参数化粒子基Vlasov-Poisson方程的自适应辛模型降阶
Adaptive symplectic model order reduction of parametric particle-based Vlasov-Poisson equation
AI总结：
本文针对参数化粒子基Vlasov-Poisson方程，提出结合自适应参数采样、DMD和DEIM的非线性辛降阶方法，在保留哈密顿结构的同时显著降低多查询模拟的计算代价。
摘要：
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，因此往往在计算上难以处理。在多查询模拟中，问题依赖于一组参数，这一困难更加突出。在本工作中，我们针对参数化Vlasov-Poisson方程的几何粒子网格（particle-in-cell）近似所得到的半离散哈密顿系统推导降阶模型。由于该问题的非耗散性和高度非线性特性使其只能在时间上局部可约化，我们采用一种非线性降阶基方法，使降阶相空间随时间演化。该策略能够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的求值仍然计算昂贵。我们提出一种新的非线性项约化方法，将自适应参数采样与超降阶（hyper-reduction）技术相结合来解决这一问题。所提出的方法能够将代价依赖于粒子数量的运算与代价依赖于所需参数实例的运算解耦。特别地，在每个时间步中，电势通过动态模式分解（DMD）近似，粒子到网格映射通过离散经验插值方法（DEIM）近似。这些近似由过去时间窗口内在少数选定参数值处获得的数据构造，以保证计算高效的自适应性。所得的DMD-DEIM降阶动力系统保留了全模型的哈密顿结构，能够提供良好的解近似，并且可以以降低的计算代价求解。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
