This work proposes a hyper-reduction method for nonlinear parametric dynamical systems characterized by gradient fields such as Hamiltonian systems and gradient flows. The gradient structure is associated with conservation of invariants or with dissipation and hence plays a crucial role in the description of the physical properties of the system. Traditional hyper-reduction of nonlinear gradient fields yields efficient approximations that, however, lack the gradient structure. We focus on Hamiltonian gradients and we propose to first decompose the nonlinear part of the Hamiltonian, mapped into a suitable reduced space, into the sum of d terms, each characterized by a sparse dependence on the system state. Then, the hyper-reduced approximation is obtained via discrete empirical interpolation (DEIM) of the Jacobian of the derived d-valued nonlinear function. The resulting hyper-reduced model retains the gradient structure and its computationally complexity is independent of the size of the full model. Moreover, a priori error estimates show that the hyper-reduced model converges to the reduced model and the Hamiltonian is asymptotically preserved. Whenever the nonlinear Hamiltonian gradient is not globally reducible, i.e. its evolution requires high-dimensional DEIM approximation spaces, an adaptive strategy is performed. This consists in updating the hyper-reduced Hamiltonian via a low-rank correction of the DEIM basis. Numerical tests demonstrate the applicability of the proposed approach to general nonlinear operators and runtime speedups compared to the full and the reduced models.

翻译：本文针对哈密顿系统与梯度流等具有梯度场的非线性参数化动力系统，提出一种超降阶方法。梯度结构与不变量守恒或耗散相关，因此在系统物理特性描述中起关键作用。传统非线性梯度场的超降阶虽能获得高效近似，但会丢失梯度结构。我们聚焦哈密顿梯度，首先将映射至合适降阶空间的哈密顿量非线性部分分解为d项之和，每项对系统状态具有稀疏依赖性；随后通过对推导所得d值非线性函数的雅可比矩阵实施离散经验插值(DEIM)获得超降阶近似。最终超降阶模型保留梯度结构，且计算复杂度与全阶模型规模无关。此外，先验误差估计表明超降阶模型可收敛至降阶模型，哈密顿量具有渐近保持性。当非线性哈密顿梯度不具备全局可降阶性（即其演化需要高维DEIM逼近空间）时，执行自适应策略：通过DEIM基的低秩修正更新超降阶哈密顿量。数值实验验证了该方法对一般非线性算子的适用性，并展示了相较于全阶模型与降阶模型的运行时加速效果。