Reduced Basis Preconditioners for Parametrized Differential Problems (RB-Prec)

• Research unit: COST
• Project number: C14.0068

Key words (English)

Numerical approximation of Partial Differential Equations; parametrized problems; Reduced Basis method; Preconditioners; vascular flows

Research programs (English)

COST-Action TD1307 - EUropean MOdel Reduction NETwork (EU-MORNET)

Short description (English)

The project deals with a Reduced Basis (RB) approach for the numerical solution of linear systems associated with high-fidelity models for the approximation of parametrized Partial Differential Equations (PDEs). Specifically, the project aims at introducing, analyzing and testing a class of preconditioners for the efficient, robust and possibly scalable resolution of linear systems by means of iterative methods in a parameter dependent framework. The idea consists in using the RB method, originally developed for the model order reduction of parametrized PDEs, to speed up the resolution of the original full scale linear system associated to the parametrized high-fidelity model. Indeed, while the RB method is known to perform efficiently and reliably for certain classes of parametrized problems, these properties are not necessarily granted for others of more practical interests. In this project, we aim at solving the large scale high-fidelity model without any computational reduction in order to address the issue of reliability, using however suitably developed RB preconditioners to efficiently solve the linear system stemming from the finite element approximation of the parametrized PDEs. RB preconditioners will therefore speed up the convergence to the high-fidelity solution by providing a very ad-hoc 'coarse level' component to standard domain decomposition preconditioners. As application, we consider vascular problems in Haemodynamics for patient-specific configurations.

Further information (English)

Full name of research-institution/enterprise: EPF Lausanne EPFL SB MATHICSE Chaire de modélisation et calcul scientifique

Partners and International Organizations (English)

AT; BE; HR; CY; FR; MK; DE; IS; IE; IL; IT; LU; NL; PL; PT; RO; ES; SE; UK

Abstract (English)

In this doctoral project we aim at employing reduced order modeling (ROM) techniques to develop efficient preconditioners for the solution of large-scale linear Systems arising from the discretization of parametrized partial différentiel équations (PDEs). Our emphasis is in particular parametrized PDEs that are hardiy reducible. Essential ingrédients of the project are (the current state of the art of successfui) ROM and preconditioning techniques, with emphasis on i) the reduced basis (RB) method and ii) 1-level and 2-level domain décomposition (DD) and multigrid (MG) methods. The research project deals with: - the development of a methodology able to combine ROM techniques with existing well-known 1-level preconditioners, e.g. domain décomposition methods, in order to form 2-level preconditioners for affine elliptic parametrized PDEs; - the analysis of the computational efficiency of the proposed techniques in terms of scalability and optimality, and a comparison with existing preconditioning methods, when facing linear three-dimensional parametrized PDEs; - the extension of thèse techniques to nonaffine and saddie-point problems.

References in databases (English)

Swiss Database: COST-DB of the State Secretariat for Education and Research Hallwylstrasse 4 CH-3003 Berne, Switzerland Tel. [phone] Swiss Project-Number: C14.0068