2023/10/08 by Malena Sabaté Landman, Landman, Malena Sabate, Jiahua Jiang +5 · 1 citation
Engineering · Computer Science · Mathematics · #Sparse and Compressive Sensing Techniques #Gaussian Processes and Bayesian Inference #Statistical and numerical algorithms
paper · pdf · doi:10.48550/arxiv.2310.05285
This paper presents two new augmented flexible (AF)-Krylov subspace methods, AF-GMRES and AF-LSQR, to compute solutions of large-scale linear discrete ill-posed problems that can be modeled as the sum of two independent random variables, exhibiting smooth and sparse stochastic characteristics respectively. Following a Bayesian modelling approach, this corresponds to adding a covariance-weighted quadratic term and a sparsity enforcing ℓ1 term in the original least-squares minimization scheme. To handle the ℓ1 regularization term, the proposed approach constructs a sequence approximating quadratic problems that are partially solved using augmented flexible Krylov-Tikhonov methods. Compared to other traditional methods used to solve this minimization problem, such as those based on iteratively reweighted norm schemes, the new algorithms build a single (augmented, flexible) approximation (Krylov) subspace that encodes information about the different regularization terms through adaptable "preconditioning". The solution space is then expanded as soon as a new problem within the sequence is defined. This also allows for the regularization parameters to be chosen on-the-fly at each iteration. Compared to most recent work on generalized flexible Krylov methods, our methods offer theoretical assurance of convergence and a more stable numerical performance. The efficiency of the new methods is shown through a variety of experiments, including a synthetic image deblurring problem, a synthetic atmospheric transport problem, and fluorescence molecular tomography reconstructions using both synthetic and real-world experimental data.