Abstract This paper studies the numerical approximation of the density of the stochastic heat equation driven by space-time white noise via the accelerated exponential Euler scheme. The existence and smoothness of the density of the numerical solution are proved by means of Malliavin calculus. Based on a priori estimates of the numerical solution we present a test-function-independent weak convergence analysis, which is crucial to show the convergence of the density. The convergence order of the density in uniform convergence topology is shown to be exactly $1/2$ in the nonlinear drift case and nearly $1$ in the affine drift case. As far as we know, this is the first result on the existence and convergence of density of the numerical solution to the stochastic partial differential equation.

中文翻译：

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随机热方程的加速指数欧拉格式：密度的收敛速度

摘要 本文通过加速指数欧拉格式研究时空白噪声驱动的随机热方程的密度数值逼近。用Malliavin演算证明了数值解密度的存在性和平滑性。基于数值解的先验估计，我们提出了与测试函数无关的弱收敛分析，这对于显示密度的收敛至关重要。均匀收敛拓扑中密度的收敛阶在非线性漂移情况下正好是 $1/2$，在仿射漂移情况下接近 $1$。据我们所知，这是关于随机偏微分方程数值解的密度存在性和收敛性的第一个结果。