Abstract One of the most challenging problems in applied mathematics is the approximate solution of nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministic approximation methods like finite differences or finite elements suffer from the curse of dimensionality in the sense that the computational effort grows exponentially in the dimension. In this work we overcome this difficulty in the case of reaction–diffusion type PDEs with a locally Lipschitz continuous coervice nonlinearity (such as Allen–Cahn PDEs) by introducing and analyzing truncated variants of the recently introduced full-history recursive multilevel Picard approximation schemes.

中文翻译：

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通过截断的全历史递归多级 Picard 近似克服 Allen-Cahn 偏微分方程数值近似中的维数灾难

摘要 应用数学中最具挑战性的问题之一是高维非线性偏微分方程 (PDE) 的近似解。标准的确定性近似方法（如有限差分或有限元）遭受维度灾难的影响，因为计算工作量在维度上呈指数增长。在这项工作中，我们通过引入和分析最近引入的全历史递归多级皮卡德近似方案的截断变体，克服了具有局部 Lipschitz 连续矫顽非线性的反应扩散型 PDE（例如 Allen-Cahn PDE）的困难。