SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1171-1190, June 2024.
Abstract. We prove a new numerical approximation result for the solutions of semilinear parabolic stochastic partial differential equations, driven by additive space-time white noise in dimension 1. The temporal discretization is performed using an accelerated exponential Euler scheme, and we show that, under appropriate regularity conditions on the nonlinearity, the total variation distance between the distributions of the numerical approximation and of the exact solution at a given time converges to 0 when the time-step size vanishes, with order of convergence [math]. Equivalently, weak error estimates with order [math] are thus obtained for bounded measurable test functions. This is an original and major improvement compared with the performance of the standard linear implicit Euler scheme or exponential Euler methods, which do not converge in the sense of total variation when the time-step size vanishes. Equivalently weak error estimates for the standard schemes require twice differentiable test functions. The proof of the total variation error bounds for the accelerated exponential Euler scheme exploits some regularization property of the associated infinite-dimensional Kolmogorov equations.

中文翻译：

---

抛物线半线性 SPDE 的加速指数欧拉方案逼近的总变分误差界


《SIAM 数值分析杂志》，第 62 卷，第 3 期，第 1171-1190 页，2024 年 6 月。

抽象的。我们证明了由 1 维加性时空白噪声驱动的半线性抛物型随机偏微分方程解的新数值近似结果。使用加速指数欧拉方案进行时间离散化，并且我们表明，在适当的规律下在非线性条件下，当时间步长消失时，给定时间的数值近似分布和精确解分布之间的总变异距离收敛到 0，收敛顺序为 [数学]。等价地，对于有界可测量测试函数，获得具有阶数[math]的弱误差估计。与标准线性隐式欧拉方案或指数欧拉方法的性能相比，这是一个原始且重大的改进，当时间步长消失时，标准线性隐式欧拉方案或指数欧拉方法在总变差的意义上不收敛。标准方案的同等弱误差估计需要两次可微测试函数。加速指数欧拉方案的总变分误差界的证明利用了相关无限维柯尔莫哥洛夫方程的一些正则化性质。