Multilevel Picard iterations for solving smooth semilinear parabolic heat equations
For researchers in computational mathematics and finance, this provides a method with provable complexity bounds for high-dimensional semilinear PDEs, though the analysis is limited to gradient-independent nonlinearities.
The paper introduces a new family of numerical algorithms for solving high-dimensional semilinear parabolic PDEs at single space-time points, achieving computational complexity bounded by O(d ε^{-(4+δ)}) for any δ>0, where d is dimension and ε is accuracy.
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 very satisfactory results. Analytical tools needed for the analysis of such algorithms, including a semilinear Feynman-Kac formula, a new class of semi-norms and their recursive inequalities, are also introduced. They allow us to prove for semilinear heat equations with gradient-independent nonlinearity that the computational complexity of the proposed algorithm is bounded by $O(d\,\varepsilon^{-(4+δ)})$ for any $δ\in (0,\infty)$ under suitable assumptions, where $d\in \mathbb{N}$ is the dimensionality of the problem and $\varepsilon\in(0,\infty)$ is the prescribed accuracy.