Martin Skovgaard Andersen

Fabio Matti, Martin Skovgaard Andersen, Tianshi Chen et al.

We propose a novel Krylov subspace method for estimating the finite impulse response (FIR) of a one-dimensional linear time-invariant systems. The method approximates the system's FIR using a kernel-based formulation combined with hyperparameter selection based on maximum likelihood estimation (MLE), which requires repeated evaluation of two terms: The data fit $\boldsymbol{y}^{\top} (λ\boldsymbol{I} + \boldsymbol{A})^{-1} \boldsymbol{y}$ and the model complexity $\log(\det (λ\boldsymbol{I} + \boldsymbol{A}))$, where $\boldsymbol{A}$ is a certain positive semidefinite matrix that admits fast matrix--vector products and $λ> 0$ is a regularization parameter. Instead of approximating these two quantities separately, we jointly approximate them using a single augmented Krylov subspace for $\boldsymbol{A}$. One major benefit of augmentation is that we obtain accelerated convergence when approximating the data fit quadratic form, through implicit preconditioning. Thanks to the shift invariance of Krylov subspaces, the extracted approximations can be used to evaluate the MLE objective for many values of $λ$ at little additional cost. We derive error bounds for the approximations, reflecting the benefits of augmentation demonstrated through multiple numerical experiments.

Remi Laumont, Yiqiu Dong, Martin Skovgaard Andersen

This paper studies two classes of sampling methods for the solution of inverse problems, namely Randomize-Then-Optimize (RTO), which is rooted in sensitivity analysis, and Langevin methods, which are rooted in the Bayesian framework. The two classes of methods correspond to different assumptions and yield samples from different target distributions. We highlight the main conceptual and theoretical differences between the two approaches and compare them from a practical point of view by tackling two classical inverse problems in imaging: deblurring and inpainting. We show that the choice of the sampling method has a significant impact on the quality of the reconstruction and that the RTO method is more robust to the choice of the parameters.

Hari Om Aggrawal, Martin Skovgaard Andersen, Sean Rose et al.

Classical methods for X-ray computed tomography are based on the assumption that the X-ray source intensity is known, but in practice, the intensity is measured and hence uncertain. Under normal operating conditions, when the exposure time is sufficiently high, this kind of uncertainty typically has a negligible effect on the reconstruction quality. However, in time- or dose-limited applications such as dynamic CT, this uncertainty may cause severe and systematic artifacts known as ring artifacts. By carefully modeling the measurement process and by taking uncertainties into account, we derive a new convex model that leads to improved reconstructions despite poor quality measurements. We demonstrate the effectiveness of the methodology based on simulated and real data sets.