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In this article we study the problem of recovering the unknown solution of a linear ill-posed problem, via iterative regularization methods. We review the problem of projection-regularization from a statistical point of view. A basic…
In this paper, we extend the previous method for solving inverse problems for steady-state equations using the Generalized Collage Theorem by searching for an approximation that not only minimizes the collage error but also maximizes the…
We consider a pencil of matrix Sturm-Liouville operators on a finite interval. We study properties of its spectral characteristics and inverse problems that consist in recovering of the pencil by the spectral data: eigenvalues and…
We study scalar multivariate non-stationary subdivision schemes with a general dilation matrix. We characterize the capability of such schemes to reproduce exponential polynomials in terms of simple algebraic conditions on their symbols.…
The affine inverse eigenvalue problem consists of identifying a real symmetric matrix with a prescribed set of eigenvalues in an affine space. Due to its ubiquity in applications, various instances of the problem have been widely studied in…
This paper presents an iterative method suitable for inverting semilinear problems which are important kernels in many numerical applications. The primary idea is to employ a parametrization that is able to reduce semilinear problems into…
The problems of computational data processing involving regression, interpolation, reconstruction and imputation for multidimensional big datasets are becoming more important these days, because of the availability of data and their widely…
We deal with the problem of description of nonsingular pairs of compatible flat metrics for the general $N$-component case. We describe the scheme of the integrating the nonlinear equations describing nonsingular pairs of compatible flat…
We describe a novel algorithm for solving general parametric (nonlinear) eigenvalue problems. Our method has two steps: first, high-accuracy solutions of non-parametric versions of the problem are gathered at some values of the parameters;…
Boundary problem for linear partial differential algebraic equations system with multiple characteristic curves is considered. It is supposed that matrix-functions pencil of the system under consideration is smoothly equivalent to special…
We define and solve classes of sparse matrix problems that arise in multilevel modeling and data analysis. The classes are indexed by the number of nested units, with two-level problems corresponding to the common situation in which data on…
We give a method for constructing a regularizing decomposition of a matrix pencil, which is formulated in terms of the linear mappings. We prove that two pencils are topologically equivalent if and only if their regularizing decompositions…
We consider a matrix pencil whose coefficients depend on a positive parameter $\epsilon$, and have asymptotic equivalents of the form $a\epsilon^A$ when $\epsilon$ goes to zero, where the leading coefficient $a$ is complex, and the leading…
In this paper, we present a generalized version of the matrix chain algorithm to generate efficient code for linear algebra problems, a task for which human experts often invest days or even weeks of works. The standard matrix chain problem…
The theory of matrix splitting is a useful tool for finding solution of rectangular linear system of equations, iteratively. The purpose of this paper is two-fold. Firstly, we revisit theory of weak regular splittings for rectangular…
Linear algebra's main concerns are sets of vectors, linear functions, subspaces, linear systems, matrices and concepts about those, such as whether the solution of linear system exists or is unique; a set of vectors is linearly independent…
In this paper, we develop a nonlinear reduction framework based on our recently introduced extended group finite element method. By interpolating nonlinearities onto approximation spaces defined with the help of finite elements, the…
A number of applications require the computation of the trace of a matrix that is implicitly available through a function. A common example of a function is the inverse of a large, sparse matrix, which is the focus of this paper. When the…
The inverse spectral problem for the second-order differential pencil with quadratic dependence on the spectral parameter is studied. We obtain sufficient conditions for the global solvability of the inverse problem, prove its local…
In high-dimensional linear regression, the goal pursued here is to estimate an unknown regression function using linear combinations of a suitable set of covariates. One of the key assumptions for the success of any statistical procedure in…