For a long time it has been well-known that high-dimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort which grows polynomially both in the dimension and in the reciprocal of the prescribed accuracy. In other words, linear PDEs do not suffer from the curse of dimensionality. For general semilinear PDEs with Lipschitz coefficients, however, it remained an open question whether these suffer from the curse of dimensionality. In this paper we partially solve this open problem. More precisely, we prove in the case of semilinear heat equations with gradient-independent and globally Lipschitz continuous nonlinearities that the computational effort of a variant of the recently introduced multilevel Picard approximations grows at most polynomially both in the dimension and in the reciprocal of the required accuracy.

中文翻译：

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克服半线性抛物型偏微分方程数值逼近中的维数灾难

长期以来，众所周知，高维线性抛物线偏微分方程 (PDE) 可以通过蒙特卡罗方法逼近，计算量在维度和规定精度的倒数上呈多项式增长。换句话说，线性偏微分方程不会受到维数灾难的影响。然而，对于具有 Lipschitz 系数的一般半线性偏微分方程，它们是否受到维数灾难的影响仍然是一个悬而未决的问题。在本文中，我们部分解决了这个开放问题。更确切地说，