hIPPYlib: An Extensible Software Framework for Large-Scale Inverse Problems Governed by PDEs; Part I: Deterministic Inversion and Linearized Bayesian Inference

Abstract

We present an extensible software framework, hIPPYlib, for solution of large-scale deterministic and Bayesian inverse problems governed by partial differential equations (PDEs) with (possibly) 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 that can be used to educate researchers and practitioners who are new to inverse problems and the Bayesian inference framework.

Fractional time stepping and adjoint based gradient computation in an inverse problem for a fractionally damped wave equation

“…so that after integrating, tn+1 tn−1 dt, the continuous system ( 7) can be reformulated analogously to (26) as…”

“…analogously to the previous subsection. Subtracting (26) from (32) and making use of the identity of ( 25) and ( 26) with v i , d i replaced by v(t i ), u(t i ), respectively, we obtain the following…”

“…that is, (26) with d 0 = 0, v replaced by e, and a different right hand side. Hence we are in the position to apply Theorem 2.1, after estimating the discrete L 1 (0, T ) norm of the right hand side in (35).…”

“…We perform the reconstructions in two space dimensions as relevant in medical imaging. To make use of Bayesian inference as described in section 1.1 we employ the Python based software FEniCS [25] and hIPPYlib [26]. We use the model (1) with chosen parameters c 2 = 1 and b = 0.1 and different values of α, as well as observations on the boundary of a circle Σ, which is fully contained in the domain D.…”

“…Theorem 2.2. For the time discretization scheme (26) with coefficients b n l = bn l according to (16) with f n+1 = f (t n+1 ), there exists a constant C(T ) depending only on T such that the global error defined by (33), (34) satisfies…”