标题： 广义多级Picard近似 显示英文标题

标题： Generalised multilevel Picard approximations

摘要： 在应用数学中，近似求解高维偏微分方程（PDEs）是最具挑战性的问题之一。 特别是，在科学文献中研究的大多数数值逼近方案都受到维度灾难的影响，即为了计算误差不超过$ \varepsilon > 0 $的近似值所需的计算操作数量至少随着PDE的维度$ d \in \mathbb{N} $或者$ \varepsilon $的倒数呈指数增长。 最近，所谓的全历史递归多级皮卡（MLP）逼近方法被引入，以解决近似求解高维PDEs的问题。 据我们所知，目前MLP逼近方法是唯一一类对于具有通用时间范围和通用初始条件的抛物型半线性PDEs，有严格证明能够真正克服维度灾难的方法。 本文的主要目的是更深入地研究MLP逼近方法，更清晰地揭示这些方法如何克服维度灾难，并提出一个广义的MLP逼近方案类，该类可以涵盖之前分析过的MLP逼近方案作为特例。 特别是，我们开发了一个抽象框架，可以在其中表述和分析这一类广义的MLP逼近，并随后应用这个抽象框架，推导出适用于半线性热方程的适当MLP逼近的计算复杂度结果。 这些针对半线性热方程的MLP逼近本质上是对之前引入的半线性热方程MLP逼近的推广。 显示英文摘要

摘要： It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). In particular, most of the numerical approximation schemes studied in the scientific literature suffer under the curse of dimensionality in the sense that the number of computational operations needed to compute an approximation with an error of size at most $ \varepsilon > 0 $ grows at least exponentially in the PDE dimension $ d \in \mathbb{N} $ or in the reciprocal of $ \varepsilon $. Recently, so-called full-history recursive multilevel Picard (MLP) approximation methods have been introduced to tackle the problem of approximately solving high-dimensional PDEs. MLP approximation methods currently are, to the best of our knowledge, the only methods for parabolic semi-linear PDEs with general time horizons and general initial conditions for which there is a rigorous proof that they are indeed able to beat the curse of dimensionality. The main purpose of this work is to investigate MLP approximation methods in more depth, to reveal more clearly how these methods can overcome the curse of dimensionality, and to propose a generalised class of MLP approximation schemes, which covers previously analysed MLP approximation schemes as special cases. In particular, we develop an abstract framework in which this class of generalised MLP approximations can be formulated and analysed and, thereafter, apply this abstract framework to derive a computational complexity result for suitable MLP approximations for semi-linear heat equations. These resulting MLP approximations for semi-linear heat equations essentially are generalisations of previously introduced MLP approximations for semi-linear heat equations.