Carlo Marcati: Catalogue data in Autumn Semester 2019 |
Name | Dr. Carlo Marcati |
URL | https://people.math.ethz.ch/~cmarcati/ |
Department | Mathematics |
Relationship | Lecturer |
Number | Title | ECTS | Hours | Lecturers | |
---|---|---|---|---|---|
401-3660-69L | Numerical Analysis Seminar: Model Order Reduction and Reduced Bases for PDEs Number of participants limited to 5. Consent of Instructor needed. | 4 credits | 2S | C. Marcati | |
Abstract | Reduced Basis (RB) methods provide a technique to reduce the computational cost of problems described by partial differential equations which involve a wide range of parameters (parametric PDEs). Such problems are ubiquitous in science and engineering, both in the analysis of physical phenomena and in the design of new objects. | ||||
Objective | The aim of this seminar is to review recent mathematical results on theoretical aspects of Reduced Basis methods and to learn how model-order reduction techniques can be used to lower computational cost in the solution of parametric PDEs. | ||||
Content | Reduced Basis (RB) methods provide a technique to reduce the computational cost of problems described by partial differential equations which involve a wide range of parameters (parametric PDEs). Such problems are ubiquitous in science and engineering, both in the analysis of physical phenomena and in the design of new objects. Building on top of classical finite element approximations, RB methods split the work into a computationally heavy offline phase and an online phase—where only a reduced-order model needs to be solved—that can be executed almost in real-time. The first phase involves computing the high-fidelity solutions to the PDE on a carefully selected "training" set of parameters (so-called snapshots). The snapshots are then used as a reduced basis (hence the name of the method) for the solution of problems on new parameters. The methods used for the (quasi-)optimal selection of the basis are of independent interest and shared with other model-order reduction techniques in statistics, approximation, and data science. The estimates on a priori RB errors are linked with the approximability of the classes of solutions to the equations; furthermire, reduced approximations can be used as a theoretical tool in the analysis of other reduction techniques. | ||||
Literature | Introductory textbooks. [1] Jan S. Hesthaven, Gianluigi Rozza, and Benjamin Stamm, Certified reduced basis methods for parametrized partial differential equations, SpringerBriefs in Mathemat- ics, Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao, 2016. [2] Alfio Quarteroni, Andrea Manzoni, and Federico Negri, Reduced basis methods for partial differential equations, Unitext, vol. 92, Springer, Cham, 2016. | ||||
Prerequisites / Notice | Format of the seminar The seminar format will be oral student presentations, combined with a written report. Student presentations will be based on a recent research paper selected in two meetings at the start of the semester. |