A priori Convergence Theory for Reduced-Basis Approximations of Single-Parametric Elliptic Partial Differential Equations

Yvon Maday (1, 2) , Anthony T. Patera (3) , Gabriel Turinici (4)

1 LJLL - Laboratoire Jacques-Louis Lions
2 DAM - Division of Applied Mathematics
3 MIT-MECHE - Department of Mechanical Engineering [Massachusetts Institute of Technology]
4 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

We consider "Lagrangian" reduced-basis methods for single-parameter symmetric coercive elliptic partial differential equations. We show that, for a logarithmic-(quasi-)uniform distribution of sample points, the reduced-basis approximation converges exponentially to the exact solution uniformly in parameter space. Furthermore, the convergence rate depends only weakly on the continuity-coercivity ratio of the operator: thus very low-dimensional approximations yield accurate solutions even for very wide parametric ranges. Numerical tests (reported elsewhere) corroborate the theoretical predictions.

Domaines: Analyse numérique [math.NA]

Yvon Maday, Anthony T. Patera, Gabriel Turinici. A priori Convergence Theory for Reduced-Basis Approximations of Single-Parametric Elliptic Partial Differential Equations. Journal of Scientific Computing, 2002, 17 (1-4), pp.437-446. ⟨hal-00798410⟩