The recently proposed statistical finite element (statFEM) approach synthesises measurement data with finite element models and allows for making predictions about the true system response. We provide a probabilistic error analysis for a prototypical statFEM setup based on a Gaussian process prior under the assumption that the noisy measurement data are generated by a deterministic true system response function that satisfies a second-order elliptic partial differential equation for an unknown true source term. In certain cases, properties such as the smoothness of the source term may be misspecified by the Gaussian process model. The error estimates we derive are for the expectation with respect to the measurement noise of the $L^2$-norm of the difference between the true system response and the mean of the statFEM posterior. The estimates imply polynomial rates of convergence in the numbers of measurement points and finite element basis functions and depend on the Sobolev smoothness of the true source term and the Gaussian process model. A numerical example for Poisson's equation is used to illustrate these theoretical results.
翻译:最近提议的统计有限要素(STATFEM)方法将测量数据与有限要素模型合成,并允许对系统的真实反应作出预测。我们提供了一种概率错误分析,用于在Gaussian程序之前的假设下建立原型的SatFEM, 假设噪音的测量数据是由确定性真实系统响应功能生成的,该功能满足了二级椭圆偏差部分等分方程式,而该等值对于一个未知的真实源术语是一个未知的真实源术语。在某些情况下,源术语的平滑性等属性可能被Gaussian进程模型错误地描述。我们得出的错误估计是为了预测对真实系统响应与StagFEM后端函数平均值之间差异的测量噪音($L2$-norm)的预期值。估计意味着测量点数和有限元素基函数数的多重趋同率,并取决于真实源术语和高斯进程模型的索波列夫斯平滑度。Poisson的等式数字示例用于说明这些理论结果。