  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.
