Parameter Space: The Final Frontier. Certified Reduced Basis Methods for Real-Time Reliable Solution of Parametrized Partial Differential Equations

2007 ◽  
Author(s):  
Anthony T. Patera
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Real Time ◽  
Parameter Space ◽  
Reduced Basis ◽  
Partial Differential ◽  
Reduced Basis Methods ◽  
Parametrized Partial Differential Equations ◽  
Reliable Solution

scholarly journals Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods

2001 ◽  
Vol 124 (1) ◽  
pp. 70-80 ◽  
Author(s):  
C. Prud’homme ◽  
D. V. Rovas ◽  
K. Veroy ◽  
L. Machiels ◽  
Y. Maday ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Real Time ◽  
Galerkin Projection ◽  
Real Time Control ◽  
Reduced Basis ◽  
Partial Differential ◽  
Approximation Process ◽  
Essential Components ◽  
On Line

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations—Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation—relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage in which, given a new parameter value, we calculate the output of interest and associated error bound, depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.

scholarly journals RBniCS - reduced order modelling in FEniCS

2015 ◽  
Author(s):  
Francesco Ballarin ◽  
Alberto Sartori ◽  
Gianluigi Rozza
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
High Performance ◽  
Theoretical Physics ◽  
Reduced Order Models ◽  
Reduced Basis ◽  
Partial Differential ◽  
Reduced Order ◽  
Reduced Order Modelling ◽  
Reduced Basis Methods

RBniCS (freely available under the GNU LGPL, version 3, at http://mathlab.sissa.it/rbnics) is a python-based library, developed on top of FEniCS (http://fenicsproject.org), aimed at the developement of reduced order models in the FEniCS environment. In particular, reduced order techniques such as the certified reduced basis method and proper orthogonal decomposition-Galerkin methods are implemented. The FEniCS project allows RBniCS to take advantage of the high-level (e.g., human readable) code used for the automated solution of partial differential equations. Thanks to the features of FEniCS the final user needs to prepare a short code (around 100 lines) to carry out a reduced order simulation. It is ideally suited for novice users willing to learn reduced basis methods and reduced order modelling, thanks to an object-oriented approach and an intuitive and versatile python interface. Indeed, it is a companion of the introductory reduced basis handbook [Hesthaven, Rozza, Stamm. Certified Reduced Basis Methods for Parametrized Partial Differential Equations. SpringerBriefs in Mathematics, Springer International Publishing, 2015], and has been already used in doctoral classes within the "Mathematical Analysis, Modelling, and Applications" PhD course at SISSA, as well as for courses within the "Master in High Performance Computing" held by SISSA and International Centre for Theoretical Physics (ICTP). RBniCS can also be used as a basis for more advanced projects that would like to assess the capability of reduced order models in their existing FEniCS-based software, thanks to the availability of several reduced order methods and algorithms in the library.

Sign in / Sign up

Export Citation Format

Share Document