# hIPPYfire : an inexact Newton-CG method for solving inverse problems governed by PDE forward models

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This study presents the implementation of hIPPYfire, a library for solving large-scale deterministic inverse problems. These inverse problems are governed by partial differential equations (PDEs) with infinite-dimensional parameter fields that become high-dimensional after discretization. It utilizes the inexact Newton Conjugate Gradient (Newton-CG) method for the computation of the maximum a posteriori (MAP) point. This algorithm exploits the fact that several PDE models of physical systems have a low-dimensional solution manifold. hIPPYfire computes the solution of the inverse problem at a cost independent of the parameter dimension, when measured in terms of the number of linearized PDE solves. However, unlike hIPPYlib (which is built on FEniCS), hIPPYfire uses Firedrake to solve the PDE governing the forward problem. Firedrake presents a unique modular structure that clearly distinguishes between the programming and mathematical aspects of the library—thereby enabling contributions from programmers and mathematicians alike and ensuring its consistent development. The functionalities of hIPPYfire are illustrated by solving an inverse problem that is governed by an elliptic PDE. The major components of the inverse problem, namely the forward problem, misfit, and prior functionals, are clearly defined and used to compute the MAP point using the inexact Newton-CG method. The design of hIPPYfire follows that of hIPPYlib, an extensible Python library for the solution of deterministic and Bayesian inverse problems governed by PDEs.