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Fast numerical algorithm for a high-precision 6D electromagnetic positioning navigation system

TURKISH JOURNAL OF PHYSICS 2014年第2期38卷 165-173页

作者： Xiang, Qian Wang, Shiqing Liu, Haifeng Southwestern Inst Phys Chengdu Sichuan Peoples R China Chengdu Univ Technol Engn & Tech Coll Inst Nucl Engn & Technol Leshan Peoples R China

A fast numerical algorithm is proposed for the 6 degrees of freedom electromagnetic navigation system, which can be used to improve surgical operations guided by medical images. Both the rotation transformation technique and optimization technique are adopted in the algorithm. The results from simulations show that the calculating time is improved greatly under the required precision in the surgery. A navigation system based on the algorithm can make the medical device's dimensions miniaturized and the structure simplified and portable. The performance and application scope of the electromagnetic positioning navigation system can be improved.

关键词： Six degrees of freedom electromagnetic positioning navigation numerical algorithm

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Game Theory and Dual Approach to the Dynamic Programming on the Example of the COVID-19 Pandemic in Poland Described by Mathematical Model With Three-Dose Vaccinated

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OPTIMAL CONTROL APPLICATIONS & METHODS 2025年第2期46卷 647-664页

作者： Matusik, Radoslaw Univ Lodz Fac Math & Comp Sci Lodz Poland

In this paper, a new approach to the disease transmission dynamics of the COVID-19 pandemic is presented, involving the use of game theory and dual dynamic programming. A new compartmental model that describes these dynamics is introduced. New classes have been added to this model to account for the portion of the population vaccinated with one dose, two doses, or three doses. Pandemic costs are also included. Time-dependent parameters (strategies) are employed, allowing for the consideration of different behavior variants and decisions made by policymakers. Sufficient conditions for a dual epsilon-closed-loop Nash equilibrium, are formulated in the form of a verification theorem. A numerical algorithm is constructed, and numerical simulations are performed. A comparison between real pandemic data for Poland and the data obtained from the model is made.

关键词： COVID-19 dual epsilon-closed-loop Nash equilibrium dual dynamic programming game model of the pandemic numerical algorithm

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Extreme values of solution of Caputo-Hadamard uncertain fractional differential equation and applications

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2024年第6期47卷 4105-4121页

作者： Liu, Hanjie Zhu, Yuanguo He, Liu Nanjing Univ Sci & Technol Sch Math & Stat Nanjing Peoples R China

Uncertain fractional differential equation (UFDE) is a useful tool for studying complex systems in uncertain environments. The mathematical characteristics of solution of an UFDE are also widely used in various fields. In this paper, we give the extreme value theorems of solution of Caputo-Hadamard UFDE and applications. A numerical algorithm for obtaining the inverse uncertainty distributions (IUDs) for extreme values of solution of Caputo-Hadamard UFDE is presented;the stability and feasibility of the proposed algorithm are validated by numerical experiments. As an application of extreme value theorems in uncertain financial market, the pricing formulas of the American option based on the new uncertain stock model are given. Besides, the algorithms for computing the price of the American option without explicit pricing formulas based on the Simpson's rule are designed. Finally, the price fluctuation of the American option is illustrated by numerical experiments.

关键词： American option extreme value theorems numerical algorithm Simpson's rule uncertain fractional differential equation

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Fast Toeplitz eigenvalue computations, joining interpolation-extrapolation matrix-less algorithms and simple-loop theory: The preconditioned setting

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APPLIED MATHEMATICS AND COMPUTATION 2024年 466卷

作者： Bogoyaa, M. Serra-Capizzano, S. Vassalos, P. Univ Valle Dept Matemat Cali Colombia Univ Insubria Dipartimento Sci & Alta Tecnol Como Italy Uppsala Univ Dept Informat Technol Div Sci Comp Uppsala Sweden Athens Univ Econ & Business Sch Informat Sci & Technol Dept Informat Athens Greece

Under appropriate technical assumptions, the simple-loop theory allows to derive various types of asymptotic expansions for the eigenvalues of Toeplitz matrices T-n (f) generated by a function f. Unfortunately, such a theory is not available in the preconditioning setting, that is for matrices of the formT(n)(-1)(g)T-n(l) with g,l real-valued, g nonnnegative and not identically zero almost everywhere. Independently and under the milder hypothesis that f = l/g is even and monotonic over [0, pi], matrix-less algorithms have been developed for the fast eigenvalue computation of large preconditioned matrices of the type above, within a linear complexity in the matrix order: behind the high efficiency of such algorithms there are the expansions as in the case g = 1, combined with the extrapolation idea, and hence we conjecture that the simple-loop theory has to be extended in such a new setting, as the numerics strongly suggest. Here we focus our attention on a change of variable, followed by the asymptotic expansion of the new variable, and we consider new matrix-less algorithms ad hoc for the current case. numerical experiments show a much higher accuracy till machine precision and the same linear computational cost, when compared with the matrix-less procedures already proposed in the literature.

关键词： Toeplitz matrix Spectra Preconditioned matrix Asymptotic expansion numerical algorithm

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A Sensitivity-Based algorithm Approach in Reconstructing Images in Electrical Impedance Tomography

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IEEE ACCESS 2024年 12卷 146560-146574页

作者： Alas, Rose Anne D. Velasco, Arrianne Crystal T. Univ Philippines Diliman Inst Math Quezon City 1101 Philippines

Electrical impedance tomography (EIT) is a medical imaging technique used to reconstruct images inside the domain of interest. EIT collects data on the boundary of the domain to infer the conductivity distribution inside the domain. The conductivity distribution will then be used to produce a tomographic image of the inside of the domain. This paper aims to recover geometric properties of a spherical perturbation in the conductivity inside a domain using sensitivity values of the electric potential on the boundary of the domain. The continuum model for EIT is first considered, as it holds more boundary information compared to other models of EIT. A change on the conductivity inside the domain is applied, and the impact on the electric potential is studied. The inverse EIT problem is then solved by formulating relations between the sensitivity values on the boundary and the geometric properties of the spherical perturbation: the radius and the projection onto the boundary and depth of its center. A reconstruction method using these relations is proposed and the method is examined by performing numerical simulations on different domains to model the head and the thorax. Lastly, the proposed method is applied to the complete electrode model of the EIT problem to analyze the performance of the method when the boundary data is limited on the electrodes.

关键词： Electrical impedance tomography gateaux derivative image reconstruction numerical algorithm sensitivity gateaux derivative image reconstruction numerical algorithm sensitivity

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A numerical algorithm and a Variational Iteration Technique for Solving Higher Order Fuzzy Integro-differential Equations

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FUNDAMENTA INFORMATICAE 2014年第4期133卷 421-431页

作者： Narayanamoorthy, S. Murugan, K. Bharathiar Univ Dept Appl Math Coimbatore 641046 Tamil Nadu India

In this paper, we present an algorithmic form of the variational iteration method (VIM) to handle both linear and nonlinear higher order fuzzy integro-differential equations. Using parametric form of fuzzy numbers to convert higher order fuzzy integro-differential equation to a system of higher order integro-differential equations in crisp case. By using the variational integration method we find the approximate solution of this system and consequently we obtain an approximation for fuzzy solution of the higher order fuzzy integro-differential equations. The numerical results are examined.

关键词： numerical algorithm fuzzy number fuzzy integro-differential equations variational iteration method convergence

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Unconditionally stable algorithm with unique solvability for image inpainting using a penalized Allen-Cahn equation

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COMMUNICATIONS IN NONLINEAR SCIENCE AND numerical SIMULATION 2025年 142卷

作者： Su, Sheng Yang, Junxiang Macau Univ Sci & Technol Fac Innovat Engn Sch Comp Sci & Engn Cotai Macau Peoples R China

Image inpainting is a technique that utilizes information from surrounding areas to restore damaged or missing parts. To achieve binary image inpainting with mathematical tools and numerical techniques, an effective mathematical model and an efficient, stable numerical solver are essential. This work aims to propose a practical and unconditionally stable numerical algorithm for image inpainting. A penalized Allen-Cahn equation is derived from a free energy using a variational approach. The proposed mathematical model achieves inpainting by eliminating the damaged region with the constraint of surrounding image values. The operator splitting strategy is used to split the original model into two subproblems. The first one is the classical Allen-Cahn equation, and the second one is a penalization equation. For the Allen-Cahn equation, a linear and strong stability-preserving factorization scheme is adopted to calculate the intermediate solution. Then, the final solution is explicitly updated from a simple correction step. The governing equation is discretized in space using the finite difference method. We analytically prove that the proposed algorithm is unconditionally stable and uniquely solvable. In the numerical simulations, we first verify the efficiency and stability via several simple benchmarks. The capability of binary image inpainting is validated by comparing the present and previous results. By slightly adjusting the governing equation, the proposed method can work well in achieving image inpainting of various gray-valued images. Finally, the proposed method is extended into three-dimensional space to show its effectiveness in restoring damaged 3D objects. The main scientific contributions are: (i) an efficient and practical numerical method is developed for image inpainting;(ii) the unconditional stability and unique solvability have been analytically estimated;(iii) extensive numerical experiments are implemented to validate the stability and capability

关键词： Image inpainting numerical algorithm Penalized Allen-Cahn equation Unconditional stability

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Fast convergence of the primal-dual dynamical system and corresponding algorithms for a nonsmooth bilinearly coupled saddle point problem

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2025年第1期90卷 151-192页

作者： Ding, Ke-wei Fliege, Jorg Vuong, Phan Tu Southwest Minzu Univ Sch Math Chengdu 610041 Sichuan Peoples R China Univ Southampton Math Sci Southampton SO17 1BJ England

This paper studies the convergence rate of a second-order dynamical system associated with a nonsmooth bilinearly coupled convex-concave saddle point problem, as well as the convergence rate of its corresponding discretizations. We derive the convergence rate of the primal-dual gap for the second-order dynamical system with asymptotically vanishing damping term. Based on an implicit discretization scheme, we propose a primal-dual algorithm and provide a non-ergodic convergence rate under a general setting for the inertial parameters when one objective function is continuously differentiable and convex and the other is a proper, convex and lower semicontinuous function. For this algorithm we derive a O1/k2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O\left( 1/k<^>2 \right) $$\end{document} convergence rate under three classical rules proposed by Nesterov, Chambolle-Dossal and Attouch-Cabot without assuming strong convexity, which is compatible with the results of the continuous-time dynamic system. For the case when both objective functions are continuously differentiable and convex, we further present a primal-dual algorithm based on an explicit discretization. We provide a corresponding non-ergodic convergence rate for this algorithm and show that the sequence of iterates generated weakly converges to a primal-dual optimal solution. Finally, we present numerical experiments that indicate the superior numerical performance of both algorithms.

关键词： Saddle point problem Primal-dual dynamical system Convergence rate numerical algorithm Nesterov's accelerated gradient method Iterates convergence

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Surface treatment of metal by combined particle beam

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INTERNATIONAL JOURNAL OF ENGINEERING SCIENCE 2024年 205卷

作者： Parfenova, Elena S. Knyazeva, Anna G. Russian Acad Sci Siberian Branch Inst Strength Phys & Mat Sci 2-4 Pr Akad Skii Tomsk 634055 Russia

This work is related to modeling of metal surface modification process by combined particles beam. On the basis of thermodynamics of irreversible processes, including equations of state in differential form, a nonlinear model is formulated. The model takes into account the interaction of thermal, diffusion and mechanical waves and finiteness of relaxation times of thermal and diffusion processes. For the combined particle flow such model is proposed for the first time. The numerical algorithm is based on implicit difference schemes. The study of the interaction of waves of different nature is carried out on the example of a copper target treated with nickel and gold particles. It is shown that deformations take the maximal value at the left boundary, which is directly related to the presence of impurity concentration gradients. Depending on the pulse duration, the difference between the extrema on the elastic wave becomes less significant. With increasing temperature, obviously, the diffusion process accelerates. The propagation velocities of the interacting waves are different. The character of distributions of concentrations of introduced particles directly depends on the value of parameters proportional to relaxation times.

关键词： Diffusion Stress wave Heat/mass flux relaxation time Coupled model numerical algorithm Thermoelastic diffusion theory

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Development of Mathematical Modeling Methods and Solution of Present-Day Problems in Geomechanics at the Institute of Mining SB RAS

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JOURNAL OF MINING SCIENCE 2024年第4期60卷 533-555页

作者： Lavrikov, S. V. Russian Acad Sci Chinakal Inst Min Siberian Branch Novosibirsk 630091 Russia

This article commemorates a double jubilee date on February 8, 2024-the 300th anniversary of the Russian Academy of Sciences and the 80th anniversary of the Institute of Mining SB RAS. The author reviews the research and findings of the Institute's scientists over the last 10-15 years in the area of mathematical modeling and numerical solution of present-day problems in geomechanics.

关键词： rock underground opening mathematical model numerical algorithm boundary value problem software system stress pattern

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