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The problem of generating random samples of high-dimensional posteriordistributions is considered. The main results consist of non-asymptoticcomputational guarantees for Langevin-type MCMC algorithms which scalepolynomially in key quantities such as the dimension of the model, the desiredprecision level, and the number of available statistical measurements. As adirect consequence, it is shown that posterior mean vectors as well asoptimisation based maximum a posteriori (MAP) estimates are computable inpolynomial time, with high probability under the distribution of the data.These results are complemented by statistical guarantees for recovery of theground truth parameter generating the data.  Our results are derived in a general high-dimensional non-linear regressionsetting (with Gaussian process priors) where posterior measures are notnecessarily log-concave, employing a set of local `geometric' assumptions onthe parameter space, and assuming that a good initialiser of the algorithm isavailable. The theory is applied to a representative non-linear example fromPDEs involving a steady-state Schr\"odinger equation.

On polynomial-time computation of high-dimensional posterior measures by Langevin-type algorithms