Computational methods for parameterized inverse problems

Mentors:
Lead: Alen Alexanderian (Mathematics, NC State), Collaborator: Joseph Hart (Sandia National Labs)

Outline:
Inverse problems are central to data-informed computational modeling in sciences and engineering. In
such problems, we formulate a mathematical model and utilize data to estimate unknown quantities that
cannot be observed directly. Examples include estimation of rate constants in epidemic models,
characterization of material properties in porous media flow, and contaminant source localization. In most
applications, the governing models contain additional parameters, called auxiliary parameters, that are not
being estimated but are themselves uncertain. For example, in a contaminant source identification
problem, one often has uncertainties regarding boundary conditions. We consider parameterized
variational inverse problems that are constrained by PDEs. We seek to efficiently compute the solution of
the inverse problem when auxiliary parameters appearing in the governing PDEs are varied. Solving the
inverse problem for different auxiliary parameters is crucial for uncertainty quantification (UQ).
However, this is computationally expensive – it requires solving many inverse problems. In this project,
we use tools from post-optimality sensitivity analysis and pseudo-time continuation [1] to enable fast UQ
for parameterized inverse problems.

Objectives:
(i) Implement the methods on inverse problems with low-dimensional parameters. (ii) Investigate
strategies to traverse the auxiliary parameter domain that enable reuse of model evaluations for different
auxiliary parameter realizations. (iii) Implement the methods for a high-dimensional inverse problem
constrained by PDEs.alman filtering.

Outcomes:
The students will present a poster summarizing their research and submit a paper for publication in a
peer-reviewed journal.

References:
[1] J. Hart, A. Alexanderian, and B. van Bloemen Waanders. Preconditioned pseudo-time
continuation for parameterized inverse problems. arXiv.org, vol. 2508.21155, 2025, doi:
10.48550/arXiv.2508.21155.