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We present a model inversion algorithm, CKLEMAP, for data assimilation andparameter estimation in partial differential equation models of physicalsystems with spatially heterogeneous parameter fields. These fields areapproximated using low-dimensional conditional Karhunen-Lo\'{e}ve expansions,which are constructed using Gaussian process regression models of these fieldstrained on the parameters' measurements. We then assimilate measurements of thestate of the system and compute the maximum a posteriori estimate of the CKLEcoefficients by solving a nonlinear least-squares problem. When solving thisoptimization problem, we efficiently compute the Jacobian of the vectorobjective by exploiting the sparsity structure of the linear system ofequations associated with the forward solution of the physics problem. TheCKLEMAP method provides better scalability compared to the standard MAP method.In the MAP method, the number of unknowns to be estimated is equal to thenumber of elements in the numerical forward model. On the other hand, inCKLEMAP, the number of unknowns (CKLE coefficients) is controlled by thesmoothness of the parameter field and the number of measurements, and is ingeneral much smaller than the number of discretization nodes, which leads to asignificant reduction of computational cost with respect to the standard MAPmethod. To show its advantage in scalability, we apply CKLEMAP to estimate thetransmissivity field in a two-dimensional steady-state subsurface flow model ofthe Hanford Site by assimilating synthetic measurements of transmissivity andhydraulic head. We find that the execution time of CKLEMAP scales nearlylinearly as $N^{1.33}$, where $N$ is the number of discretization nodes, whilethe execution time of standard MAP scales as $N^{2.91}$. The CKLEMAP methodimproved execution time without sacrificing accuracy when compared to thestandard MAP.

Gaussian process regression and conditional Karhunen-Lo\'{e}ve models for data assimilation in inverse problems