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Vol 65, No 7 (2025)
K 70-letiyu E.E. Tyrtyshnikova
Computational Mathematics and Mathematical Physics. 2025;65(7):1059
1059
General numerical methods
Eigenvalues of non-Hermitian banded Toeplitz matrices approaching simple points of the limiting set
Abstract
For large non-Hermitian banded Toeplitz matrices, it is well known that their eigenvalues cluster along a limiting set, which is formed by a finite union of closed analytic arcs. We consider general non-Hermitian banded Toeplitz matrices and extend the simple-loop method to obtain individual asymptotic expansions for eigenvalues approaching simple and non-degenerate points of the limiting set as the matrix order increases to infinity. We also develop an algorithm to effectively compute these expansions.
Computational Mathematics and Mathematical Physics. 2025;65(7):1060-1076
1060-1076
Tensor cross interpolation for global discrete optimization with application to Bayesian network inference
Abstract
Global discrete optimization is notoriously difficult due to the lack of gradient information and the curse of dimensionality, making exhaustive search infeasible. Tensor cross approximation is an efficient technique to approximate multivariate tensors (and discretized functions) by tensor product decompositions based on a small number of tensor elements, evaluated on adaptively selected fibers of the tensor, that intersect on submatrices of (nearly) maximum volume. The submatrices of maximum volume are empirically known to contain large elements, hence the entries selected for cross interpolation can also be good candidates for the globally maximal element within the tensor. In this paper we consider evolution of epidemics on networks, and infer the contact network from observations of network nodal states over time. By numerical experiments we demonstrate that the contact network can be inferred accurately by finding the global maximum of the likelihood using tensor cross interpolation. The proposed tensor product approach is flexible and can be applied to global discrete optimization for other problems, e.g. discrete hyperparameter tuning.
Computational Mathematics and Mathematical Physics. 2025;65(7):1077-1090
1077-1090
A FAST NUMERICAL METHOD FOR THE SOURCE RECONSTRUCTION IN THE COAGULATION-FRAGMENTATION EQUATION
Abstract
A fast numerical method is proposed for the problem of restoring the source function in the Smoluchowski coagulation-fragmentation equation. The proposed method is based on the earlier work with a more detailed description of the transition from the coagulation-fragmentation equation to the final system of variational equations and the iterative process. Exploitation of the low-rank matrices has been introduced into this process to reduce the computational complexity of each iteration. The proposed methodology allows speeding up the calculations by thousands of times without losing the accuracy of the original approach.
Computational Mathematics and Mathematical Physics. 2025;65(7):1091-1109
1091-1109
SYMMETRIC TRIANGULAR DECOMPOSITION FOR CONSTRUCTING APPROXIMATIONS TO SOLVING THE QUADRATIC ASSIGNMENT PROBLEM
Abstract
The permutation matrices that arise in the process of triangular decomposition of shifted symmetric matrices with the choice of the maximum modulo leading element on the diagonal are used as initial approximations for a series of elementary permutations that improve the target value of the quadratic assignment problem. The results of testing the proposed method on 128 test tasks from QAPLIB are presented.
Computational Mathematics and Mathematical Physics. 2025;65(7):1110-1117
1110-1117
ADAPTIVE BLOCK ALGEBRAIC MULTIGRID METHOD FOR MULTIPHYSICS PROBLEMS
Abstract
We propose the adaptive block algebraic method to solve the multiphysics problems arising from the collocated finite volume discretization methods. The method is specifically designed to solve multiphysics problems featuring various physics in various parts of the domain, resulting in block-structured saddle-point linear algebraic systems with variable block size. The adaptive algebraic multigrid method uses available information on the eigenvectors of the problem to construct prolongation and restriction operators. The information on the distribution of degrees of freedom within the blocks to form an initial set of vectors is used. It was shown that the arising linear systems are amenable to the solution with the proposed method. Various approaches to strong point selection, coarse space refinement, and bootstrapping the test vectors are discussed and analysed. In this work, we address the systems arising from coupled problems of free-flow and poroelasticity, frictional rigid body contact mechanics, and poroplasticity with fractures. All of the problems are of saddle-point nature.
Computational Mathematics and Mathematical Physics. 2025;65(7):1118-1142
1118-1142
K-OPTIMAL PRECONDITIONERS BASED ON APPROXIMATIONS OF INVERSE MATRICES
Abstract
The problem of constructing preconditioners of a special kind for solving systems of linear algebraic equations is considered. A new approach to the construction of preconditioners based on minimizing the K-number of conditionality for the A−1P matrix is proposed, where A is the initial matrix of the system, P is the preconditioner. It is proved that for circulant matrices, this approach is equivalent to constructing an optimal Chen circulant for the inverse matrix. Numerical experiments have been carried out on a series of test problems with Toeplitz matrices, showing that the proposed approach makes it possible to significantly reduce the number of iterations of the conjugate gradient method compared with the classical approach. The results obtained open up new possibilities for constructing effective preconditioners in other classes of matrices.
Computational Mathematics and Mathematical Physics. 2025;65(7):1143-1155
1143-1155
ADAPTIVE PRIMAL–DUAL METHODS WITH AN INEXACT ORACLE FOR RELATIVELY SMOOTH OPTIMIZATION PROBLEMS AND THEIR APPLICATIONS TO RECOVERING LOW-RANK MATRICES
Abstract
The article is devoted to adaptive primal-dual first-order methods for relatively smooth optimization problems with constraint inequalities, as well as their applications to problems of reconstruction of low-rank matrices. It is shown that for a certain class of relatively smooth problems of reconstruction of low–rank matrices, a triangular scaled property with a scaling coefficient of γ = 2 can be applied, which opened up the possibility of applying accelerated methods and methods of the Frank-Wolfe type and the results on their computational guarantees for such problems. An adaptive version of the similar triangles method is proposed for smooth problems with respect to the Bregman divergence with a triangular scaled property with a scaling coefficient of γ = 2. Non-accelerated and accelerated primal-dual adaptive methods with an inaccurate oracle for relatively smooth problems are also proposed. The accelerated primal-dual method is also an analogue of the method of similar triangles and uses the triangular scaled property of the Bregman divergence with a scaling coefficient γ = 2. The key feature of the methods studied in the article is the possibility of using inaccurate information in iterations and taking into account the inaccuracy of solving auxiliary subtasks in iterations of methods. This is natural due to the complication of such subtasks due to the use of the Bregman divergence instead of the square of the Euclidean norm. In particular, this led to a variant of the Frank–Wolfe method for a distinguished class of relatively smooth problems. For all the proposed methods, theoretical results on the quality of the solution were obtained.
Computational Mathematics and Mathematical Physics. 2025;65(7):1156-1177
1156-1177
APPLICATION OF THE MOSAIC-SKELETON MATRIX APPROXIMATION METHOD IN ELECTROMAGNETIC SCATTERING PROBLEMS
Abstract
Algorithms for solving electromagnetic wave scattering problems in the frequency domain using the method of integral equations, as well as using a model of physical optics that takes into account the re-reflection of waves, are considered. In both cases, the main computational costs, both in terms of calculation time and in terms of the required machine memory, are associated with storing dense matrices of the interaction of discrete elements and performing operations with these matrices. The features of applying the mosaic-skeleton approximation method to such matrices and the possibilities of this method in this class of problems are analyzed.
Computational Mathematics and Mathematical Physics. 2025;65(7):1178-1195
1178-1195
BLOCK GENERALIZED METHOD OF MINIMAL INCONSISTENCIES
Abstract
This paper presents a block extension of the generalized minimum residual method (GMRES) with a new block reduction technology. Unlike the currently known methods, a block can be reduced not only when it has degenerated, but also when a part of the residuals converges with the required accuracy or when the residuals become linearly dependent with a given accuracy. In addition, the method makes it possible to continue the process when adding new right-hand sides. At the same time, after reducing the block and adding new right-hand sides, the method retains its compact form and low complexity. This makes it possible to use it in cases where not all the right-hand sides are known in advance. It also makes it possible to limit the maximum block size, thus balancing between performance and the final dimension of the space, i.e. the required memory. Numerical experiments confirm the high efficiency of the method in comparison with the non-block extension of GMRES and its naive block generalization.
Computational Mathematics and Mathematical Physics. 2025;65(7):1196-1210
1196-1210
Ordinary differential equations
RATIONAL COEFFICIENTS OF ORTHOGONAL DECOMPOSITIONS OF CERTAIN FUNCTIONS
Abstract
Decompositions of many elementary and special functions into series by orthogonal polynomials have coefficients known explicitly. However, these coefficients are almost always irrational. Therefore, any numerical method gives these coefficients approximately when calculating in any arithmetic. This also applies to spectral methods that provide efficient approximations of holonomic functions. However, in some exceptional cases, the expansion coefficients obtained by the spectral method turn out to be rational and are calculated exactly in rational arithmetic. We consider such decompositions with respect to some classical orthogonal polynomials. It is shown that in this way it is possible to obtain an infinite set of linear forms for some irrationalities, in particular, for Euler’s constant.
Computational Mathematics and Mathematical Physics. 2025;65(7):1211-1224
1211-1224
SOLVABILITY AND PROPERTIES OF CRITICAL POINTS OF LINEAR VOLTERRA INTEGRO-ALGEBRAIC EQUATIONS
Abstract
Systems of Volterra linear integral equations with an identically degenerate matrix in the domain of definition with a principal term are considered. Such systems are now commonly referred to as integro-algebraic equations. The concept of a simple structure of integro-algebraic equations is introduced and the issues of solvability are investigated. In particular, systems are considered when there are critical points in the domain of definition. The article formalizes the concept of a critical point of such systems. A number of examples illustrating the theoretical results are given.
Computational Mathematics and Mathematical Physics. 2025;65(7):1225-1240
1225-1240
1241-1248
Partial Differential Equations
APPLICATION OF QUADRATURE FORMULAS FOR THE SIMPLE LAYER POTENTIAL TO THE EXTERNAL NEUMANN PROBLEM
Abstract
A method for numerically solving the external Neumann problem is proposed based on new quadrature formulas for the simple layer potential constructed using analytical calculation of integrals. The method is tested on the Neumann problem for the Laplace equation outside of an ellipsoid, for which explicit solutions are found. It is shown that the numerical solution of the problem obtained by the proposed method uniformly approximates the exact solution and provides a lower error and faster convergence than the numerical solution obtained using standard quadrature formulas based on numerical integration. The dependence of the numerical solutions on the ellipsoid parameters are discussed.
Computational Mathematics and Mathematical Physics. 2025;65(7):1249-1264
1249-1264
ON VARIATIONAL SETTINGS OF THE INVERSE COEFFICIENT PROBLEMS IN MAGNETIC HYDRODYNAMICS
Abstract
The uniqueness of the inverse coefficient problem in the framework of magnetic hydrodynamics is considered. Relating theorem is proved provided that zeroth approximations of velocity field and magnetic field are known. The viscosity coefficient is uniquely determined if "full" magnetic field is given.
Computational Mathematics and Mathematical Physics. 2025;65(7):1265-1276
1265-1276
Mathematical physics
STUDY OF RESONANCE PROPERTIES OF PAIRED NANOPARTICLES WITH MESOSCOPIC BOUNDARY CONDITIONS BY THE DISCRETE SOURCE METHOD
Abstract
The discrete source method was adapted to calculate the field intensity in a nanometer gap of a pair of plasmonic nanoparticles taking into account quantum effects described by mesoscopic boundary conditions with Feibelman parameters. Based on the computational experiment, it was found that for particles made of noble metals, taking into account the quantum effect leads to blue shift of the plasmon resonance and a damping its amplitude. In the case of an alkali metal, taking into account the quantum effect leads to red shift of the plasmon resonance, and when the gap is reduced to 1-2 nm, an enhancement of the intensity in the gap is observed. Analysis of the intensity distribution over the particle surface made it possible to determine that its highest values are achieved at the ends of the particles, with the absolute maximum observed at the ends facing inside the gap. In addition, it was found that the field intensity along the particle surface can vary by four orders of magnitude over a length of only 12 nm, which is only 1.5% of the wavelength of external excitation.
Computational Mathematics and Mathematical Physics. 2025;65(7):1277-1285
1277-1285
ON THE AMPLIFICATION OF THE GAS SUSPENSION MODEL WITH A GENERAL PRESSURE
Abstract
A method is presented for amplifying a model of a multi-velocity multicomponent heterogeneous mixture with a general pressure consisting of various gases and one incompressible component by introducing a parameter into the model equations. The characteristic analysis of the equations of the modified model is carried out and their hyperbolicity at the parameter value is established. It is shown that with a certain choice of e, the “a” movement of the individual components of the mixture is stopped. When integrating a hyperbolic system of equations, a multidimensional nodal method of characteristics is applied, based on splitting the initial system of equations in coordinate directions into a number of one-dimensional subsystems, each of which is solved using the inverse method of characteristics. Using this approach, a number of one-dimensional and two-dimensional model problems have been calculated.
Computational Mathematics and Mathematical Physics. 2025;65(7):1286-1300
1286-1300