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Symplectic dynamical low rank approximation of wave equations with random parameters

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Abstract

In this paper we propose a dynamical low-rank strategy for the approximation of second order wave equations with random parameters. The governing equation is rewritten in Hamiltonian form and the approximate solution is expanded over a set of 2S dynamical symplectic-orthogonal deterministic basis functions with time-dependent stochastic coefficients. The reduced (low rank) dynamics is obtained by a symplectic projection of the governing Hamiltonian system onto the tangent space to the approximation manifold along the approximate trajectory. The proposed formulation is equivalent to recasting the governing Hamiltonian system in complex setting and looking for a dynamical low rank approximation in the low dimensional manifold of all complex-valued random fields with rank equal to S. Thanks to this equivalence, we are able to properly define the approximation manifold in the real setting, endow it with a differential structure and obtain a proper parametrization of its tangent space, in terms of orthogonal constraints on the dynamics of the deterministic modes. Finally, we derive the Symplectic Dynamically Orthogonal reduced order system for the evolution of both the stochastic coefficients and the deterministic basis of the approximate solution. This consists of a system of S deterministic PDEs coupled to a reduced Hamiltonian system of dimension 2S. As a result, the approximate solution preserves the mean energy over the flow.

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Acknowledgements

This work has been supported by the Swiss National Science Foundation under the Project No. 146360 “Dynamical low rank approximation of evolution equations with random parameters” and Project No. 172678 “Uncertainty Quantification Techniques for PDE constrained optimization and random evolution equation”. The authors also acknowledge the support from the Center for ADvanced MOdeling Science (CADMOS).

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  1. Institute of Mathematics, CSQI, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Eleonora Musharbash, Fabio Nobile & Eva Vidličková

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  1. Eleonora Musharbash
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  2. Fabio Nobile
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  3. Eva Vidličková
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Correspondence to Fabio Nobile.

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Communicated by David Cohen.

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Musharbash, E., Nobile, F. & Vidličková, E. Symplectic dynamical low rank approximation of wave equations with random parameters. Bit Numer Math 60, 1153–1201 (2020). https://doi.org/10.1007/s10543-020-00811-6

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  • DOI: https://doi.org/10.1007/s10543-020-00811-6

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