Interpolation of inverse operators for preconditioning parameter-dependent equations

Olivier Zahm (1) , Anthony Nouy (2, 1)

1 GeM - Institut de Recherche en Génie Civil et Mécanique
2 GdR MASCOT-NUM - Méthodes d'Analyse Stochastique des Codes et Traitements Numériques

We propose a method for the construction of preconditioners of parameter-dependent matrices for the solution of large systems of parameter-dependent equations. The proposed method is an interpolation of the matrix inverse based on a projection of the identity matrix with respect to the Frobenius norm. Approximations of the Frobenius norm using random matrices are introduced in order to handle large matrices. The resulting statistical estimators of the Frobenius norm yield quasi optimal projections that are controlled with high probability. Strategies for the adaptive selection of interpolation points are then proposed for different objectives in the context of projection-based model order reduction methods: the improvement of residual-based error estimators, the improvement of the projection on a given reduced approximation space, and the reuse of computations for sampling-based model order reduction methods.

Domaines: Analyse numérique [math.NA]

Olivier Zahm, Anthony Nouy. Interpolation of inverse operators for preconditioning parameter-dependent equations. SIAM Journal on Scientific Computing, 2016, 38 (2), pp.A1044-A1074. ⟨10.1137/15M1019210⟩. ⟨hal-01262424⟩