Related papers: Least squares solvers for ill-posed PDEs that are …
We construct least squares formulations of PDEs with inhomogeneous essential boundary conditions, where boundary residuals are not measured in unpractical fractional Sobolev norms, but which formulations nevertheless are shown to yield a…
In this paper, based on the theory of adjoint operators and dual norms, we define condition numbers for a linear solution function of the weighted linear least squares problem. The explicit expressions of the normwise and componentwise…
Backward parabolic equations, such as the backward heat equation, are classical examples of ill-posed problems where solutions may not exist or depend continuously on the data. In this work, we study a least squares finite element method to…
When solving rank-deficient or discrete ill-posed problems by regularization methods, the choice of the regularization parameter is crucial. It is also of interest, the regularization norm used in the selection of the solution. In this…
The condition number of a linear function of the indefinite least squares solution is called the partial condition number for the indefinite least squares problem. In this paper, based on a new and very general condition number which can be…
Sketch-and-precondition techniques are efficient and popular for solving large least squares (LS) problems of the form $Ax=b$ with $A\in\mathbb{R}^{m\times n}$ and $m\gg n$. This is where $A$ is ``sketched" to a smaller matrix $SA$ with…
In this paper, the normwise condition number of a linear function of the equality constrained linear least squares solution called the partial condition number is considered. Its expression and closed formulae are first presented when the…
In this paper, within a unified framework of the condition number theory we present the explicit expression of the projected condition number of the equality constrained indefinite least squares problem. By setting specific norms and…
In this paper, a well-posed simultaneous space-time First Order System Least Squares formulation is constructed of the instationary incompressible Stokes equations with slip boundary conditions. As a consequence of this well-posedness, the…
In this paper we formulate and solve a robust least squares problem for a system of linear equations subject to quantization error in the data matrix. Ordinary least squares fails to consider uncertainty in the operator, modeling all noise…
Tilt stability plays a pivotal role in understanding how local solutions of an optimization problem respond to small, targeted perturbations of the objective. Although quadratic bundles are a powerful tool for capturing second-order…
Optimization problems constrained by partial differential equations (PDEs) naturally arise in scientific computing, as those constraints often model physical systems or the simulation thereof. In an implicitly constrained approach, the…
In this paper, we study a class of approximation problems, appearing in data approximation and signal processing. The approximations are constructed as combinations of polynomial splines (piecewise polynomials), whose parameters are subject…
In this paper, conditional stability estimates are derived for unique continuation and Cauchy problems associated to the Poisson equation in ultra-weak variational form. Numerical approximations are obtained as minima of regularized least…
Unlike ordinary differential equations (ODEs), linear partial differential equations (PDEs) admit multiple non-equivalent notions of stability. This variety makes interpretation of Lyapunov stability results challenging. \blue{To simplify…
In this paper, we present proofs of the coerciveness of first-order system least-squares methods for general (possibly indefinite) second-order linear elliptic PDEs under a minimal uniqueness assumption. For general linear second-order…
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,…
Recently, collocation based radial basis function (RBF) partition of unity methods (PUM) for solving partial differential equations have been formulated and investigated numerically and theoretically. When combined with stable evaluation…
This paper studies least-square regression penalized with partly smooth convex regularizers. This class of functions is very large and versatile allowing to promote solutions conforming to some notion of low-complexity. Indeed, they force…
This study shows how to obtain least-squares solutions to initial and boundary value problems to nonhomogeneous linear differential equations with nonconstant coefficients of any order. However, without loss of generality, the approach has…