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Fast local reduced basis updates for the efficient reduction of nonlinear systems with hyper-reduction. (English) Zbl 1331.65094
Summary: Projection-based model reduction techniques rely on the definition of a small dimensional subspace in which the solution is approximated. Using local subspaces reduces the dimensionality of each subspace and enables larger speedups. Transitions between local subspaces require special care and updating the reduced bases associated with each subspace increases the accuracy of the reduced-order model. In the present work, local reduced basis updates are considered in the case of hyper-reduction, for which only the components of state vectors and reduced bases defined at specific grid points are available. To enable local reduced basis updates, two comprehensive approaches are proposed. The first one is based on an offline/online decomposition. The second approach relies on an approximated metric acting only on those components where the state vector is defined. This metric is computed offline and used online to update the local bases. An analysis of the error associated with this approximated metric is then conducted and it is shown that the metric has a kernel interpretation. Finally, the application of the proposed approaches to the model reduction of two nonlinear physical systems illustrates their potential for achieving large speedups and good accuracy.

MSC:
65L05 Numerical methods for initial value problems
34A34 Nonlinear ordinary differential equations and systems, general theory
35Q53 KdV equations (Korteweg-de Vries equations)
35Q35 PDEs in connection with fluid mechanics
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
78M34 Model reduction in optics and electromagnetic theory
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