We consider the recovery of an unknown function $f$ from a noisy observation of the solution $u_f$ to a partial differential equation that can be written in the form $\mathcal{L} u_f=c(f,u_f)$, for a differential operator $\mathcal{L}$ that is rich enough to recover $f$ from $\mathcal{L} u_f$. Examples include the time-independent Schr\"odinger equation $\Delta u_f = 2u_ff$, the heat equation with absorption term $(\partial_t -\Delta_x/2) u_f=fu_f$, and the Darcy problem $\nabla\cdot (f \nabla u_f) = h$. We transform this problem into the linear inverse problem of recovering $\mathcal{L} u_f$ under the Dirichlet boundary condition, and show that Bayesian methods with priors placed either on $u_f$ or $\mathcal{L} u_f$ for this problem yield optimal recovery rates not only for $u_f$, but also for $f$. We also derive frequentist coverage guarantees for the corresponding Bayesian credible sets. Adaptive priors are shown to yield adaptive contraction rates for $f$, thus eliminating the need to know the smoothness of this function. The results are illustrated by numerical experiments on synthetic data sets.

翻译：我们考虑从偏微分方程解$u_f$的噪声观测中恢复未知函数$f$的问题，该方程可写为$\mathcal{L} u_f=c(f,u_f)$的形式，其中微分算子$\mathcal{L}$需足够丰富以从$\mathcal{L} u_f$中恢复$f$。典型示例包括稳态薛定谔方程$\Delta u_f = 2u_ff$、含吸收项的热方程$(\partial_t -\Delta_x/2) u_f=fu_f$以及达西问题$\nabla\cdot (f \nabla u_f) = h$。我们将该问题转化为狄利克雷边界条件下恢复$\mathcal{L} u_f$的线性反问题，并证明对此问题在$u_f$或$\mathcal{L} u_f$上设置先验的贝叶斯方法不仅能获得$u_f$的最优恢复率，也能获得$f$的最优恢复率。我们还推导了相应贝叶斯可信集的频率派覆盖保证。研究表明自适应先验能为$f$产生自适应收缩率，从而无需预知该函数的光滑性。通过合成数据集的数值实验对结果进行了验证。