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At present, only some special differential equations have explicit analytical solutions. In general, no one thinks that it is possible to analytically find the exact solution of nonlinear equations. In this article based on the idea that…
A representation of an arbitrary system of strict linear inequalities in R^n as a system of points is proposed. The representation is obtained by using a so-called polarity. Based on this representation an algorithm for constructing a…
This paper introduces an abstract framework for randomized subspace correction methods for convex optimization, which unifies and generalizes a broad class of existing algorithms, including domain decomposition, multigrid, and block…
This paper introduces a new method of partitioning the solution space of a multi-objective optimisation problem for parallel processing, called Efficient Projection Partitioning. This method projects solutions down into a single dimension,…
We describe an exact and highly efficient numerical algorithm for solving a special but important class of convection-diffusion equations. These equations occur in many problems in physics, chemistry, or biology, and they are usually hard…
In this article we determine several theorems and methods for solving linear congruences and systems of linear congruences, and we find the number of distinct solutions. Many examples of solving congruences are given.
In recent years, various subspace algorithms have been developed to handle large-scale optimization problems. Although existing subspace Newton methods require fewer iterations to converge in practice, the matrix operations and full…
A novel orthogonalization-free method together with two specific algorithms are proposed to solve extreme eigenvalue problems. On top of gradient-based algorithms, the proposed algorithms modify the multi-column gradient such that earlier…
In this paper we study the convergence of an iterative algorithm for finding zeros with constraints for not necessarily monotone set-valued operators in a reflexive Banach space. This algorithm, which we call the proximal-projection method…
The numerical methods for differential equation solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods have the restricted class of…
Linear systems are the bedrock of virtually all numerical computation. Machine learning poses specific challenges for the solution of such systems due to their scale, characteristic structure, stochasticity and the central role of…
Solving linear systems of equations is an important problem in science and engineering. Many quantum algorithms, such as the Harrow-Hassidim-Lloyd (HHL) algorithm (for quantum-gate computers) and the box algorithm (for quantum-annealing…
Ordinary least squares (OLS) is the default method for fitting linear models, but is not applicable for problems with dimensionality larger than the sample size. For these problems, we advocate the use of a generalized version of OLS…
Linear differential equations are ubiquitous in science and engineering. Quantum computers can simulate quantum systems, which are described by a restricted type of linear differential equations. Here we extend quantum simulation algorithms…
Various versions of the Dynamical Systems Method (DSM) are proposed for solving linear ill-posed problems with bounded and unbounded operators. Convergence of the proposed methods is proved. Some new results concerning discrepancy principle…
This is the first in a series of papers which deal with the development of novel methods for solving a system of linear algebraic equations with a time complexity lower than existing algorithms. The NxN system of linear equations, Ax = b,…
This paper introduces new algorithm for line extraction from laser range data including methodology for efficient computation. The task is cast to series of one dimensional problems in various spaces. A fast and simple specialization of…
A cumbersome operation in numerical analysis and linear algebra, optimization, machine learning and engineering algorithms; is inverting large full-rank matrices which appears in various processes and applications. This has both numerical…
This paper presents a solution for efficiently and accurately solving separable least squares problems with multiple datasets. These problems involve determining linear parameters that are specific to each dataset while ensuring that the…
In a recent paper, a distributed algorithm was proposed for solving linear algebraic equations of the form $Ax = b$ assuming that the equation has at least one solution. The equation is presumed to be solved by $m$ agents assuming that each…