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Publication# Symplectic dynamical low rank approximation of wave equations with random parameters

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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Related concepts (16)

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Approximation

An approximation is anything that is intentionally similar but not exactly equal to something else. The word approximation is derived from Latin approximatus, from proximus meaning very near and the prefix ad- (ad- before p becomes ap- by assimilation) meaning to. Words like approximate, approximately and approximation are used especially in technical or scientific contexts. In everyday English, words such as roughly or around are used with a similar meaning. It is often found abbreviated as approx.

Tangent space

In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold. In differential geometry, one can attach to every point of a differentiable manifold a tangent space—a real vector space that intuitively contains the possible directions in which one can tangentially pass through .

Parameter

A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when identifying the system, or when evaluating its performance, status, condition, etc. Parameter has more specific meanings within various disciplines, including mathematics, computer programming, engineering, statistics, logic, linguistics, and electronic musical composition.

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