  Bayesian statistical inverse problems are often solved with Markov chain
Monte Carlo (MCMC)-type schemes. When the problems are governed by large-scale
discrete nonlinear partial differential equations (PDEs), they are
computationally challenging because one would then need to solve the forward
problem at every sample point. In this paper, the use of the discrete empirical
interpolation method (DEIM) is considered for the forward solves within an MCMC
routine. A preconditioning strategy for the DEIM model is also proposed to
accelerate the forward solves. The preconditioned DEIM model is applied to a
finite difference discretization of a nonlinear PDE in the MCMC model.
Numerical experiments show that this approach yields accurate forward results.
Moreover, the computational cost of solving the associated statistical inverse
problem is reduced by more than 70%.
