For $\mathcal{O}$ a bounded domain in $\mathbb{R}^d$ and a given smooth function $g:\mathcal{O}\to\mathbb{R}$, we consider the statistical nonlinear inverse problem of recovering the conductivity $f>0$ in the divergence form equation $$ \nabla\cdot(f\nabla u)=g\ \textrm{on}\ \mathcal{O}, \quad u=0\ \textrm{on}\ \partial\mathcal{O}, $$ from $N$ discrete noisy point evaluations of the solution $u=u_f$ on $\mathcal O$. We study the statistical performance of Bayesian nonparametric procedures based on a flexible class of Gaussian (or hierarchical Gaussian) process priors, whose implementation is feasible by MCMC methods. We show that, as the number $N$ of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate $N^{-\lambda}, \lambda>0,$ for the reconstruction error of the associated posterior means, in $L^2(\mathcal{O})$-distance.

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椭圆逆问题中贝叶斯推理与高斯过程先验的一致性

对于在 $\mathbb{R}^d$ 中的有界域 $\mathcal{O}$ 和给定的平滑函数 $g:\mathcal{O}\to\mathbb{R}$，我们考虑统计非线性逆问题在发散形式方程中恢复电导率 $f>0$ $$ \nabla\cdot(f\nabla u)=g\ \textrm{on}\ \mathcal{O}, \quad u=0\ \textrm{ on}\ \partial\mathcal{O}, $$ 来自 $N$ 离散噪声点对 $\mathcal O$ 上的解决方案 $u=u_f$ 的评估。我们研究了基于灵活类高斯（或分层高斯）过程先验的贝叶斯非参数程序的统计性能，其实现是可行的 MCMC 方法。我们表明，随着测量数量 $N$ 的增加，产生的后验分布集中在生成数据的真实参数周围，并得出收敛率 $N^{-\lambda}, \lambda>0，