  We propose a novel method for sampling and optimization tasks based on a
stochastic interacting particle system. We explain how this method can be used
for the following two goals: (i) generating approximate samples from a given
target distribution; (ii) optimizing a given objective function. The approach
is derivative-free and affine invariant, and is therefore well-suited for
solving inverse problems defined by complex forward models: (i) allows
generation of samples from the Bayesian posterior and (ii) allows determination
of the maximum a posteriori estimator. We investigate the properties of the
proposed family of methods in terms of various parameter choices, both
analytically and by means of numerical simulations. The analysis and numerical
simulation establish that the method has potential for general purpose
optimization tasks over Euclidean space; contraction properties of the
algorithm are established under suitable conditions, and computational
experiments demonstrate wide basins of attraction for various specific
problems. The analysis and experiments also demonstrate the potential for the
sampling methodology in regimes in which the target distribution is unimodal
and close to Gaussian; indeed we prove that the method recovers a Laplace
approximation to the measure in certain parametric regimes and provide
numerical evidence that this Laplace approximation attracts a large set of
initial conditions in a number of examples.
