论文标题
具有Lipschitz非线性的高维半二阶PDE的多级PICARD近似
Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities
论文作者
论文摘要
最近引入的全历史递归多级PICARD（MLP）近似方法已在高维非线性PDE的溶液的数值近似中非常成功。特别是，文献中存在数学收敛的结果，证明MLP的近似方法确实克服了非线性二阶PDE的数值近似值的诅咒，这是在提出的MLP近似方法的计算方法数量中，在两种情况下，在两种情况下都在$ 1/$ $ 1/$ $ 1/$ 1 $ 1/amper y的计算方法中生长。 pde dimension $ d \ in \ mathbb {n} = \ {1,2,3，\ ldots \} $。但是，在文献中MLP近似方法的每个收敛结果中，假定二阶差异操作员面前的系数函数是仿射线性的。特别是，直到今天，科学文献尚无结果证明，任何具有一般时间范围内的半线性二阶PDE，并且在二阶差异算子面前的非仿射线性系数函数都可以在没有维度的诅咒的情况下近似。这是本文克服这一障碍的关键贡献，并为半二阶PDE提出和分析一种新型的MLP近似方法，并在二阶差异操作员面前可能具有非线性系数功能。特别是，本文的主要结果证明，这种新的MLP近似方法确实确实克服了半线性二阶PDE的数值近似中维度的诅咒。
The recently introduced full-history recursive multilevel Picard (MLP) approximation methods have turned out to be quite successful in the numerical approximation of solutions of high-dimensional nonlinear PDEs. In particular, there are mathematical convergence results in the literature which prove that MLP approximation methods do overcome the curse of dimensionality in the numerical approximation of nonlinear second-order PDEs in the sense that the number of computational operations of the proposed MLP approximation method grows at most polynomially in both the reciprocal $1/ε$ of the prescribed approximation accuracy $ε>0$ and the PDE dimension $d\in \mathbb{N}=\{1,2,3, \ldots\}$. However, in each of the convergence results for MLP approximation methods in the literature it is assumed that the coefficient functions in front of the second-order differential operator are affine linear. In particular, until today there is no result in the scientific literature which proves that any semilinear second-order PDE with a general time horizon and a non affine linear coefficient function in front of the second-order differential operator can be approximated without the curse of dimensionality. It is the key contribution of this article to overcome this obstacle and to propose and analyze a new type of MLP approximation method for semilinear second-order PDEs with possibly nonlinear coefficient functions in front of the second-order differential operators. In particular, the main result of this article proves that this new MLP approximation method does indeed overcome the curse of dimensionality in the numerical approximation of semilinear second-order PDEs.
