Related papers: A fast and well-conditioned spectral method
Markov expanding maps, a class of simple chaotic systems, are commonly used as models for chaotic dynamics, but existing numerical methods to study long-time statistical properties such as invariant measures have a poor trade-off between…
This paper explores preconditioning the normal equation for non-symmetric square linear systems arising from PDE discretization, focusing on methods like CGNE and LSQR. The concept of ``normal'' preconditioning is introduced and a strategy…
We give a specific method to solve with quadratic complexity the linear systems arising in known algorithms to deal with the sign determination problem. In particular, this enable us to improve the complexity bound for sign determination in…
We analyze the fundamental quantum limit of the resolution of an optical imaging system from the perspective of the detection problem of deciding whether the optical field in the image plane is generated by one incoherent on-axis source…
Oscillatory second order linear ordinary differential equations arise in many scientific calculations. Because the running times of standard solvers increase linearly with frequency when they are applied to such problems, a variety of…
Sequences of parametrized Lyapunov equations can be encountered in many application settings. Moreover, solutions of such equations are often intermediate steps of an overall procedure whose main goal is the computation of…
An exact discretization method is being developed for solving linear systems of ordinary fractional-derivative differential equations with constant matrix coefficients (LSOFDDECMC). It is shown that the obtained linear discrete system in…
This paper introduces a new method for discretizing and solving integral equation formulations of Maxwell's equations which achieves spectral accuracy for smooth surfaces. The approach is based on a hybrid Nystr\"om-collocation method using…
Two combined numerical methods for solving semilinear differential-algebraic equations (DAEs) are obtained and their convergence is proved. The comparative analysis of these methods is carried out and conclusions about the effectiveness of…
In this paper we establish new renormalized oscillation theorems for discrete symplectic eigenvalue problems with Dirichlet boundary conditions. These theorems present the number of finite eigenvalues of the problem in arbitrary interval…
Non-self-adjoint second-order ordinary differential operators on a finite interval with complex weights are studied. Properties of spectral characteristics are established and the inverse problem of recovering operators from their spectral…
This work aims to accelerate the convergence of proximal gradient methods used to solve regularized linear inverse problems. This is achieved by designing a polynomial-based preconditioner that targets the eigenvalue spectrum of the normal…
The aim of the present work is to introduce a method based on Chebyshev polynomials for the numerical solution of a system of Cauchy type singular integral equations of the first kind on a finite segment. Moreover, an estimation error is…
In this paper we present an algebraic study concerning the general second order linear differential equation with polynomial coefficients. By means of Kovacic's algorithm and asymptotic iteration method we find a degree independent…
This study aims to construct a stable, high-order compact finite difference method for solving Sobolev-type equations with Dirichlet boundary conditions in one-space dimension. Approximation of higher-order mixed derivatives in some…
Recently, a flexible and stable algorithm was introduced for the computation of 2D unstable manifolds of periodic solutions to systems of ordinary differential equations. The main idea of this approach is to represent orbits in this…
Second order linear non-autonomous differential equations with negative stiffness are considered. Using Chetaev-like (Lyapunov-like) functions, necessary (sufficient) conditions are found for the solutions to be bounded for all initial…
We propose a new method for preconditioning Kaczmarz method by sketching. Kaczmarz method is a stochastic method for solving overdetermined linear systems based on a sampling of matrix rows. The standard approach to speed up convergence of…
A practical and simple stable method for calculating Fourier integrals is proposed, effective both at low and at high frequencies. An approach based on the fruitful idea of Levin, to use of the collocation method to approximate the slowly…
We present and experimentally realize a quantum algorithm for efficiently solving the following problem: given an $N\times N$ matrix $\mathcal{M}$, an $N$-dimensional vector $\textbf{\emph{b}}$, and an initial vector $\textbf{\emph{x}}(0)$,…