The Bayesian method has proven to be a powerful way of modeling inverse problems.
The solution to Bayesian inverse problems is the posterior distribution of estimated
parameters which can provide not only estimates for the inferred parameters
but also the uncertainty of these estimations. Markov chain Monte Carlo (MCMC)
is a useful technique to sample the posterior distribution and information can be
extracted from the sampled ensemble. However, MCMC is very expensive to compute,
especially in inverse problems where the underlying forward problems involve
solving differential equations. Even worse, MCMC is difficult to parallelize due to
its sequential nature|that is, under the current framework, we can barely accelerate
MCMC with parallel computing.
We develop a new framework of parallel MCMC algorithms-the Markov chain
preconditioned Monte Carlo (MCPMC) method-for sampling Bayesian inverse problems.
With the help of a fast auxiliary MCMC chain running on computationally
cheaper approximate models, which serves as a stochastic preconditioner to the target
distribution, the sampler randomly selects candidates from the preconditioning
chain for further processing on the accurate model. As this accurate model processing
can be executed in parallel, the algorithm is suitable for parallel systems.
We implement it using a modified master-slave architecture, analyze its potential
to accelerate sampling and apply it to three examples. A two dimensional Gaussian
mixture example shows that the new sampler can bring statistical efficiency in
addition to increasing sampling speed. Through a 2D inverse problem with an elliptic
equation as the forward model, we demonstrate the use of an enhanced error
model to build the preconditioner. With a 3D optical tomography problem we use
adaptive finite element methods to build the approximate model. In both examples,
the MCPMC successfully samples the posterior distributions with multiple processors,
demonstrating efficient speedups comparing to traditional MCMC algorithms.
In addition, the 3D optical tomography example shows the feasibility of applying
MCPMC towards real world, large scale, statistical inverse problems
