  We consider the problem of estimating the uncertainty in statistical inverse
problems using Bayesian inference. When the probability density of the noise
and the prior are Gaussian, the solution of such a statistical inverse problem
is also Gaussian. Therefore, the underlying solution is characterized by the
mean and covariance matrix of the posterior probability density. However, the
covariance matrix of the posterior probability density is full and large.
Hence, the computation of such a matrix is impossible for large dimensional
parameter spaces. It is shown that for many ill-posed problems, the Hessian
matrix of the data misfit part has low numerical rank and it is therefore
possible to perform a low-rank approach to approximate the posterior covariance
matrix. For such a low-rank approximation, one needs to solve a forward partial
differential equation (PDE) and the adjoint PDE in both space and time. This in
turn gives $\mathcal{O}(n_x n_t)$ complexity for both, computation and storage,
where $n_x$ is the dimension of the spatial domain and $n_t$ is the dimension
of the time domain. Such computations and storage demand are infeasible for
large problems. To overcome this obstacle, we develop a new approach that
utilizes a recently developed low-rank in time algorithm together with the
low-rank Hessian method. We reduce both the computational complexity and
storage requirement from $\mathcal{O}(n_x n_t)$ to $\mathcal{O}(n_x + n_t)$. We
use numerical experiments to illustrate the advantages of our approach.
