Home – Home – Jornals – Numerical Analysis and Applications 2023 number 2

In this work, we study a new implicit method to compute the solutions of ill-posed linear operator equations of the first kind under the setting of compact operators. The regularization theory can be used to demonstrate the stability and convergence of this scheme. Furthermore, we obtain convergence results and effective stopping criteria according to Morozov's discrepancy principle. Numerical performances are calculated to show the validity of our implicit method and demonstrate its applicability to deblurring problems.

The system of linear acoustic equations is hyperbolic. It describes the process of the acoustic wave propagation in deformable media. An important property of the schemes used for the numerical solution is their high approximation order. This property allows one to simulate the perturbation propagation process over sufficiently large distances. Another important property is monotonicity of the schemes used, which prevents the appearance of non-physical solution oscillations. In this paper, we present linear quasi-monotone and hybrid grid-characteristic schemes for a linear transport equation and a one-dimensional acoustic system. They are constructed by a method of analysis in the space of unknown coefficients proposed by A.S. Kholodov and a grid-characteristic monotonicity criterion. Wide spatial stencils with five to seven nodes of the computational grid are considered. Reflection of a longitudinal wave with a sharp front from the interface between media with different parameters is used to compare the numerical solutions.

In this study, we introduce multi-step collocation methods (MSCM) for solving the Volterra integral equation (VIE) of the auto-convolution type such that without increasing the computational cost, the order of convergence of the proposed one-step collocation methods will be increased. A convergence analysis of the MSCM is investigated using the Peano theorems for interpolation and, finally, two numerical examples are introduced to clarify the significant advantage of the MSCM.

The demands of accuracy in measurements and engineering models today render the condition number of problems larger. While a corresponding increase in the precision of floating point numbers ensured a stable computing, the uncertainty in convergence when using residue as a stopping criterion has increased. We present an analysis of the uncertainty in convergence when using relative residue as a stopping criterion for iterative solution of linear systems, and the resulting over/under computation for a given tolerance in error. This shows that error estimation is significant for an efficient or accurate solution even when the condition number of the matrix is not large. An Ο(1) error estimator for iterations of the CG algorithm was proposed more than two decades ago. Recently, an Ο(κ²) error estimator was described for the GMRES algorithm which allows for non-symmetric linear systems as well, where κ is the iteration number. We suggest a minor modification in this GMRES error estimation for increased stability. In this work, we also propose an Ο(n) error estimator for A-norm and l₂-norm of the error vector in Bi-CG algorithm. The robust performance of these estimates as a stopping criterion results in increased savings and accuracy in computation, as condition number and size of problems increase.

An equilibrium problem of a two-dimensional body with a thin defect whose properties are characterized by a fracture parameter is considered. The problem is discretized, and an approximation accuracy theorem is proved. To solve the problem, a dual method based on a modified Lagrange functional is used. In computational experiments, when solving the direct problem, a generalized Newton's method is used with a step satisfying Armijo's condition.

The relation between complex matrices H and A, given by the equality H A = ĀH is called the pseudo-commutation. The set S_H of all A that pseudo-commute with a nonsingular n × n matrix H is called the pseudo-commutation class defined by H . Every class S_H is a subspace of the space M_n(C) interpreted as a real vector space of dimension 2n². Under the assumption dim_R S_H = n², we find a necessary and sufficient condition for the possibility to decomplexify all the matrices in S_H by one and the same similarity transformation.

In the article, formulas for exact calculation of the approximation error of multiple Itô stochastic integrals based on their orthogonal expansion are obtained. As an example, stochastic Itô integrals with multiplicities 2-4 are considered, which are used in the numerical methods for solving stochastic differential equations with orders of strong convergence 1-2.

The paper discusses the question of why intervals, which are the main object of Interval Analysis, have exactly the form that we know well and habitually use, and not some other. In particular, we investigate why traditional intervals are closed, i.e. contain their endpoints, and also what is wrong with an empty interval. A second question considered in the work is how expedient it is to expand the set of traditional intervals by some other objects. We show that improper («reversed») intervals and the arithmetic of such intervals (the Kaucher complete interval arithmetic) are very useful from many different points of view.