  Bayesian methods have been widely used in the last two decades to infer
statistical properties of spatially variable coefficients in partial
differential equations from measurements of the solutions of these equations.
Yet, in many cases the number of variables used to parameterize these
coefficients is large, and obtaining meaningful statistics of their values is
difficult using simple sampling methods such as the basic Metropolis-Hastings
(MH) algorithm -- in particular if the inverse problem is ill-conditioned or
ill-posed. As a consequence, many advanced sampling methods have been described
in the literature that converge faster than MH, for example by exploiting
hierarchies of statistical models or hierarchies of discretizations of the
underlying differential equation.
  At the same time, it remains difficult for the reader of the literature to
quantify the advantages of these algorithms because there is no commonly used
benchmark. This paper presents a benchmark for the Bayesian inversion of
inverse problems -- namely, the determination of a spatially-variable
coefficient, discretized by 64 values, in a Poisson equation, based on point
measurements of the solution -- that fills the gap between widely used simple
test cases (such as superpositions of Gaussians) and real applications that are
difficult to replicate for developers of sampling algorithms. We provide a
complete description of the test case, and provide an open source
implementation that can serve as the basis for further experiments. We have
also computed $2\times 10^{11}$ samples, at a cost of some 30 CPU years, of the
posterior probability distribution from which we have generated detailed and
accurate statistics against which other sampling algorithms can be tested.
