Strong $L^p$-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations

Abstract

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 ∈ (0, ∞). 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 highdimensional semilinear PDEs with the approximation error measured in the L p -sense with p ∈ (0, ∞).

Multilevel Picard Approximations of High-Dimensional Semilinear Parabolic Differential Equations with Gradient-Dependent Nonlinearities

“…To the best of our knowledge the only approximation method which has been mathematically proved to overcome the curse of dimensionality for certain semilinear PDEs is the full history recursive multilevel Picard (MLP) method introduced in [20] and analyzed, e.g., in [45,48,53,46,51,6,50,27,6]. In this article we extend the analysis of MLP approximations to the case of semilinear PDEs with locally monotone coefficient functions and globally Lipschitz continuous, gradient-independent nonlinearities.…”

Multilevel Picard approximations for high-dimensional decoupled forward-backward stochastic differential equations

“…The assumptions of Proposition 3.3 are collected in the following Setting 3.1 and include global Lipschitz continuity of the nonlinearity f , local Lipschitz continuity of the terminal condition g whose local Lipschitz constant grows at most like the Lyapunov-type function V , and strong regularity estimates (37) and (38) for the forward diffusion. Proposition 3.3 extends the analysis of [35] where the foward diffusion is Brownian motion. Moreover, Corollary 3.4 shows that MLP method approximates semilinear PDEs with a computatinal effort which is of order 2+ in the reciprocal accuracy 1 /ε and at most of polynomial order in the dimension.…”

Generic bounds on the approximation error for physics-informed (and) operator learning

“…In what follows, we will provide a definition of a particular kind of MLP approximation (cf. [30]) and a theorem that quantifies the accuracy of the approximation. First, we rigorously introduce the setting of the nonlinear parabolic PDE (4.3) that is under consideration, cf.…”

“…We choose the d-dimensional torus T d = [0, 2π) d as domain and impose periodic boundary conditions. This setting allows us to use the results of [30], which are set in R d , and yet still consider a bounded domain so that the error can be quantified using an uniform probability measure.…”

“…In this setting, multilevel Picard approximations can be introduced. We follow the definition of [30].…”