Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing

By Mario Ohlberger, Stephan Rave. 2013

128 citations recorded by the source corpus.

Abstract We present a new method for the nonlinear approximation of the solution manifolds of parameterized nonlinear evolution problems, in particular in hyperbolic regimes with moving discontinuities. Given the action of a Lie group on the solution space, the original problem is reformulated as a partial differential algebraic equation system by decomposing the solution into a group component and a spatial shape component, imposing appropriate algebraic constraints on the decomposition. The system is then projected onto a reduced basis space. We show that efficient online evaluation of the scheme is possible and study a numerical example showing its strongly improved performance in comparison to a scheme without freezing.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1b98d134732ba607ae97a442fbee93377bf524a9

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In Computational Mechanics

In Computational Theory and Mathematics

In Statistical and Nonlinear Physics

Citations in the corpus

Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders (2018, methodology): “Such methods include separating transport dynamics via ‘freezing’ [54], applying a coordinate transformation to the trial basis [39, 10], shifting the POD basis [60], transforming the physical domain of the snapshots [74, 73], constructing the trial basis on a Lagrangian formulation of the governing”