Related papers: A finite dimensional analog of the Krein formula

In this paper we establish a comparison result through symmetrization for solutions to some boundary value problems involving the fractional Laplacian. This allows to get sharp estimates for the solutions, obtained by comparing them with…

In the present paper, we consider large-scale differential Lyapunov matrix equations having a low rank constant term. We present two new approaches for the numerical resolution of such differential matrix equations. The first approach is…

We prove low frequency estimates for the boundary values of the resolvent of long range perturbations of the flat Laplacian in divergence form.

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem. We write an equivalent characterization as a thin obstacle problem. In this way…

We describe a simple way of constructing exponentially growing solutions of the second order systems with the Laplacian as the principal term.

For the fractional Laplacian of variable order, an efficient and accurate numerical evaluation in multi-dimension is a challenge for the nature of a singular integral. We propose a simple and easy-to-implement finite difference scheme for…

We prove an analog of Krein's resolvent formula expressing the resolvents of self-adjoint extensions in terms of boundary conditions. Applications to quantum graphs and systems with point interactions are discussed.

The thesis studies linear and semilinear Dirichlet problems driven by different fractional Laplacians. The boundary data can be smooth functions or also Radon measures. The goal is to classify the solutions which have a singularity on the…

In the present paper we extend results of M.G. Krein associated to the spectral problem for Krein systems to systems with matrix valued accelerants with a possible jump discontinuity at the origin. Explicit formulas for the accelerant are…

We study here a singular perturbation problem of biLaplacian type, which can be seen as the biharmonic counterpart of classical combustion models. We provide different results, that include the convergence to a free boundary problem driven…

The paper looks at a scaled variant of the stochastic gradient descent algorithm for the matrix completion problem. Specifically, we propose a novel matrix-scaling of the partial derivatives that acts as an efficient preconditioning for the…

When approximating elliptic problems by using specialized approximation techniques, we obtain large structured matrices whose analysis provides information on the stability of the method. Here we provide spectral and norm estimates for…

The purpose of this work is to introduce and analyze a numerical scheme to efficiently solve boundary value problems involving the spectral fractional Laplacian. The approach is based on a reformulation of the problem posed on a…

The efficient numerical integration of large-scale matrix differential equations is a topical problem in numerical analysis and of great importance in many applications. Standard numerical methods applied to such problems require an unduly…

We propose an extremely versatile approach to address a large family of matrix nearness problems, possibly with additional linear constraints. Our method is based on splitting a matrix nearness problem into two nested optimization problems,…

In this article we establish an approximation result involving the Laplacian with Robin boundary conditions. It informs about the weak solutions dependence from the input function on the boundary.

Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…

We revisit a formula that connects the minimal ranks of triangular parts of a matrix and its inverse and relate the result to structured rank matrices. We also address the generic minimal rank problem.

This paper is concerned with the calmness of a partial perturbation to the composite rank constraint system, an intersection of the rank constraint set and a general closed set, which is shown to be equivalent to a local Lipschitz-type…

In this paper we consider approximations of Neumann problems for the integral fractional Laplacian by continuous, piecewise linear finite elements. We analyze the weak formulation of such problems, including their well-posedness and…