标题： 双曲系统的快速伴随响应方法最优响应 显示英文标题

标题： Optimal Response for Hyperbolic Systems by the fast adjoint response method

摘要： 在均匀双曲系统中，我们考虑从一个可行的集合$P$中找到最优的无穷小扰动，以最大程度地增加给定观测函数的期望值。 我们通过与接近单位映射的微分同胚复合，或者通过向动力学中添加确定性扰动来扰动系统。 在这两种情况下，使用快速伴随响应公式，我们证明了线性响应算子，该算子将期望值对动力学扰动的响应联系起来，在扰动的$C^{1,\alpha}$范数方面是有界的。 在假设$P$是一个严格凸的、闭合的希尔伯特空间$\cH$的子集，并且可以连续映射到我们相空间上的$C^3$向量场空间的情况下，我们证明了在$P$中存在唯一的最优扰动，它最大化给定观测函数的增加量。 此外，由于响应算子由 $\cH$的一个元素 $v$表示，当可行集 $P$是 $\cH$的单位球时，最优扰动是 $v/||v||_{\cH}$。 我们还展示了如何在不同情况下计算傅里叶展开 $v$。 我们的方法甚至可以在高维系统上使用。 我们在二维、三维和二十一维的数值例子中演示了我们的方法。 显示英文摘要

摘要： In a uniformly hyperbolic system, we consider the problem of finding the optimal infinitesimal perturbation to apply to the system, from a certain set $P$ of feasible ones, to maximally increase the expectation of a given observation function. We perturb the system both by composing with a diffeomorphism near the identity or by adding a deterministic perturbation to the dynamics. In both cases, using the fast adjoint response formula, we show that the linear response operator, which associates the response of the expectation to the perturbation on the dynamics, is bounded in terms of the $C^{1,\alpha}$ norm of the perturbation. Under the assumption that $P$ is a strictly convex, closed subset of a Hilbert space $\cH$ that can be continuously mapped in the space of $C^3$ vector fields on our phase space, we show that there is a unique optimal perturbation in $P$ that maximizes the increase of the given observation function. Furthermore since the response operator is represented by a certain element $v$ of $\cH$, when the feasible set $P$ is the unit ball of $\cH$, the optimal perturbation is $v/||v||_{\cH}$. We also show how to compute the Fourier expansion $v$ in different cases. Our approach can work even on high dimensional systems. We demonstrate our method on numerical examples in dimensions 2, 3, and 21.