两类随机微分方程数值方法的强收敛和弱收敛性研究

As an important type of stochastic differential equation (SDE), stochastic evolution equation (SEE) has found a wide utilization in many science fields including biology, control engineering and neurophysiology. However, the analytic solutions of the most SEEs can not be obtained, it is thus necessary to find the numerical solutions to SEEs. As a result of the existence of the nonlinear function in the multiplicative noise, some standard convergence theorys for SEEs with additive noise do not apply to SEEs driven by multiplicative noise. The present project is to analyze the errors of the numerical methods for SEEs driven by multiplicative noise. First of all, we study the optimal strong convergence of the finite element method for a semilinear stochastic parabolic equation with a nonselfadjoint operator, and then research the weak convergence of numerical methods for it. Furthermore, the project will focus on the weak convergence of the finite element methods, spectral methods and finite difference schemes for a semilinear stochastic strongly damped wave equation, and then offer efficient numerical algorithms for these methods. The topic not only enriches the content of the numerical methods for SDEs, but also provides a guide to construct practical and efficient numerical algorithms.

作为一类重要的随机微分方程，随机发展方程在许多科学领域中(如生物学、控制工程及神经生理学等)有着重要的应用价值，然而大部分随机发展方程解析解无法获得，因此其数值求解显得十分必要。由于乘性噪声项是关于方程解的非线性随机函数，许多适合于带有加性噪声项的随机发展方程的经典收敛理论已经不适用于带有乘性噪声项的方程。本项目拟研究带有乘性噪声项的随机发展方程数值方法的收敛性：针对含有非自伴算子的半线性随机抛物方程，研究有限元方法的最优强收敛性, 在此基础上进一步探索其数值方法的弱收敛理论，并得到其收敛阶；针对带有强阻尼项半线性随机双曲方程，研究有限元方法、谱方法及差分法等数值方法的弱收敛性，并在此基础上提出弱收敛的数值算法。本项目拟研究的内容不仅会丰富随机微分方程数值方法的研究成果，还将为随机微分方程数值算法提供理论依据。

带有噪声项的随机偏微分方程在许多科学领域中(如生物学、控制工程及神经生理学.等)有着重要的应用价值，然而大部分的随机偏微分方程解析解无法获得，因.此其数值求解显得十分必要。本项目拟研究带有噪声项的随机偏微分方程数值方法的收敛性：针对一类非线性随机抛物方程，研究了有限元方法的最优强收敛性，得到了最优收敛阶；针对带有强阻尼项的半线性的随机双曲方程，研究了有限元方法并得到了数值方法的的收敛性和收敛阶。本项目所研究的内容不仅丰富了随机偏微分方程数值.方法的研究成果，还将为随机偏微分方程数值算法提供理论依据。.在国际重要学术期刊上发表SCI收录论文4篇，专著一本。