  Many scientific and engineering problems require to perform Bayesian
inferences for unknowns of infinite dimension. In such problems, many standard
Markov Chain Monte Carlo (MCMC) algorithms become arbitrary slow under the mesh
refinement, which is referred to as being dimension dependent. To this end, a
family of dimensional independent MCMC algorithms, known as the preconditioned
Crank-Nicolson (pCN) methods, were proposed to sample the infinite dimensional
parameters. In this work we develop an adaptive version of the pCN algorithm,
where the covariance operator of the proposal distribution is adjusted based on
sampling history to improve the simulation efficiency. We show that the
proposed algorithm satisfies an important ergodicity condition under some mild
assumptions. Finally we provide numerical examples to demonstrate the
performance of the proposed method.

Computation;Numerical Analysis