hIPPYlib: An Extensible Software Framework for Large-scale Inverse Problems

We present an Inverse Problem PYthon library (hIPPYlib) for solving large-scale deterministic and Bayesian inverse problems governed by partial differential equations (PDEs). hIPPYlib implements state-of-the-art scalable algorithms that exploit the structure of the problem, notably the Hessian of the log posterior. The key property of the algorithms implemented in hIPPYlib is that the solution is computed at a cost, measured in 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 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. hIPPYlib makes 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 that can be used to educate students and practitioners who are new to inverse problems and to the Bayesian inference framework