Large-scale linear least-squares problems arise in many areas of computational science and data analysis, where efficiency and scalability are crucial. In this talk, we introduce a randomized preconditioning framework for iterative solvers based on low-rank approximations of small sketches of the original problem. The key idea is to iteratively construct low-rank preconditioners that reshape the singular value distribution in a favorable way. By tightly coupling the preconditioning and Krylov solving phases within an iterative CUR decomposition -- a low-rank approximation built from selected of columns and rows of the original matrix -- the proposed algorithm achieves faster and earlier convergence than existing methods. The algorithm performs particularly well on problems that are large in both dimensions, as well as on sparse and ill-conditioned systems. This is a joint work with Coralia Cartis and Yuji Nakatsukasa.

Charles Parker II will be talking about: 'Structure-preserving finite elements and the convergence of augmented Lagrangian methods'

Problems with physical constraints, such as the incompressibility constraint for mass conservation in fluids or Gauss's laws for electric and magnetic fields, result in generalized saddle point systems. So-called structure-preserving finite elements respect the constraints pointwise, resulting in more physically accurate solutions that are typically robust with respect to some problem parameters. However, constructing these finite elements may involve complicated spaces for the Lagrange multiplier variables. Augmented Lagrangian methods (ALMs) provide one process to compute the solution without the need for an explicit basis for the Lagrange multiplier space. In this talk, we present new convergence estimates for a standard ALM method, sometimes called the iterated penalty method, applied to structure-preserving discretizations of linear saddle point systems.