The Backward Problem of Stochastic Convection-Diffusion Equation

In this paper, we consider a backward problem for the stochastic convection-diffusion equation. The source term is driven by the fraction Brownian motion. We illustrate the regularity of the mild solution and prove the instability of this problem. In order to overcome the ill-posedness, we apply a truncated regularization method to obtain a stable numerical approximation to u(x, t). Convergence estimates are presented under the a-priori parameter choice rule. Finally, some numerical experiments are given to show the effectivity of the regularization method.

Automated summary

In this paper, a backward problem for the stochastic convection-diffusion equation with source term driven by the fraction Brownian motion is considered. The regularity of the mild solution is illustrated and the instability of the problem is proved. A truncated regularization method is applied to overcome ill-posedness and obtain a stable numerical approximation to u(x, t), with convergence estimates presented under the a-priori parameter choice rule. Numerical experiments are conducted to demonstrate the effectiveness of the regularization method.