

2023 year, number 2
Tahar Bechouat
Mohammed Cherif Messaadia University, Souk Ahras, Algeria
Keywords: illposed problem, operator equation of first kind, iterative regularization, image deblurring
Abstract >>
In this work, we study a new implicit method to compute the solutions of illposed 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.

E.K. Guseva, V.I. Golubev, I.B. Petrov
Moscow Institute of Physics and Technology (National Research University), Dolgoprudny, Russia
Keywords: gridcharacteristic method, monotonicity criterion, hybrid schemes, acoustic waves
Abstract >>
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 nonphysical solution oscillations. In this paper, we present linear quasimonotone and hybrid gridcharacteristic schemes for a linear transport equation and a onedimensional acoustic system. They are constructed by a method of analysis in the space of unknown coefficients proposed by A.S. Kholodov and a gridcharacteristic 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.

Parviz Darania, Saeed Pishbin, Azam Ebadi
Urmia University, Urmia, Iran
Keywords: autoconvolution Volterra integral equation, Convergence analysis, Multistep collocation methods
Abstract >>
In this study, we introduce multistep collocation methods (MSCM) for solving the Volterra integral equation (VIE) of the autoconvolution type such that without increasing the computational cost, the order of convergence of the proposed onestep 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.

Puneet Jain, Krishna Manglani, Murugesan Venkatapathi
Indian Institute of Science, Bangalore, India
Keywords: error, stopping criteria, condition number, Conjugate Gradients, BiCG, GMRES
Abstract >>
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(κ^{2}) error estimator was described for the GMRES algorithm which allows for nonsymmetric 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 Anorm and l_{2}norm of the error vector in BiCG 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.

A.V. Zhiltsov^{1}, N.N. Maksimova^{2}
^{1}Far Eastern State Transport University, Khabarovsk,Russia ^{2}Amur State University, Blagoveshchensk, Russia
Keywords: body with defect, finite element method, duality methods, Lagrange functionals, generalized Newton’s method, Armijo’s condition
Abstract >>
An equilibrium problem of a twodimensional 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.

Kh.D. Ikramov
Lomonosov Moscow State University, Moscow, Russia
Keywords: centrohermitian matrices, crossmatrices, block quaternion, consimilarity, Schur's lemma
Abstract >>
The relation between complex matrices H and A, given by the equality H A = ĀH is called the pseudocommutation. The set S_{H} of all A that pseudocommute with a nonsingular n × n matrix H is called the pseudocommutation 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 2n^{2}. Under the assumption dim_{R} S_{H} = n^{2}, 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.

K.A. Rybakov
Moscow Aviation Institute, Moscow, Russia
Keywords: approximation, orthogonal expansion, multiple stochastic integral, numerical method, stochastic differential equations
Abstract >>
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 24 are considered, which are used in the numerical methods for solving stochastic differential equations with orders of strong convergence 12.

S.P. Shary^{1,2}
^{1}Federal Research Center for Information and Computational Technologies, Novosibirsk, Russia ^{2}Novosibirsk State University, Novosibirsk, Russia
Keywords: interval analysis, interval, nontraditional intervals, classical interval arithmetic, Kaucher interval arithmetic
Abstract >>
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.

