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In this article, we study Bayesian inverse problems with multi-layered Gaussian priors. We first describe the conditionally Gaussian layers in terms of a system of stochastic partial differential equations. We build the computational…

In this paper, we design preconditioners for the matrix-free solution of high-order continuous and discontinuous Galerkin discretizations of elliptic problems based on FEM-SEM equivalence and additive Schwarz methods. The high-order…

We propose a geometry-aware strategy for training neural preconditioners tailored to parametrized linear systems arising from the discretization of mixed-dimensional partial differential equations (PDEs). These systems are typically…

Despite hundreds of papers on preconditioned linear systems of equations, there remains a significant lack of comprehensive performance benchmarks comparing various preconditioners for solving symmetric positive definite (SPD) systems. In…

Low rank approximation is a commonly occurring problem in many computer vision and machine learning applications. There are two common ways of optimizing the resulting models. Either the set of matrices with a given rank can be explicitly…

This paper presents a couple of preconditioning techniques that can be used to enhance the performance of iterative regularization methods applied to image deblurring problems with a variety of point spread functions (PSFs) and boundary…

We consider using the preconditioned-Krylov subspace method to solve the system of linear equations with a three-by-three block structure. By making use of the three-by-three block structure, eight inexact block factorization…

We present a new paradigm for fine-tuning large-scale visionlanguage pre-trained models on downstream task, dubbed Prompt Regularization (ProReg). Different from traditional fine-tuning which easily overfits to the downstream task data,…

Recent work in deep learning has opened new possibilities for solving classical algorithmic tasks using end-to-end learned models. In this work, we investigate the fundamental task of solving linear systems, particularly those that are…

Estimating the values of unknown parameters from corrupted measured data faces a lot of challenges in ill-posed problems. In such problems, many fundamental estimation methods fail to provide a meaningful stabilized solution. In this work,…

We study non-linear Bayesian inverse problems arising from semilinear partial differential equations (PDEs) that can be transformed into linear Bayesian inverse problems. We are then able to extend the early stopping for Ensemble…

Preconditioning for overdetermined least-squares problems has received comparatively little attention, and designing methods that are both effective and memory-efficient remains challenging. We propose a class of ILU-based preconditioners…

We present a simple discretization scheme for the hypersingular integral representation of the fractional Laplace operator and solver for the corresponding fractional Laplacian problem. Through singularity subtraction, we obtain a…

This paper proposes a matrix-free residual evaluation technique for the hybridizable discontinuous Galerkin method requiring a number of operations scaling only linearly with the number of degrees of freedom. The method results from…

Motivated by the theory of self-duality which provides a variational formulation and resolution for non self-adjoint partial differential equations \cite{G1, G2}, we propose new templates for solving large non-symmetric linear systems. The…

This paper focuses on a challenging class of inverse problems that is often encountered in applications. The forward model is a complex non-linear black-box, potentially non-injective, whose outputs cover multiple decades in amplitude.…

Uniform preconditioners for operators of negative order discretized by (dis)continuous piecewise polynomials of any order are constructed from a boundedly invertible operator of opposite order discretized by continuous piecewise linears.…

We introduce a preconditioner for a hybridizable discontinuous Galerkin discretization of the linearized Navier-Stokes equations at high Reynolds number. The preconditioner is based on an augmented Lagrangian approach of the full…

The solution of matrices with $2\times 2$ block structure arises in numerous areas of computational mathematics, such as PDE discretizations based on mixed-finite element methods, constrained optimization problems, or the implicit or steady…

This paper is concerned with the design, analysis and implementation of preconditioning concepts for spectral Discontinuous Galerkin discretizations of elliptic boundary value problems. While presently known techniques realize a growth of…