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.
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Acknowledgments
We also thank K. Smetana and the anonymous referees for valuable comments.
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Open Access funding provided by Projekt DEAL. The authors would like to acknowledge the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart.
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Communicated by: Anthony Nouy
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This article belongs to the Topical Collection: Model Reduction of Parametrized Systems
Guest Editors: Anthony Nouy, Peter Benner, Mario Ohlberger, Gianluigi Rozza, Karsten Urban and Karen Willcox
Appendices
Appendix A. Proof of Theorem 1
In the following, we make frequent use of the identity
which is a direct application of the fundamental theorem of calculus.
Let \(H: {X} \rightarrow {X}\) be defined via \(H(x) := x - \left . \text {DG}\right |_{\hat {x}}^{-1}(G(x))\). The proof works by showing the existence of a fixed point x∗∈ X of the mapping H in the vicinity of the approximate solution \({\hat {x}}\). It is an easy observation that G(x) = 0 ⇔ H(x) = x, which motivates the application of Banach’s fixed-point theorem. To this end, we define the set \(M=\overline {B_{2\varepsilon }}({\hat {x}}) := \{ x \in {X} | \|{x-{\hat {x}}}\|_{X} \leq 2 \varepsilon \}\), i.e. the closed ball around the approximate solution \({\hat {x}}\) with radius 2ε. In order to be able to apply Banach’s fixed-point theorem to H in M, we have to prove that H is a self-mapping and a contraction in M.
Let x ∈ M. Consider
Since \({\hat {x}} + t(x - {\hat {x}}) \in M\) for t ∈ [0, 1] we get the estimate
which shows that H(x) ∈ M for x ∈ M. Thus, H is a self-mapping in M. By making use of (19), we obtain the bound
which proves the contraction property.
Hence, we can apply Banach’s fixed-point theorem and prove the existence of x∗∈ M with G(x∗) = 0. We furthermore directly get the bound \(\| x^{\ast } - \hat {x}\|_{X} \leq 2\epsilon \). However, the bound can be slightly refined by considering for x ∈ M
from which we get for \(x = {\hat {x}} \in M\) the final estimate
Appendix B. Proof of Lemma 1
We only have to slightly modify the proof of Theorem 1. We consider the set \(M=\overline {B_{\alpha }({\hat {x}})} := \{ x \in X | \|{x-{\hat {x}}}\|_{X} \leq \alpha \}\) and determine the minimal radius α such that H is a contracting self-mapping on M. Analogous to the proof of Theorem 1, we get
Solving the resulting quadratic inequality, we have that α is contained in the interval [α−,α+], where
Hence, the smallest α for which H is a self-mapping is given by \(\alpha _{-} = \frac {2}{1 + \sqrt { 1 - 4\gamma C \varepsilon }} \varepsilon \). For the proof of the contracting property, no further modification of the proof of Theorem 1 is necessary. Finally, it follows that
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Schmidt, A., Wittwar, D. & Haasdonk, B. Rigorous and effective a-posteriori error bounds for nonlinear problems—application to RB methods. Adv Comput Math 46, 32 (2020). https://doi.org/10.1007/s10444-020-09741-x
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DOI: https://doi.org/10.1007/s10444-020-09741-x