  We present an extensible software framework, hIPPYlib, for solution of
large-scale deterministic and Bayesian inverse problems governed by partial
differential equations (PDEs) with infinite-dimensional parameter fields (which
are high-dimensional after discretization). hIPPYlib overcomes the prohibitive
nature of Bayesian inversion for this class of problems by implementing
state-of-the-art scalable algorithms for PDE-based inverse problems that
exploit the structure of the underlying operators, notably the Hessian of the
log-posterior. The key property of the algorithms implemented in hIPPYlib is
that the solution of the deterministic and linearized Bayesian inverse problem
is computed at a cost, measured in linearized forward PDE solves, that is
independent of the parameter dimension. The mean of the posterior is
approximated by the MAP point, which is found by minimizing the negative
log-posterior. This deterministic nonlinear least-squares optimization problem
is solved with an inexact matrix-free Newton-CG method. The posterior
covariance is approximated by the inverse of the Hessian of the negative log
posterior evaluated at the MAP point. This Gaussian approximation is exact when
the parameter-to-observable map is linear; otherwise, its logarithm agrees to
two derivatives with the log-posterior at the MAP point, and thus it can serve
as a proposal for Hessian-based MCMC methods. The construction of the posterior
covariance is made tractable by invoking a low-rank approximation of the
Hessian of the log-likelihood. Scalable tools for sample generation are also
implemented. hIPPYlib makes all of these advanced algorithms easily accessible
to domain scientists and provides an environment that expedites the development
of new algorithms. hIPPYlib is also a teaching tool to educate researchers and
practitioners who are new to inverse problems and the Bayesian inference
framework.
