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Many scientific and engineering problems require to perform Bayesianinferences for unknowns of infinite dimension. In such problems, many standardMarkov Chain Monte Carlo (MCMC) algorithms become arbitrary slow under the meshrefinement, which is referred to as being dimension dependent. To this end, afamily of dimensional independent MCMC algorithms, known as the preconditionedCrank-Nicolson (pCN) methods, were proposed to sample the infinite dimensionalparameters. In this work we develop an adaptive version of the pCN algorithm,where the covariance operator of the proposal distribution is adjusted based onsampling history to improve the simulation efficiency. We show that theproposed algorithm satisfies an important ergodicity condition under some mildassumptions. Finally we provide numerical examples to demonstrate theperformance of the proposed method.

On an adaptive preconditioned Crank-Nicolson MCMC algorithm for infinite dimensional Bayesian inferences