Title: Multilevel Sequential${}^2$ Monte Carlo for Bayesian Inverse Problems,
Abstract: The identification of parameters in mathematical models using noisy
observations is a common task in uncertainty quantification. We employ the
framework of Bayesian inversion: we combine monitoring and observational data
with prior information to estimate the posterior distribution of a parameter.
Specifically, we are interested in the distribution of a diffusion coefficient
of an elliptic PDE. In this setting, the sample space is high-dimensional, and
each sample of the PDE solution is expensive. To address these issues we
propose and analyse a novel Sequential Monte Carlo (SMC) sampler for the
approximation of the posterior distribution. Classical, single-level SMC
constructs a sequence of measures, starting with the prior distribution, and
finishing with the posterior distribution. The intermediate measures arise from
a tempering of the likelihood, or, equivalently, a rescaling of the noise. The
resolution of the PDE discretisation is fixed. In contrast, our estimator
employs a hierarchy of PDE discretisations to decrease the computational cost.
We construct a sequence of intermediate measures by decreasing the temperature
or by increasing the discretisation level at the same time. This idea builds on
and generalises the multi-resolution sampler proposed in [P.S. Koutsourelakis,
J. Comput. Phys., 228 (2009), pp. 6184-6211] where a bridging scheme is used to
transfer samples from coarse to fine discretisation levels. Importantly, our
choice between tempering and bridging is fully adaptive. We present numerical
experiments in 2D space, comparing our estimator to single-level SMC and the
multi-resolution sampler.