High-resolution simulations of particle-based kinetic plasma models typically require a high number of particles and thus often become computationally intractable. This is exacerbated in multi-query simulations, where the problem depends on a set of parameters. In this work, we derive reduced order models for the semi-discrete Hamiltonian system resulting from a geometric particle-in-cell approximation of the parametric Vlasov-Poisson equations. Since the problem's non-dissipative and highly nonlinear nature makes it reducible only locally in time, we adopt a nonlinear reduced basis approach where the reduced phase space evolves in time. This strategy allows a significant reduction in the number of simulated particles, but the evaluation of the nonlinear operators associated with the Vlasov-Poisson coupling remains computationally expensive. We propose a novel reduction of the nonlinear terms that combines adaptive parameter sampling and hyper-reduction techniques to address this. The proposed approach allows decoupling the operations having a cost dependent on the number of particles from those that depend on the instances of the required parameters. In particular, in each time step, the electric potential is approximated via dynamic mode decomposition (DMD) and the particle-to-grid map via a discrete empirical interpolation method (DEIM). These approximations are constructed from data obtained from a past temporal window at a few selected values of the parameters to guarantee a computationally efficient adaptation. The resulting DMD-DEIM reduced dynamical system retains the Hamiltonian structure of the full model, provides good approximations of the solution, and can be solved at a reduced computational cost.

翻译：基于粒子的动理学等离子体模型的高分辨率模拟通常需要大量粒子，因此往往计算上难以处理。在问题依赖于一组参数的多查询模拟中，这一问题尤为突出。本文针对参数化Vlasov-Poisson方程的几何粒子云网格近似产生的半离散哈密顿系统，推导了降阶模型。由于问题非耗散且高度非线性的特性使其仅在时间上局部可降阶，我们采用一种非线性降阶基方法，其中降阶相空间随时间演化。该策略可显著减少模拟粒子的数量，但与Vlasov-Poisson耦合相关的非线性算子评估仍计算量巨大。我们提出了一种结合自适应参数采样和超降阶技术处理非线性项的新型方法。该方法允许将依赖于粒子数量的操作与依赖于所需参数实例的操作解耦。具体而言，在每个时间步，电场势通过动态模式分解（DMD）逼近，粒子到网格映射通过离散经验插值法（DEIM）逼近。这些近似基于过去时间窗口内少数选定参数值的数据构建，以保证计算高效的自适应性。得到的DMD-DEIM降阶动力系统保留了全模型的哈密顿结构，能提供良好的解近似，且可在降低的计算成本下求解。