Full-history recursive multilevel Picard (MLP) approximation schemes have been shown to overcome the curse of dimensionality in the numerical approximation of high-dimensional semilinear partial differential equations (PDEs) with general time horizons and Lipschitz continuous nonlinearities. However, each of the error analyses for MLP approximation schemes in the existing literature studies the $L^2$-root-mean-square distance between the exact solution of the PDE under consideration and the considered MLP approximation and none of the error analyses in the existing literature provides an upper bound for the more general $L^p$-distance between the exact solution of the PDE under consideration and the considered MLP approximation. It is the key contribution of this article to extend the $L^2$-error analysis for MLP approximation schemes in the literature to a more general $L^p$-error analysis with $p\in (0,\infty)$. In particular, the main result of this article proves that the proposed MLP approximation scheme indeed overcomes the curse of dimensionality in the numerical approximation of high-dimensional semilinear PDEs with the approximation error measured in the $L^p$-sense with $p \in (0,\infty)$.

翻译：已经显示,全历史递归多级Picard(MLP)近似方案克服了高维半线性局部偏差方程(PDEs)数字近似和一般时空和Lipschitz连续非线性等离线性数字近似中的维度诅咒。然而,现有文献中MLP近似方案每次错误分析分析都研究了正在审议的PDE的精确解决方案与考虑的MLP近似(MLP)近似(MLP)的准确解决方案和考虑的MLP近似(PDEs)的准确解决方案之间较一般的值$Lp$-p$的距离。将文献中MLP($2$-eror)对MLP(MLP)近似方案的分析扩展为更一般的$L%p美元(0.0,/infty)美元的分析。特别是,这一文章的主要结果证明,拟议的MLP(P)近似方案确实克服了所考虑的P-decolentyal-alimal $(美元) IMSirimalimalisimal-Ils-IL.