论文标题
准蒙特卡洛逆变换采样，以节省拉格朗日粒子方法和欧拉 - 拉格朗日耦合
Quasi Monte Carlo inverse transform sampling for phase space conserving Lagrangian particle methods and Eulerian-Lagrangian coupling
论文作者
论文摘要
本文介绍了针对拉格朗日和欧拉斯夫·弗拉索夫 - 波森求解器的几何整合，低拨言抽样和代码耦合之间的新颖而实际的有用的联系。低discrepancy序列，也称为准随机序列（Quasi Monte Carlo），提供的收敛速率接近$ \ Mathcal {o}（n^{ - 1}）$，它们比（伪）随机数（Monte carlo）仅在$ \ \ \ \ \ Mathcal {o}（o）{n^o}（n^n^^^=} $中，它仅优于（伪）随机数（monte carlo）。 Lagrangian粒子方法（例如PIC）取决于蒙特卡洛整合。粒子分布受到特征后的正向流动的非线性干扰。因此，这仍然是一个问题，即粒子方法是否可以从这种准随机序列中受益。任何非线性量度保留图都不会影响QMC序列的低分配，因此会收敛顺序仍然存在。本文表明，相连的几何粒子方法的正向流动自然地诱导了这种量度保留的图，这是它们在新框架中重要性的基础。在这种情况下，观察到Hardy Krause变化首次增加了Vlasov-Poisson系统的增加。与线性相。同样，为整个PIC模拟提供了星形差异。另一方面，Eulerian和Lagrangian求解器具有不同的优点和劣势，因此我们提出了一种从弗拉索夫 - 波西森系统到PIC模拟的新型方式。这是通过较高维度的逆变换采样（Rosenblatt-Mück变换）来实现的。以这种方式，马尔可夫链蒙特卡洛技术被规避，允许使用伪和准随机数。在后一种情况下，可以在线性和非线性相中观察到更好的收敛速率。
This article presents a novel and practically useful link between geometric integration, low-discrepancy sampling and code coupling for Lagrangian and Eulerian Vlasov-Poisson solvers. Low-discrepancy sequences, also called quasi-random sequences (Quasi Monte Carlo), provide convergence rates close to $\mathcal{O}( N^{-1})$ which are far superior to (pseudo) random numbers (Monte Carlo) settling in at only $\mathcal{O}(N^{-0.5})$. Lagrangian particle methods such as PIC rely on Monte Carlo integration. The particle distributions are nonlinearly perturbed by the forward flow following the characteristics. Hence it remains the question of whether particle methods can benefit from such quasi-random-sequences. Any nonlinear measure-preserving map does not affect the low-discrepancy of a QMC sequence such that the order of convergence remains. This article shows that the forward flow of phase space-conserving geometric particle methods induces naturally such a measure-preserving map underlying their importance in a new framework. In this context the Hardy Krause Variation is observed to increase in the Vlasov-Poisson system for the first time. with the linear phase. Also the star discrepancy is presented for an entire PIC simulation. On the other hand, Eulerian and Lagrangian solvers have different strengths and weaknesses, such that we present a novel way of transiting from a spectral discretization of the Vlasov--Poisson system to a PIC simulation. This is achieved by higher dimensional inverse transform sampling (Rosenblatt-Mück transform). In this way Markov Chain Monte Carlo techniques are circumvented which allows the use of pseudo and quasi-random numbers. In the latter case better convergence rates can be observed both in the linear and nonlinear phase.
