Multilevel Picard iterations for solving smooth semilinear parabolic heat equations

We introduce a new family of numerical algorithms for approximating solutions of general high-dimensional semilinear parabolic partial differential equations at single space-time points. The algorithm is obtained through a delicate combination of the Feynman–Kac and the Bismut–Elworthy–Li formulas, and an approximate decomposition of the Picard fixed-point iteration with multilevel accuracy. The algorithm has been tested on a variety of semilinear partial differential equations that arise in physics and finance, with satisfactory results. Analytical tools needed for the analysis of such algorithms, including a semilinear Feynman–Kac formula, a new class of seminorms and their recursive inequalities, are also introduced. They allow us to prove for semilinear heat equations with gradient-independent nonlinearities that the computational complexity of the proposed algorithm is bounded by O(d\,ε -(4+δ )) O ( d ε - ( 4 + δ ) ) for any δ ∈ (0,∞ ) δ ∈ ( 0 , ∞ ) under suitable assumptions, where d∈ N d ∈ N is the dimensionality of the problem and ε ∈ (0,∞ ) ε ∈ ( 0 , ∞ ) is the prescribed accuracy. Moreover, the introduced class of numerical algorithms is also powerful for proving high-dimensional approximation capacities for deep neural networks.