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 2 S 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 2 S . As a result, the approximate solution preserves the mean energy over the flow.

中文翻译：

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具有随机参数的波动方程的辛动力学低秩逼近

在本文中，我们提出了一种用于逼近具有随机参数的二阶波动方程的动态低秩策略。控制方程以哈密顿形式重写，近似解扩展到一组具有时间相关随机系数的 2 S 动态辛正交确定性基函数。通过将控制哈密顿系统辛投影到逼近流形沿近似轨迹的切线空间来获得简化的（低秩）动力学。所提出的公式等效于在复杂设置中重铸控制哈密顿系统，并在秩等于 S 的所有复值随机场的低维流形中寻找动态低秩近似。由于这种等价性，我们能够在真实环境中正确定义近似流形，赋予其微分结构并获得其切线空间的适当参数化，根据对确定性模式的动力学的正交约束。最后，我们为随机系数和近似解的确定性基础的演化推导出辛动态正交降阶系统。这包括一个 S 确定性偏微分方程系统，耦合到维度为 2 S 的简化哈密顿系统。因此，近似解保留了流动的平均能量。我们为随机系数和近似解的确定性基础的演化推导出辛动态正交降阶系统。这包括一个 S 确定性偏微分方程系统，耦合到维度为 2 S 的简化哈密顿系统。因此，近似解保留了流动的平均能量。我们为随机系数和近似解的确定性基础的演化推导出辛动态正交降阶系统。这包括一个 S 确定性偏微分方程系统，耦合到维度为 2 S 的简化哈密顿系统。因此，近似解保留了流动的平均能量。