SIAM/ASA Journal on Uncertainty Quantification, Volume 8, Issue 1, Page 374-413, January 2020.
We consider PDE constrained nonparametric regression problems in which the parameter $f$ is the unknown coefficient function of a second order elliptic partial differential operator $L_f$, and the unique solution $u_f$ of the boundary value problem $L_fu=g_1$ on $\mathcal O, u=g_2$ on $\partial \mathcal O,$ is observed corrupted by additive Gaussian white noise. Here $\mathcal O$ is a bounded domain in $\mathbb R^d$ with smooth boundary $\partial \mathcal O$, and $g_1, g_2$ are given functions defined on $\mathcal O, \partial \mathcal O$, respectively. Concrete examples include $L_fu=\Delta u-2fu$ (Schrödinger equation with attenuation potential $f$) and $L_fu=\text{div} (f\nabla u)$ (divergence form equation with conductivity $f$). In both cases, the parameter space $\mathcal F=\{f\in H^\alpha(\mathcal O)| f > 0\}, \alpha>0$, where $H^\alpha(\mathcal O)$ is the usual order $\alpha$ Sobolev space, induces a set of nonlinearly constrained regression functions $\{u_f: f \in \mathcal F\}$. We study Tikhonov-type penalized least squares estimators $\hat f$ for $f$. The penalty functionals are of squared Sobolev-norm type and thus $\hat f$ can also be interpreted as a Bayesian “maximum a posteriori” estimator corresponding to some Gaussian process prior. We derive rates of convergence of $\hat f$ and of $u_{\hat f}$, to $f, u_f$, respectively. We prove that the rates obtained are minimax-optimal in prediction loss. Our bounds are derived from a general convergence rate result for nonlinear inverse problems whose forward map satisfies a modulus of continuity condition, a result of independent interest that is applicable also to linear inverse problems, illustrated in an example with the Radon transform.

中文翻译：

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PDE约束回归问题中惩罚最小二乘估计的收敛速度

SIAM / ASA不确定性量化期刊，第8卷，第1期，第374-413页，2020年1月。
我们考虑PDE约束的非参数回归问题，其中参数$ f $是二阶椭圆偏微分算子$ L_f $的未知系数函数，并且边值问题$ L_fu = g_1 $的唯一解$ u_f $观察到\ mathcal O，在$ \ partial \ mathcal O，$上的u = g_2 $被加性高斯白噪声破坏。这里$ \ mathcal O $是$ \ mathbb R ^ d $中的边界域，具有平滑边界$ \ partial \ mathcal O $，并且$ g_1，g_2 $被赋予在$ \ mathcal O，\ partial \ mathcal O上定义的函数$。具体示例包括$ L_fu = \ Delta u-2fu $（具有衰减势$ f $的Schrödinger方程）和$ L_fu = \ text {div}（f \ nabla u）$（具有电导率$ f $的发散形式方程）。在两种情况下，参数空间$ \ mathcal F = \ {f \ in H ^ \ alpha（\ mathcal O）| f> 0 \}，\ alpha> 0 $，其中$ H ^ \ alpha（\ mathcal O）$是通常的顺序$ \ alpha $ Sobolev空间，产生一组非线性约束的回归函数$ \ {u_f：f \ in \ mathcal F \} $。我们研究Tikhonov型惩罚最小二乘估计子$ \ hat f $为$ f $。罚函数是平方的Sobolev范数，因此$ \ hat f $也可以解释为对应于某些高斯过程先验的贝叶斯“最大后验”估计量。我们分别得出$ \ hat f $和$ u _ {\ hat f} $到$ f，u_f $的收敛速度。我们证明所获得的速率在预测损失中是最小最大最优的。我们的界线是根据非线性反问题的一般收敛速度结果得出的，该非线性反问题的前向映射满足连续模数条件，这是一个独立感兴趣的结果，也适用于线性反问题，