We present a model inversion algorithm, CKLEMAP, for data assimilation and parameter estimation in partial differential equation models of physical systems with spatially heterogeneous parameter fields. These fields are approximated using low-dimensional conditional Karhunen-Lo\'{e}ve expansions, which are constructed using Gaussian process regression models of these fields trained on the parameters' measurements. We then assimilate measurements of the state of the system and compute the maximum a posteriori estimate of the CKLE coefficients by solving a nonlinear least-squares problem. When solving this optimization problem, we efficiently compute the Jacobian of the vector objective by exploiting the sparsity structure of the linear system of equations associated with the forward solution of the physics problem. The CKLEMAP method provides better scalability compared to the standard MAP method. In the MAP method, the number of unknowns to be estimated is equal to the number of elements in the numerical forward model. On the other hand, in CKLEMAP, the number of unknowns (CKLE coefficients) is controlled by the smoothness of the parameter field and the number of measurements, and is in general much smaller than the number of discretization nodes, which leads to a significant reduction of computational cost with respect to the standard MAP method. To show its advantage in scalability, we apply CKLEMAP to estimate the transmissivity field in a two-dimensional steady-state subsurface flow model of the Hanford Site by assimilating synthetic measurements of transmissivity and hydraulic head. We find that the execution time of CKLEMAP scales nearly linearly as $N^{1.33}$, where $N$ is the number of discretization nodes, while the execution time of standard MAP scales as $N^{2.91}$. The CKLEMAP method improved execution time without sacrificing accuracy when compared to the standard MAP.

翻译：我们提出了一种模型反演算法CKLEMAP，用于具有空间异质性参数场的物理系统偏微分方程模型中的数据同化与参数估计。这些参数场通过基于高斯过程回归模型训练的测量数据构建的低维条件Karhunen-Loéve展开进行近似。随后，我们同化系统状态的测量数据，并通过求解非线性最小二乘问题计算CKLE系数的最大后验估计。在求解该优化问题时，我们利用物理问题前向解相关线性方程组的稀疏结构，高效计算向量目标函数的雅可比矩阵。与标准MAP方法相比，CKLEMAP方法具有更好的可扩展性。在MAP方法中，待估计未知量的数量等于数值前向模型中的单元数；而CKLEMAP中未知量（CKLE系数）的数量由参数场的光滑性和测量数据量控制，通常远小于离散节点数，从而显著降低计算成本。为展示其可扩展性优势，我们通过同化透射率与测压水头的合成测量数据，将CKLEMAP应用于汉福德场地二维稳态地下水流模型的透射率场估计。结果表明，CKLEMAP的执行时间随$N^{1.33}$近似线性增长（$N$为离散节点数），而标准MAP的执行时间随$N^{2.91}$增长。与标准MAP相比，CKLEMAP方法在不牺牲精度的前提下改善了执行时间。