Efficient generalized Golub-Kahan based methods for dynamic inverse problems
This work provides a computationally efficient approach for Bayesian inference in large-scale dynamic inverse problems, which is relevant for applications like tomography and deblurring.
The authors develop efficient, matrix-free iterative methods based on generalized Golub-Kahan bidiagonalization for dynamic inverse problems, enabling automatic regularization and variance estimation. They demonstrate scalability by solving a problem with 43,000 measurements and 7.8 million unknowns in under 40 seconds on a standard desktop.
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.