Generalised multilevel Picard approximations

Abstract

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.