A natural-norm Successive Constraint Method for inf-sup lower bounds

By Dinh Bao Phuong Huynh, David J. Knezevic, Yanlai Chen, Jan S. Hesthaven, Anthony T. Patera. 2010

110 citations recorded by the source corpus.

We present a new approach for the construction of lower bounds for the inf-sup stability constants required in a posteriori error analysis of reduced basis approximations to anely parametrized partial dierential equations. We combine the \linearized" inf-sup statement of the natural{ norm approach with the approximation procedure of the Successive Constraint Method (SCM): the former (natural{norm) provides an economical parameter expansion and local concavity in parameter | a small(er) optimization problem which enjoys intrinsic lower bound properties; the latter (SCM) provides a systematic optimization framework | a Linear Program (LP) relaxation which readily incorporates continuity and stability constraints. The natural{norm SCM requires a parameter domain decomposition: we propose a greedy algorithm for selection of the SCM control points as well as adaptive construction of the optimal subdomains. The ecacy of the natural{norm SCM is illustrated through numerical results for two types of non-coercive problems: the Helmholtz equation (for acoustics, elasticity, and electromagnetics), and the convection{diusi on equation.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f5cbbd4dd461d5da5f267e84cd76fe38e1a4e21b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Statistical and Nonlinear Physics

In Statistics and Probability

Citations in the corpus

A static condensation reduced basis element method: Complex problems (2013, unknown): “…however, it is possible to improve the current error bound further by simply constructing a better lower bound for the inf-sup parameter, such as by the successive constraint method [24] [25] Figure 3: System configuration for System 1 and the magnitude of pressure field solution for k = 2.”
Certified reduced basis approximation for parametrized partial differential equations and applications (2011, methodology): “where β N (μ) is a lower bound of inf-sup constant (66) and can be computed by means of the same SCM procedure used for the lower bound of coercivity constants [34,40].”
Stabilized reduced basis method for parametrized advection-diffusion PDEs (2014, background): “Moreover, several improvements for the SCM have been recently proposed [6, 23,33].”
Model Order Reduction in Fluid Dynamics: Challenges and Perspectives (2014, methodology): “A general version using the so-called “natural norm” [110] has been analyzed in [61], where it has been applied to noncoercive problems such as Helmholtz equations – the simpler coercive case can be seen as a particular instance where the stability factor is just the coercivity constant.”
An efficient computational framework for reduced basis approximation and a posteriori error estimation of parametrized Navier–Stokes flows (2014, methodology): “Concerning the Offline stage, we rely on a P2−P1 FE approximation, and first we run the SCM algorithm.”