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:粒子动力等离子体模型的高分辨率模拟通常需要大量粒子,因此往往会变得难以计算。在多式模拟中,问题取决于一系列参数,而这一问题在多式模拟中更加严重。在这项工作中,我们从参数Vlasov-Poisson等式的几何分辨粒子-细胞近似法中为半分辨汉密尔顿系统得出减少的订单模型。由于问题非分解性和高度非线性的性质使得它只能在当地及时复制,我们采用了非线性降低基础方法,即阶段空间缩小后会及时演变。这一战略使得模拟粒子的数量大为减少,但对于与Vlasov-Poisson等离子组合相关的非线性操作者的评估仍然计算成本高昂。我们建议对非线性术语进行新的削减,将适应性参数取样和超减精减技术结合起来解决这个问题。拟议方法使得操作的成本可以从依赖所需参数的粒子数量中解析,我们采用了非线性降低的基础基基基数。 特别是,在动态模型中,从动态模型到动态模型的精确度计算方法的精确度计算方法,使电流流递减了一个方向的模型的模型的模型,这些模型的计算法系的精确度计算法系的精确度计算法系的精确度计算法系的精确度计算法是通过动态模型的模型的精确度计算法系。