This paper introduces SPDE bridges with observation noise and contains an analysis of their spatially semidiscrete approximations. The SPDEs are considered in the form of mild solutions in an abstract Hilbert space framework suitable for parabolic equations. They are assumed to be linear with additive noise in the form of a cylindrical Wiener process. The observational noise is also cylindrical and SPDE bridges are formulated via conditional distributions of Gaussian random variables in Hilbert spaces. A general framework for the spatial discretization of these bridge processes is introduced. Explicit convergence rates are derived for a spectral and a finite element based method. It is shown that for sufficiently rough observation noise, the rates are essentially the same as those of the corresponding discretization of the original SPDE.

翻译：本文介绍SPDE桥及其观测噪音,并分析其空间半分立近似值。SPDE以轻度解决办法的形式,在适合抛物线方程式的抽象Hilbert空间框架内加以考虑。它们被假定为线性,以圆柱形维纳进程的形式具有添加噪音。观测噪音也是圆柱形的,SPDE桥通过Hilbert空间高西亚随机变量的有条件分布而形成。引入了这些桥梁过程的空间分解总框架。光谱和以有限元素为基础的方法的明显汇合率是推导出来的。在足够粗略的观测噪音中,这些比率基本上与原SPDE的相应离散率相同。