标题： 在Vlasov-Poisson系统中，应优化哪种度量以抑制不稳定性？ 显示英文标题

标题： What metric to optimize for suppressing instability in a Vlasov-Poisson system?

摘要： 稳定等离子体动力学是通过核聚变生成绿色能源的重要任务。 一种常见的策略是引入外部场以防止等离子体分布发展出湍流。 然而，即使对于简化的模型，如Vlasov-Poisson（VP）系统，高效找到这样的外部场仍然是一个开放问题。 在本工作中，我们采用两种不同的方法来构建这样的场：对于第一种方法，我们利用VP系统的色散关系的解析推导，找到一组可能抑制不稳定的合理场，提供定性建议。 对于第二种方法，我们利用PDE约束优化，使用不同的损失函数获得局部最优场。 由于系统的稳定性可以通过几种不同的方式表征，目标函数需要相应地进行调整。 我们通过大量的数值测试表明，诸如相对熵（KL散度）和$L^{2}$范数的目标函数会导致高度非凸的问题，使得全局最小值难以找到。 然而，我们表明，将系统的电能作为损失函数是有优势的，因为它在全局最小值附近有一个大的凸盆地。 不幸的是，在盆地之外，电能景观由非物理的平坦局部最小值组成，因此良好的初始猜测对于优化问题的整体收敛至关重要，尤其是对于具有自适应步长的求解器。 显示英文摘要

摘要： Stabilizing plasma dynamics is an important task in green energy generation via nuclear fusion. One common strategy is to introduce an external field to prevent the plasma distribution from developing turbulence. However, finding such external fields efficiently remains an open question, even for simplified models such as the Vlasov-Poisson (VP) system. In this work, we leverage two different approaches to build such fields: for the first approach, we use an analytical derivation of the dispersion relation of the VP system to find a range of reasonable fields that can potentially suppress instability, providing a qualitative suggestion. For the second approach, we leverage PDE-constrained optimization to obtain a locally optimal field using different loss functions. As the stability of the system can be characterized in several different ways, the objective functions need to be tailored accordingly. We show, through extensive numerical tests, that objective functions such as the relative entropy (KL divergence) and the $L^{2}$ norm result in a highly non-convex problem, rendering the global minimum difficult to find. However, we show that using the electric energy of the system as a loss function is advantageous, as it has a large convex basin close to the global minimum. Unfortunately, outside the basin, the electric energy landscape consists of unphysical flat local minima, thus rendering a good initial guess key for the overall convergence of the optimization problem, particularly for solvers with adaptive steps.