# Efficient generalized Golub-Kahan based methods for dynamic inverse problems

## TL;DR

This paper introduces efficient, matrix-free iterative methods based on the generalized Golub-Kahan bidiagonalization for solving large-scale dynamic inverse problems, enabling uncertainty estimation and regularization without explicit prior covariance matrix computations.

## Contribution

The work develops novel, flexible, and scalable algorithms for dynamic inverse problems that incorporate prior information efficiently without explicit covariance matrix operations.

## Key findings

- Methods successfully applied to photoacoustic tomography, deblurring, and seismic tomography.
- Achieved rapid solutions for large-scale problems with millions of unknowns.
- Demonstrated ability to process real and synthetic data efficiently.

## Abstract

We consider efficient methods for computing solutions to and estimating uncertainties in dynamic inverse problems, where the parameters of interest may change during the measurement procedure. Compared to static inverse problems, incorporating prior information in both space and time in a Bayesian framework can become computationally intensive, in part, due to the large number of unknown parameters. In these problems, explicit computation of the square root and/or inverse of the prior covariance matrix is not possible. In this work, we develop efficient, iterative, matrix-free methods based on the generalized Golub-Kahan bidiagonalization that allow automatic regularization parameter and variance estimation. We demonstrate that these methods can be more flexible than standard methods and develop efficient implementations that can exploit structure in the prior, as well as possible structure in the forward model. Numerical examples from photoacoustic tomography, deblurring, and passive seismic tomography demonstrate the range of applicability and effectiveness of the described approaches. Specifically, in passive seismic tomography, we demonstrate our approach on both synthetic and real data. To demonstrate the scalability of our algorithm, we solve a dynamic inverse problem with approximately $43,000$ measurements and $7.8$ million unknowns in under $40$ seconds on a standard desktop.