We study \textit{rescaled gradient dynamical systems} in a Hilbert space $\mathcal{H}$, where implicit discretization in a finite-dimensional Euclidean space leads to high-order methods for solving monotone equations (MEs). Our framework can be interpreted as a natural generalization of celebrated dual extrapolation method from first order to high order via appeal to the regularization toolbox of optimization theory. In particular, we establish the global existence and uniqueness and analyze the convergence properties of solution trajectories. We also present discrete-time counterparts of our high-order continuous-time methods, and we show that the $p^{th}$-order method achieves an ergodic rate of $O(k^{-(p+1)/2})$ in terms of a restricted merit function and a pointwise rate of $O(k^{-p/2})$ in terms of a residue function. Under regularity conditions, the restarted version of $p^{th}$-order methods achieves local convergence with the order $p \geq 2$. Notably, our methods are \textit{optimal} since they have matched the lower bound established for solving the monotone equation problems.

翻译：我们在一个 Hilbert 空间 $\ mathcal{H} 中研究\ textit{ 缩放梯度动态系统} $\ mathcal{H} $。 在这种空间中,在有限维度的 Euclidean 空间中,隐含的离散导致解决单色方程式(MEs) 的高阶方法。 我们的框架可以被解释为一种自然的概括,即从一阶到高阶的已知双极外推法。 通过呼吁优化理论的正规化工具框,我们特别建立了全球存在和独特性,并分析了解决方案轨迹的趋同性。 我们还介绍了我们高阶连续时间方法的离散对应方,并且我们显示, $p{th} $- sord 方法在有限的功绩功能方面达到了1美元(k ⁇ - (p+1/2) /2} 的ergodic 率。 和 $O (k ⁇ - pp/2} 的近似速率 。 在正常条件下, $ $ 重新启用的版本的 $_trotimal 方法在本地上实现了它们与 $\geqq $ 2$ 之间的解算问题。