Rigorous and effective a-posteriori error bounds for nonlinear problems—application to RB methods

Abstract

Quantifying the error that is induced by numerical approximation techniques is an important task in many fields of applied mathematics. Two characteristic properties of error bounds that are desirable are reliability and efficiency. In this article, we present an error estimation procedure for general nonlinear problems and, in particular, for parameter-dependent problems. With the presented auxiliary linear problem (ALP)-based error bounds and corresponding theoretical results, we can prove large improvements in the accuracy of the error predictions compared with existing error bounds. The application of the procedure in parametric model order reduction setting provides a particularly interesting setup, which is why we focus on the application in the reduced basis framework. Several numerical examples illustrate the performance and accuracy of the proposed method.

A New Error Estimator for Reduced-Order Modeling of Linear Parametric Systems

“…Again, computing x r du (μ) in (12) requires solving a large system in (13). Instead, we compute the ROM of (13),Q (μ)z r du (μ) =r du (μ),…”

“…We see that instead of solving the dual-residual system (13), one can also solve the primal-residual system as below, Q(μ)x rpr (μ) = r pr (μ).…”

“…The dual system (6), the dual-residual system (13), as well as the primal-residual system (21), the primal-residual-residual system (31) are parametric systems in frequency domain, withμ = s or μ = (µ, s) being the vector of parameters. Similarly, we can compute the projection matrices for MOR of these systems either through snapshot based methods, or the multi-moment-matching method.…”

“…• Computing ∆ 2 (μ), ∆ pr 2 (μ) or ∆ 3 (μ) involves constructing the ROM of the dual system (6), and additionally the ROM of a corresponding residual system needs to be constructed: the ROM of the dual-residual system (13) or the ROM of the primal-residual system (21).…”

“…The error bound in [13] is proposed for nonlinear systems and also requires the computation of the inf-sup constant or its lower bound. Numerical issues concerning computing these quantities remain.…”