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Vol 65, No 3 (2025)
General numerical methods
SIMULTANEOUS DIAGONABILITY OF A PAIR OF MATRICES: SIMILARITY AND CONGRUENCE
Abstract
Suppose A and B are nondegenerate unitoids with simple canonical angles, and their co-squares commute. The relations that the matrices C and P must satisfy, are established. In congruence theory these relations can be considered as a kind of substitute for the usual permutability.
Computational Mathematics and Mathematical Physics. 2025;65(3):245-250
245-250
INDEXING IN THE GOOD–THOMAS FAST FOURIER TRANSFORM ALGORITHM
Abstract
The article indicates a simple principle for the formation of input and output numbering of arrays in the fast Goode–Thomas algorithm for implementing the discrete Fourier transform. Specific examples show how to construct a superposition with a fast algorithm with a constant structure. A generalization of the algorithm for three or more co-prime factors is considere.
Computational Mathematics and Mathematical Physics. 2025;65(3):251-257
251-257
Optimal control
ANTISYMMETRIC EXTREMUM MAPPING AND LINEAR DYNAMICS
Abstract
An optimal control problem is considered on a fixed time interval. Choosing a control generates a phase trajectory of this problem. The left end of the trajectory is fixed, while a finite-dimensional problem of calculating a fixed point of an extremal mapping is set up at the right end. In the optimal situation, the right end of the phase trajectory coincides with the fixed point of the mapping. In other words, the task is, by choosing a suitable control, to construct a phase trajectory in a Hilbert space that leaves the initial position at the left end of the time interval and arrives at the fixed point of the extremal mapping at the right end of the time interval. To solve the problem within the framework of the Lagrangian formalism, we propose a new approach based on saddle point sufficient optimality conditions. An iterative computational process of saddle point gradient type is investigated. The process is proved to converge strongly in phase and dual trajectories, as well as in terminal variables of the finite-dimensional boundary value problem of linear programming, and to converge weakly in controls. The emphasis is placed on the fact that only proof-based computational techniques transform a mathematical model into a tool for making a guaranteed decision.
Computational Mathematics and Mathematical Physics. 2025;65(3):258-274
258-274
SPARSE AND TRANSFERABLE UNIVERSAL SINGULAR VECTORS ATTACK
Abstract
Mounting concerns about neural networks’ safety and robustness call for a deeper understanding of models’ vulnerability and research in adversarial attacks. Motivated by this, we propose a novel universal attack that is highly efficient in terms of transferability. In contrast to the existing (p, q)-singular vectors approach, we focus on finding sparse singular vectors of Jacobian matrices of the hidden layers by employing the truncated power iteration method. We discovered that using resulting vectors as adversarial perturbations can effectively attack the original model and models with entirely different architectures, highlighting the importance of sparsity constraint for attack transferability. Moreover, we achieve results comparable to dense baselines while damaging less than 1% of pixels and utilizing only 256 samples for perturbation fitting. Our algorithm also admits higher attack magnitude without affecting the human ability to solve the task, and damaging 5% of pixels attains more than a 50% fooling rate on average across models. Finally, our findings demonstrate the vulnerability of state-of-the-art models to universal sparse attacks and highlight the importance of developing robust machine learning systems.
Computational Mathematics and Mathematical Physics. 2025;65(3):275-293
275-293
SUMMATION METHOD FOR FOURIER SERIES ASSOCIATED WITH A MIXED PROBLEM FOR THE INHOMOGENEOUS TELEGRAPH EQUATION
Abstract
A summation method for Fourier series associated with a mixed problem for the telegraph equation in a semi-infinite strip is proposed. The concept of the regularity of a summation method is introduced. It is shown that the generalized sum produced by this method can be used as a solution of a generalized mixed problem. Here, the generalized sum is specified by an exponentially converging function series. In the case of smooth initial data, this series gives a strong solution of the original problem.
Computational Mathematics and Mathematical Physics. 2025;65(3):294-300
294-300
APPLICATION OF INTERVAL SLOPES IN NONSMOOTH ONE-DIMENSIONAL OPTIMIZATION PROBLEMS
Abstract
The interval interpretation of a first-order divided difference, namely, interval slope is considered. Some properties of interval slopes, including ones for convex (concave) functions are proved. Based on the interval slope, necessary and sufficient conditions for the monotonicity of a function are formulated and proved. These criteria are used to propose an algorithm for the global optimization of a one-variable function taking into account its monotonicity. Numerical experiments are conducted that show that the developed global optimization method is applicable in the nondifferentiable case and significantly accelerates finding an approximate global optimum as compared with the basic version.
Computational Mathematics and Mathematical Physics. 2025;65(3):301-324
301-324
OPTIMAL APPROXIMATION OF AVERAGE REWARD MARKOV DECISION PROCESSES
Abstract
We continue to develop the concept of studying the ε-optimal policy for Average Reward Markov Decision Processes (AMDP) by reducing it to Discounted Markov Decision Processes (DMDP). Existing research often stipulates that the discount factor must not fall below a certain threshold. Typically, this threshold is close to one, and as is well-known, iterative methods used to find the optimal policy for DMDP become less effective as the discount factor approaches this value. Our work distinguishes itself from existing studies by allowing for inaccuracies in solving the empirical Bellman equation. Despite this, we have managed to maintain the sample complexity that aligns with the latest results. We have succeeded in separating the contributions from the inaccuracy of approximating the transition matrix and the residuals in solving the Bellman equation in the upper estimate so that our findings enable us to determine the total complexity of the epsilon-optimal policy analysis for DMDP across any method with a theoretical foundation in iterative complexity. Bybl. 17. Fig. 5. Table 1.
Computational Mathematics and Mathematical Physics. 2025;65(3):325-337
325-337
THE NEW IS THE WELL-FORGOTTEN OLD — F4 ALGORITHM OPTIMIZATION
Abstract
The Grobner basis is a fundamental concept in computational algebra. F4 is one of the fastest algorithms for computing Grobner basis. In this paper, we will discuss the process of writing effective F4. Despite the fact that this work focuses on algorithms from computational algebra, some of the results and ideas presented here may have broader applications beyond this specific subject area. In general, the theory described below can be regarded as an abstraction, as it progresses through the text. This is because the text is not actually about the F4 algorithm itself, but rather about the power of profiling, unconventional techniques, and selecting the appropriate memory model. We will provide examples of inefficient usage of the standard library, recall the fundamental principles of optimization in order to apply them as efficiently as possible to obtain the fastest F4 algorithm, using non-traditional approaches.
Computational Mathematics and Mathematical Physics. 2025;65(3):338-346
338-346
Error Analysis of Numerical Methods for Optimization Problems
Abstract
The article discusses methods for constructing solution error estimates in optimization problems, which fall into two categories: theoretical and numerical. Theoretical estimates are based on convergence analysis and are mainly useful for qualitative insights, while numerical estimates provide explicit values but are limited to certain methods. The paper introduces two new numerical error estimation methods for a broad range of optimization problems. The first method uses a three-point scheme to derive an exact error estimate from a decreasing sequence of objective function values. The second method, called the rounding method, estimates the error by tracking the increase in significant digits of the solution as iterations progress. Numerical experiments are provided to support these methods.
Computational Mathematics and Mathematical Physics. 2025;65(3):347-363
347-363
AN ADAPTIVE VARIANT OF THE FRANK–WOLFE METHOD FOR RELATIVE SMOOTH CONVEX OPTIMIZATION PROBLEMS
Abstract
This paper proposes a new variant of the adaptive Frank–Wolfe algorithm for relatively smooth convex minimization problems. It suggests using a divergence different from half of the squared Euclidean norm in the step size adjustment formula. Convergence rate estimates for this algorithm are proven for minimization problems involving relatively smooth convex functions with the triangle scaling property. We also conducted computational experiments for the Poisson linear inverse problem and SVM models. The paper also identifies the conditions under which the proposed algorithm shows a clear advantage over the adaptive proximal gradient Bregman method and its accelerated variants.
Computational Mathematics and Mathematical Physics. 2025;65(3):364-375
364-375
Partial Differential Equations
ON THE UNIQUENESS OF DISCRETE GRAVITY AND MAGNETIC POTENTIALS
Abstract
The problem of the uniqueness of the solution to finite-differenced analogues of the Laplace equation in various domains of the three-dimensional space for restoring discrete gravity and magnetic potentials is considered. This approach makes it possible to determine potentials in two cases: a) when the discrete fundamental solution is known, and b) if an additional a priori information on the boundary values of potentials is given. Our work distinguishes itself from existing studies by allowing for inaccuracies in solving the empirical Bellman equation. Despite this, we have managed to maintain the sample complexity that aligns with the latest results. We have succeeded in separating the contributions from the inaccuracy of approximating the transition matrix and the residuals in solving the Bellman equation in the upper estimate so that our findings enable us to determine the total complexity of the epsilon-optimal policy analysis for DMDP across any method with a theoretical foundation in iterative complexity.
Computational Mathematics and Mathematical Physics. 2025;65(3):376-389
376-389
CAUCHY PROBLEM FOR THE ONE-DIMENSIONAL EQUATION OF MOTION IN A METAMATERIAL
Abstract
Propagation of nonlinear longitudinal elastic waves in a metamaterial (gradient-elastic medium) is modeled using a nonlinear Sobolev-type differential equation, for which the Cauchy problem in the space of continuous functions is investigated. Conditions for the existence of a global solution and for solution blow-up on a finite time interval are considered.
Computational Mathematics and Mathematical Physics. 2025;65(3):390-400
390-400
Computer science
MATHEMATICAL RECONSTRUCTION OF SIGNALS AND IMAGES USING TEST TRIALS: A NON-BLIND APPROACH
Abstract
The paper considers a mathematical method for the non-blind recovery of regular and multidimensional signals, including images distorted in processing by linear stationary systems. Instead of transfer functions, which are often difficult to determine, this method directly utilizes the trial test signals of processing systems to non-blindly recover of the signal from the test equation. The use of test signals belonging to the class of core functions greatly simplifies the signal recovery procedure and makes it more accurate and robust. The operator approach based on the multivariate convolution equation significantly improves the speed of numerical computation. Regularization technique is used to solve incorrectly posed and ill-conditioned problems, which allows efficiently recovering of real nondeterministic signals with noise and uncertainties. The influence of the test signals type on the recovery accuracy is analyzed and a method of their formation is proposed in the paper. Numerical experiments demonstrating the stability and efficiency of the proposed algorithm when recovering one-dimensional signals and two-dimensional images at high level of noise and uncertainties in data are considered. The proposed technique is able to improve the quality of signal and image processing without the need to modify complex and expensive equipment, to expand the field of practical application of mathematical reconstruction methods. Program codes and datasets are available at: https://github.com/novikov-borodin/data-rec.
Computational Mathematics and Mathematical Physics. 2025;65(3):401-414
401-414