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, while the computational effort of the multilevel Picard approximations grow at most polynomially in both dimension $d$ and prescribed reciprocal accuracy $\epsilon$.

翻译：我们证明，在梯度依赖、Lipschitz连续的非线性项情形下，多层Picard近似能够以$L^{p}$意义（${p}\in [2,\infty)$）逼近半线性热方程的解，且多层Picard近似的计算量在维数$d$和预设精度倒数$\epsilon$上至多以多项式速度增长。