  This paper focuses on a challenging class of inverse problems that is often
encountered in applications. The forward model is a complex non-linear
black-box, potentially non-injective, whose outputs cover multiple decades in
amplitude. Observations are supposed to be simultaneously damaged by additive
and multiplicative noises and censorship. As needed in many applications, the
aim of this work is to provide uncertainty quantification on top of parameter
estimates. The resulting log-likelihood is intractable and potentially
non-log-concave. An adapted Bayesian approach is proposed to provide
credibility intervals along with point estimates. An MCMC algorithm is proposed
to deal with the multimodal posterior distribution, even in a situation where
there is no global Lipschitz constant (or it is very large). It combines two
kernels, namely an improved version of (Preconditioned Metropolis Adjusted
Langevin) PMALA and a Multiple Try Metropolis (MTM) kernel. Whenever smooth,
its gradient admits a Lipschitz constant too large to be exploited in the
inference process. This sampler addresses all the challenges induced by the
complex form of the likelihood. The proposed method is illustrated on classical
test multimodal distributions as well as on a challenging and realistic inverse
problem in astronomy.
