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 grow at most polynomially in both the dimension $d$ and the reciprocal $1/\epsilon$ of the prescribed accuracy $\epsilon$.

翻译：我们证明，对于梯度依赖且Lipschitz连续的非线性项，多层Picard近似能够在$L^{p}$意义下（${p}\in [2,\infty)$）逼近半线性热方程的解。具体而言，多层Picard近似的计算量在维度$d$和给定精度$\epsilon$的倒数$1/\epsilon$上至多以多项式速度增长。