Multilevel Picard approximations overcome the curse of dimensionality when approximating semilinear heat equations with gradient-dependent nonlinearities in Lp-sense

Abstract

We prove that multilevel Picard approximations are capable of approximating solutions of semilinear heat equations in $L^p$-sense, $p\in [2,\infty )$, in the case of gradient-dependent, Lipschitz-continuous nonlinearities, in the sense that the computational effort of the multilevel Picard approximations grows at most polynomially in both the dimension $d$ and the reciprocal $1/ϵ$ of the prescribed accuracy $ϵ$.