  We use a conditional Karhunen-Lo\`eve (KL) model to quantify and reduce
uncertainty in a stochastic partial differential equation (SPDE) problem with
partially-known space-dependent coefficient, $Y(x)$.
  We assume that a small number of $Y(x)$ measurements are available and model
$Y(x)$ with a KL expansion.
  We achieve reduction in uncertainty by conditioning the KL expansion
coefficients on measurements.
  We consider two approaches for conditioning the KL expansion: In Approach 1,
we condition the KL model first and then truncate it.
  In Approach 2, we first truncate the KL expansion and then condition it.
  We employ the conditional KL expansion together with Monte Carlo and sparse
grid collocation methods to compute the moments of the solution of the SPDE
problem.
  Uncertainty of the problem is further reduced by adaptively selecting
additional observation locations using two active learning methods.
  Method 1 minimizes the variance of the PDE coefficient, while Method 2
minimizes the variance of the solution of the PDE.
  We demonstrate that conditioning leads to dimension reduction of the KL
representation of $Y(x)$.
  For a linear diffusion SPDE with uncertain log-normal coefficient, we show
that Approach 1 provides a more accurate approximation of the conditional
log-normal coefficient and solution of the SPDE than Approach 2 for the same
number of random dimensions in a conditional KL expansion.
  Furthermore, Approach 2 provides a good estimate for the number of terms of
the truncated KL expansion of the conditional field of Approach 1.
  Finally, we demonstrate that active learning based on Method 2 is more
efficient for uncertainty reduction in the SPDE's states (i.e., it leads to a
larger reduction of the variance) than active learning using Method 2.
