  We present a non-trivial integration of dimension-independent
likelihood-informed (DILI) MCMC (Cui, Law, Marzouk, 2016) and the multilevel
MCMC (Dodwell et al., 2015) to explore the hierarchy of posterior
distributions. This integration offers several advantages: First, DILI-MCMC
employs an intrinsic likelihood-informed subspace (LIS) (Cui et al., 2014) --
which involves a number of forward and adjoint model simulations -- to design
accelerated operator-weighted proposals. By exploiting the multilevel structure
of the discretised parameters and discretised forward models, we design a
Rayleigh-Ritz procedure to significantly reduce the computational effort in
building the LIS and operating with DILI proposals. Second, the resulting
DILI-MCMC can drastically improve the sampling efficiency of MCMC at each
level, and hence reduce the integration error of the multilevel algorithm for
fixed CPU time. Numerical results confirm the improved computational efficiency
of the multilevel DILI approach.
