Overcoming the curse of dimensionality in the numerical approximation of Allen–Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations

Abstract

<jats:title>Abstract</jats:title><jats:p>One of the most challenging problems in applied mathematics is the approximate solution of nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministic approximation methods like finite differences or finite elements suffer from the curse of dimensionality in the sense that the computational effort grows exponentially in the dimension. In this work we overcome this difficulty in the case of reaction-diffusion type PDEs with a locally Lipschitz continuous coervice nonlinearity (such as Allen–Cahn PDEs) by introducing and analyzing truncated variants of the recently introduced full-history recursive multilevel Picard approximation schemes.</jats:p>

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

“…MLP approximation methods are, roughly speaking, based on the idea to (IIa) reformulate the PDE under consideration as a stochastic fixed point problem with the PDE solution being the fixed point of the stochastic fixed point equation, to (IIb) approximate the fixed point through Banach fixed point iterates (which are also referred to as Picard iterates in the context of integral fixed point equations), and to (IIc) approximate the resulting Banach fixed point iterates through suitable full-history recursive multilevel Monte Carlo approximations. In the case of MLP approximation methods there are both encouraging numerical simulation results (see [8,19]) and rigorous mathematical results which prove that MLP approximation methods do indeed overcome the curse of dimensionality in the numerical approximation of nonlinear second-order PDEs (see [5,6,18,24,35,[37][38][39]). However, in each of the convergence results for MLP approximation methods in the scientific literature it is assumed that the coefficient functions in front of the second-order differential operator are affine linear.…”

“…Especially, there are two types of approximation methods which have turned out to be quite successful in the numerical approximation of solutions of high-dimensional nonlinear secondorder PDEs, namely, (I) deep learning based approximation methods for PDEs (cf., e.g., [1-3, 9-11, 13, 14, 16, 17, 20, 22, 23, 25, 28-32, 41, 43, 46-52, 55, 56]) and (II) full-history recursive multilevel Picard approximation methods for PDEs (cf. [6,8,18,19,24,35,[37][38][39]; in the following we abbreviate full-history recursive multilevel Picard as MLP). Deep learning based approximation methods for PDEs are, roughly speaking, based on the idea to (Ia) approximate the PDE problem under consideration through a stochastic optimization problem involving deep neural networks as approximations for the solution or the derivatives of the solution of the PDE under consideration and to (Ib) apply stochastic gradient descent methods to approximately solve the resulting stochastic optimization problem.…”

“…can be approximated by the MLP approximations (5) with the number of involved function evaluations of f , u d (T, •), µ d , and σ d and the number of involved scalar random variables growing at most quintically in the reciprocal 1/ε of the prescribed approximation accuracy ε ∈ (0, ∞) and at most linearly in the PDE dimension d ∈ AE. Our proofs of Theorem 1.1 above and Theorem 4.2 below, respectively, are partially based on previous analyses for MLP approximations in the scientific literature (cf., e.g., [5,6,18,24,35,[37][38][39]) and on analyses for numerical approximations for SDEs with non-globally Lipschitz continuous coefficient functions (cf., e.g., [33]).…”

Numerical Simulations for Full History Recursive Multilevel Picard Approximations for Systems of High-Dimensional Partial Differential Equations

“…(I) Deep learning based approximation methods for PDEs; cf., e.g., [3-5, 9-12, 15-17, 19, 20, 24, 26, 28, 31, 36-40, 46, 48, 49, 52-63, 65, 66] (II) Full history recursive multilevel Picard approximation methods for PDEs; cf., e.g., [6,7,22,23,29,41,[43][44][45] (in the following we abbreviate full history recursive multilevel Picard by MLP)…”

On existence and uniqueness properties for solutions of stochastic fixed point equations

“…SFPEs of the form as in (2) have a strong connection with semilinear Kolmogorov PDEs and arise, for example, in models from the environmental sciences as well as in pricing problems from financial engineering (cf., for example, Burgard & Kjaer [4], Crépey et al [5], Duffie et al [6], and Henry-Labordère [10]). SFPEs such as (2) are also important for full-history recursive multilevel Picard approximation (MLP) methods, which were recently introduced in [11,13]; see also [1,12,14,15]. In [13,14] it has been shown that functions which satisfy SFPEs related to semilinear Kolmogorov PDEs can be approximated by MLP schemes without the curse of dimensionality.…”