For a simple HJB control example, MLP approximations have L2 error growing like d^{n/2}/(c^n sqrt(n!)) - 1, so no polynomial-in-dimension error bound can hold uniformly as the number of levels grows.
Multilevel Picard algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities
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abstract
In this paper we introduce a multilevel Picard approximation algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities whose coefficient functions do not need to be constant. We also provide a full convergence and complexity analysis of our algorithm. To obtain our main results, we consider a particular stochastic fixed-point equation (SFPE) motivated by the Feynman-Kac representation and the Bismut-Elworthy-Li formula. We show that the PDE under consideration has a unique viscosity solution which coincides with the first component of the unique solution of the stochastic fixed-point equation. Moreover, the gradient of the unique viscosity solution of the PDE exists and coincides with the second component of the unique solution of the stochastic fixed-point equation. Furthermore, we also provide a numerical example in up to $300$ dimensions to demonstrate the practical applicability of our multilevel Picard algorithm.
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Full history recursive multilevel Picard approximations suffer from the curse of dimensionality for the Hamilton-Jacobi-Bellman equation of a stochastic control problem
For a simple HJB control example, MLP approximations have L2 error growing like d^{n/2}/(c^n sqrt(n!)) - 1, so no polynomial-in-dimension error bound can hold uniformly as the number of levels grows.