hippy-colab

Selected hIPPYlib notebooks on Google Colab.

hIPPYlib implements state-of-the-art scalable adjoint-based algorithms for PDE-based deterministic and Bayesian inverse problems. It builds on FEniCS for the discretization of the PDE and on PETSc for scalable and efficient linear algebra operations and solvers.

These collection of notebook was prepared for our SIAM Annual Meeting Minitutorial on Introduction to Bayesian Inverse Problems Governed by Partial Differential Equations (PDEs): A Hands-on Tutorial using hIPPYlib.

Notebooks

1. Inverse problem prototype: An illustrative example of an ill-posed inverse problem.
2. Unconstrained Minimization: This notebook illustrates the minimization of a non-quadratic energy functional using Netwon Method.
3. Deterministic Inverse Problem: This notebook illustrates the use of FEniCS for solving an inverse problem for the coefficient field of a Poisson equation, using the Steepest Descent and inexact Newton CG method.
4. Gaussian Priors in infinite dimensions: This notebook shows how to construct PDE-based priors that lead to well-posed Bayesian inverse problems in infinite dimesions.
5. Linearized Bayesian Inverse Problem: This notebook illustrates how to solve a non-linear parameter inversion for the Poisson equation in a Bayesian setting using hIPPYlib.

Useful References

Theory and computational methods for inverse problems:

• Heinz Engl, Michael Hanke, and Andreas Neubauer, Regularization of Inverse Problems, Dordrecht, 2nd ed., 1996.

• Curtis R. Vogel, Computational Methods for Inverse Problems, SIAM, 2002.

• Guy Chavent, Nonlinear Least Squares for Inverse Problems, Springer, 2009.

• Omar Ghattas and Karen Willcox, Learning physics-based models from data: perspectives from inverse problems and model reduction, Acta Numerical, 2021.

Numerical optimization background:

• Jorge Nocedal and Stephen J. Wright, Numerical Optimization, Springer-Verlag, 1999.

• C. Tim Kelley, Iterative Methods of Optimization, SIAM, 1999.

Optimization of systems governed by PDEs:

• A. Borzi and V. Schulz, Computational Optimization of Systems Governed by Partial Differential Equations, SIAM, 2012.

• Fredi Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Studies in Mathematics Vol.~112, AMS, 2010.

• M. Hinze, R. Pinnau, M. Ulbich, and S. Ulbrich, Optimization with PDE constraints, Springer, 2009.

• Alen Alexanderian, Computational Inverse Problems Governed by PDEs, SIAM 2026.

Finite element background and FEniCS:

• Eric B. Becker, Graham F. Carey, and J. Tinsley Oden, Finite Elements: Volume I, An Introduction, Prentice Hall, 1981.

• Mark S. Gockenbach, Understanding and Implementing the Finite Element Method, SIAM, 2006.

• Howard Elman, David Silvester, and Andrew Wathen, Finite Elements and Fast Iterative Solvers, Oxford University Press, 2005.

• A. Logg, K. A. Mardal, and G. Wells, Automated solution of differential equations by the finite element method: The FEniCS book, vol. 84, Springer Science \& Business Media, 2012.

• H. P. Langtangen, A. Logg, Solving PDEs in Python: The FEniCS Tutorial Volume I, Springer, 2017.

Probabilistic approach to inverse problems:

• Albert Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM, 2005.

• Jari Kaipio and Erkki Somersalo, Statistical and Computational Inverse Problems, Springer, 2005.

• Andrew Stuart, Inverse problems, A Bayesian approach, Acta Numerica, 2010.

• Daniela Calvetti and Erkki Somersalo, Bayesian Scientific Computing, Springer, 2023.

• Johnathan Bardsley, Computational Uncertainty Quantification for Inverse Problems, SIAM, 2018.

Acknowledgements

The hIPPYlib project started in 2016 with initial support from the US National Science Foundation under awards ACI-1550593, ACI-1550547.

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