  We consider the statistical linear inverse problem of making inference on an
unknown source function in an elliptic partial differential equation from noisy
observations of its solution. We employ nonparametric Bayesian procedures based
on Gaussian priors, leading to convenient conjugate formulae for posterior
inference. We review recent results providing theoretical guarantees on the
quality of the resulting posterior-based estimation and uncertainty
quantification, and we discuss the application of the theory to the important
classes of Gaussian series priors defined on the Dirichlet-Laplacian eigenbasis
and Mat\'ern process priors. We provide an implementation of posterior
inference for both classes of priors, and investigate its performance in a
numerical simulation study.
