Related papers: Direct Solution Method for System of Linear Equati…
In a previous paper we have presented a new method for solving a class of Cauchy integral equations. In this work we discuss in detail how to manage this method numerically, when only a finite and noisy data set is available: particular…
A numerical method for solving elliptic PDEs with variable coefficients on two-dimensional domains is presented. The method is based on high-order composite spectral approximations and is designed for problems with smooth solutions. The…
We propose a novel method for a solution of a system of linear equations with the non-negativity condition. The method is based on the Tikhonov functional and has better accuracy and stability than other well-known algorithms.
An iterative scheme for solving ill-posed nonlinear operator equations with monotone operators is introduced and studied in this paper. A Dynamical Systems Method (DSM) algorithm for stable solution of ill-posed operator equations with…
We develop a one step matrix method in order to obtain approximate solutions of first order systems and non-linear ordinary differential equations, reducible to first order systems. We find a sequence of such solutions that converge to the…
This survey describes a class of methods known as "fast direct solvers". These algorithms address the problem of solving a system of linear equations $\boldsymbol{Ax}=\boldsymbol{b}$ arising from the discretization of either an elliptic PDE…
We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for…
Despite its numerical challenges, finite element method is used to compute viscous fluid flow. A consensus on the cause of numerical problems has been reached; however, general algorithms---allowing a robust and accurate simulation for any…
The numerical solution of large-scale Lyapunov matrix equations with symmetric banded data has so far received little attention in the rich literature on Lyapunov equations. We aim to contribute to this open problem by introducing two…
The standard methodology handling nonlinear PDE's involves the two steps: numerical discretization to get a set of nonlinear algebraic equations, and then the application of the Newton iterative linearization or its variants to solve the…
Multi-level numerical methods that obtain the exact solution of a linear system are presented. The methods are devised by combining ideas from the full multi-grid algorithm and perfect reconstruction filters. The problem is stated as…
This article investigates a fast and stable method to solve Henderson's mixed model equation. The proposed algorithm is stable in that it avoids inverting a matrix of a large dimension and hence is free from the curse of dimensionality.…
The work is devoted to the development of numerical methods for computing "formal solutions" of interval systems of linear algebraic equations. These solutions are found in Kaucher interval arithmetic, which extends and completes the…
We investigate modified steepest descent methods coupled with a loping Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations. We show that the proposed method is a convergent regularization…
A method based on order completion for solving general equations is presented. In particular, this method can be used for solving large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems.
The Riccati equation method is used to establish a new stability criteria for linear systems of ordinary differential equations. Two examples are presented in which the obtained result is compared with the results obtained by the Lyapunov…
Linear systems of neutral type are considered using the infinite dimensional approach. The main problems are asymptotic, non-exponential stability, exact controllability and regular asymptotic stabilizability. The main tools are the moment…
The need to estimate a positive definite solution to an overdetermined linear system of equations with multiple right hand side vectors arises in several process control contexts. The coefficient and the right hand side matrices are…
When a system of first order linear ordinary differential equations has eigenvalues of large magnitude, its solutions exhibit complicated behaviour, such as high-frequency oscillations, rapid growth or rapid decay. The cost of representing…
Integral-equation-based fast direct solvers for electromagnetic scattering can substantially reduce computational costs, especially in the presence of multiple excitations. We recently proposed a new high-frequency fast direct solver…