Overcoming the Curse of Dimensionality in the Numerical Approximation of Parabolic Partial Differential Equations with Gradient-Dependent Nonlinearities

Abstract

Partial differential equations (PDEs) are a fundamental tool in the modeling of many real-world phenomena. In a number of such real-world phenomena the PDEs under consideration contain gradient-dependent nonlinearities and are high-dimensional. Such high-dimensional nonlinear PDEs can in nearly all cases not be solved explicitly, and it is one of the most challenging tasks in applied mathematics to solve high-dimensional nonlinear PDEs approximately. It is especially very challenging to design approximation algorithms for nonlinear PDEs for which one can rigorously prove that they do overcome the so-called curse of dimensionality in the sense that the number of computational operations of the approximation algorithm needed to achieve an approximation precision of size $${\varepsilon }> 0$$ ε > 0 grows at most polynomially in both the PDE dimension $$d \in \mathbb {N}$$ d ∈ N and the reciprocal of the prescribed approximation accuracy $${\varepsilon }$$ ε . In particular, to the best of our knowledge there exists no approximation algorithm in the scientific literature which has been proven to overcome the curse of dimensionality in the case of a class of nonlinear PDEs with general time horizons and gradient-dependent nonlinearities. It is the key contribution of this article to overcome this difficulty. More specifically, it is the key contribution of this article (i) to propose a new full-history recursive multilevel Picard approximation algorithm for high-dimensional nonlinear heat equations with general time horizons and gradient-dependent nonlinearities and (ii) to rigorously prove that this full-history recursive multilevel Picard approximation algorithm does indeed overcome the curse of dimensionality in the case of such nonlinear heat equations with gradient-dependent nonlinearities.

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

“…Next, (39), Jensen's inequality, a Lyapunov-type estimate (see, e.g., [15, Lemma 2.2]) combined with (35) and (38), the fact that c ≤ ρ, and Lemma 2.1 (applied with β β − 1 in the notation of Lemma 2.1) combined with (37)-( 39), (33), the assumption that κ ∈ [0, p/(3β + 1)], and the fact that ∀ h ∈ (0, T ], x ∈ D h : ϕ(x) ≤ c(b 3 h) −κ (see (45)) imply that for all t ∈ [t 0 , T ], q ∈ [1, ∞) with βq ≤ p it holds that µ(Y θ,x 0 t 0 ,t )…”

“…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.…”

Active Learning Based Sampling for High-Dimensional Nonlinear Partial Differential Equations

“…These theoretical results have justified the application of DNNs to solve high-dimensional PDEs recently in [24,25,26,27,28,29,27,30,31]. Two main advantages of deep-learning-based methods presented in these studies are summarizes as follows: firstly, the curse of dimensionality can be weakened or even be overcome in certain classes of PDEs [32,33]; secondly, deep-learning-based PDE solvers are mesh-free without tedious mesh generation for complex domains in traditional solvers. Thus, deeplearning-based methods have shown tremendous potential to surpass other numerical methods especially in solving high-dimensional PDEs in complex domains.…”

Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations

“…MLP approximations have previously been shown to overcome the curse of dimensionality in the case of a number of semilinear PDE problems (cf. [7,8,11,12,46,47,54,[71][72][73][74][75]) and this is also the key ingredient in this article to overcome the curse of dimensionality in the numerical approximation of solution paths of BSDEs.…”