbolic system of conservation laws with source terms due to the bottom topography and friction forces. In recent years, there has been increasing interest in the design of numerical schemes with good properties for shallow water equations. The schemes are required to have high-order accuracy (both in space and time) in the regions where the solution is smooth and to capture shock discontinuities accurately. Moreover they have also to satisfy a correct balance between the flux and the source terms, in order to properly capture the water-at-rest stationary solutions. In this course, the shallow water equations will be derived in the simpler case of only one space variable. A short introduction to Godunov and Roe methods for systems of conservation laws will follow. Next, the numerical treatment of the source term will be discussed. A brief introduction to the multidimensional case and to high order extensions will be finally presented.