Journal articles: 'Numerical and computational mathematics'

1

Planitz, Max, and T. R. F. Nonweiler. "Computational Mathematics: An Introduction to Numerical Approximation." Mathematical Gazette 69, no. 447 (March 1985): 67. http://dx.doi.org/10.2307/3616478.

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Wimp, Jet, and T. R. F. Nonweiler. "Computational Mathematics, An Introduction to Numerical Approximation." Mathematics of Computation 46, no. 174 (April 1986): 761. http://dx.doi.org/10.2307/2008016.

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Temirgaliyev, N., and A. Zhubanysheva. "Approximation Theory, Computational Mathematics and Numerical Analysis in new conception of Computational (Numerical) Diameter." BULLETIN of the L.N. Gumilyov Eurasian National University. MATHEMATICS.COMPUTER SCIENCE. MECHANICS Series 124, no. 3 (2018): 8–88. http://dx.doi.org/10.32523/2616-7182/2018-124-3-8-88.

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G., W., Gunther Hammerlin, Karl-Heinz Hoffmann, and Larry Schumaker. "Numerical Mathematics." Mathematics of Computation 58, no. 198 (April 1992): 855. http://dx.doi.org/10.2307/2153223.

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Cook, Gregory B., and Saul A. Teukolsky. "Numerical relativity: challenges for computational science." Acta Numerica 8 (January 1999): 1–45. http://dx.doi.org/10.1017/s0962492900002889.

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We describe the burgeoning field of numerical relativity, which aims to solve Einstein's equations of general relativity numerically. The field presents many questions that may interest numerical analysts, especially problems related to nonlinear partial differential equations: elliptic systems, hyperbolic systems, and mixed systems. There are many novel features, such as dealing with boundaries when black holes are excised from the computational domain, or how even to pose the problem computationally when the coordinates must be determined during the evolution from initial data. The most important unsolved problem is that there is no known general 3-dimensional algorithm that can evolve Einstein's equations with black holes that is stable. This review is meant to be an introduction that will enable numerical analysts and other computational scientists to enter the field. No previous knowledge of special or general relativity is assumed.

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Vabishchevich, P. N. "Works of A.A. Samarskii on Computational Mathematics." Computational Methods in Applied Mathematics 9, no. 1 (2009): 5–36. http://dx.doi.org/10.2478/cmam-2009-0002.

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Abstract This is a review of the main results in computational mathematics that were obtained by the eminent Russian mathematician Alexander Andreevich Samarskii (February 19, 1919 – February 11, 2008). His outstanding research output addresses all the main questions that arise in the construction and justification of algorithms for the numerical solution of problems from mathematical physics. The remarkable works of A.A. Samarskii include statements of the main principles re- quired in the construction of difference schemes, rigorous mathematical proofs of the stability and convergence of these schemes, and also investigations of their algorith- mic implementation. A.A. Samarskii and his collaborators constructed and applied in practical calculations a large number of algorithms for solving various problems from mathematical physics, including thermal physics, gas dynamics, magnetic gas dynam- ics, plasma physics, ecology and other important models from the natural sciences.

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Kissane, Barry. "The Scientific Calculator and School Mathematics." Southeast Asian Mathematics Education Journal 6, no. 1 (December 27, 2016): 29–48. http://dx.doi.org/10.46517/seamej.v6i1.38.

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Scientific calculators are sometimes regarded as important only for obtaining numerical answers to computational questions, and thus in some countries regarded as inappropriate for school mathematics, lest they might undermine the school curriculum. This paper argues a contrary view that, firstly, numerical computation is not the principal purpose of scientific calculators in education, and secondly that calculators can play a valuable role in supporting students’ learning. Recent developments of calculators are outlined, noting that theirprincipal intention has been to make calculators easier to use, align their functionality with the school mathematics curriculum and represent mathematical expressions in conventional ways. A model for the educational use of calculators is described, with four key components:representation, computation, exploration and affirmation. Examples of how these might impact positively on school mathematics are presented, and suggestions are made regarding good pedagogy and curriculum with calculators in mind. The paper concludes that scientific calculators represent the best available technology to provide widespread access to some ICT in the mathematics curriculum for all students in the SEAMEO region.

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Braack, Malte, Dietmar Gallistl, Jun Hu, Guido Kanschat, and Xuejun Xu. "Sino–German Computational and Applied Mathematics." Computational Methods in Applied Mathematics 21, no. 3 (June 8, 2021): 497–99. http://dx.doi.org/10.1515/cmam-2021-0102.

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Abstract This short article serves as an epilog of the thirteen preceding papers in this special issue of CMAM. All contributions are authored by participants of the 7th Sino–German Workshop on Computational and Applied Mathematics at the Kiel University. The topics cover fourth-order problems, solvers and multilevel methods, a posteriori error control and adaptivity, and data science.

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Hoppe, Ronald H. W., Jun Hu, Malte A. Peter, Rolf Rannacher, Zhongci Shi, and Xuejun Xu. "Chinese–German Computational and Applied Mathematics." Computational Methods in Applied Mathematics 16, no. 4 (October 1, 2016): 605–8. http://dx.doi.org/10.1515/cmam-2016-0028.

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AbstractThis short article is the epilog of the 14 preceding papers in this and the previous issue of CMAM. All are extracted from the 5th Chinese–German Workshop on Computational and Applied Mathematics at Augsburg but submitted as individual papers to the journal.

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Smolensky, Paul. "Symbolic functions from neural computation." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 370, no. 1971 (July 28, 2012): 3543–69. http://dx.doi.org/10.1098/rsta.2011.0334.

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Is thought computation over ideas? Turing, and many cognitive scientists since, have assumed so, and formulated computational systems in which meaningful concepts are encoded by symbols which are the objects of computation. Cognition has been carved into parts, each a function defined over such symbols. This paper reports on a research program aimed at computing these symbolic functions without computing over the symbols. Symbols are encoded as patterns of numerical activation over multiple abstract neurons, each neuron simultaneously contributing to the encoding of multiple symbols. Computation is carried out over the numerical activation values of such neurons, which individually have no conceptual meaning. This is massively parallel numerical computation operating within a continuous computational medium. The paper presents an axiomatic framework for such a computational account of cognition, including a number of formal results. Within the framework, a class of recursive symbolic functions can be computed. Formal languages defined by symbolic rewrite rules can also be specified, the subsymbolic computations producing symbolic outputs that simultaneously display central properties of both facets of human language: universal symbolic grammatical competence and statistical, imperfect performance.

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Piqueras, M. A., R. Company, and L. Jódar. "Stable Numerical Solutions Preserving Qualitative Properties of Nonlocal Biological Dynamic Problems." Abstract and Applied Analysis 2019 (July 1, 2019): 1–7. http://dx.doi.org/10.1155/2019/5787329.

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This paper deals with solving numerically partial integrodifferential equations appearing in biological dynamics models when nonlocal interaction phenomenon is considered. An explicit finite difference scheme is proposed to get a numerical solution preserving qualitative properties of the solution. Gauss quadrature rules are used for the computation of the integral part of the equation taking advantage of its accuracy and low computational cost. Numerical analysis including consistency, stability, and positivity is included as well as numerical examples illustrating the efficiency of the proposed method.

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I., E., Heinz Rutishauser, and Walter Gautschi. "Lectures on Numerical Mathematics." Mathematics of Computation 57, no. 196 (October 1991): 869. http://dx.doi.org/10.2307/2938724.

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YÜKÇÜ, Sılay Aytaç. "The Numerical Evaluation Methods for Beta Function." Süleyman Demirel Üniversitesi Fen Edebiyat Fakültesi Fen Dergisi 17, no. 2 (November 25, 2022): 288–302. http://dx.doi.org/10.29233/sdufeffd.1128768.

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In this study, the beta function that is encountered in computational mathematics and physics is analyzed. The correct evaluation of this function also affects the accuracy of other mathematical functions in quantum mechanical calculations. Especially in recent years, there is an interest in studies related to the beta function for zero and negative p and q integers. This study, considering the neutrix limits of the beta function, presents new relations for the numerical computation of the beta function, especially for negative integers p and q. In addition, taking into account the definition of the beta function for positive p and q integer values, an algorithm is created to calculate the function for all integer values. Finally, numerical results obtained with the help of our new recurrence relations and algorithm are presented.

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Loja, Maria Amélia R., and Joaquim I. Barbosa. "Preface to Numerical and Symbolic Computation: Developments and Applications—2019." Mathematical and Computational Applications 25, no. 2 (May 11, 2020): 28. http://dx.doi.org/10.3390/mca25020028.

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This book constitutes the printed edition of the Special Issue Numerical and Symbolic Computation: Developments and Applications—2019, published by Mathematical and Computational Applications (MCA) and comprises a collection of articles related to works presented at the 4th International Conference in Numerical and Symbolic Computation—SYMCOMP 2019—that took place in Porto, Portugal, from April 11th to April 12th 2019 [...]

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Petrovskii, Sergei, and Natalia Petrovskaya. "Computational ecology as an emerging science." Interface Focus 2, no. 2 (January 5, 2012): 241–54. http://dx.doi.org/10.1098/rsfs.2011.0083.

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It has long been recognized that numerical modelling and computer simulations can be used as a powerful research tool to understand, and sometimes to predict, the tendencies and peculiarities in the dynamics of populations and ecosystems. It has been, however, much less appreciated that the context of modelling and simulations in ecology is essentially different from those that normally exist in other natural sciences. In our paper, we review the computational challenges arising in modern ecology in the spirit of computational mathematics, i.e. with our main focus on the choice and use of adequate numerical methods. Somewhat paradoxically, the complexity of ecological problems does not always require the use of complex computational methods. This paradox, however, can be easily resolved if we recall that application of sophisticated computational methods usually requires clear and unambiguous mathematical problem statement as well as clearly defined benchmark information for model validation. At the same time, many ecological problems still do not have mathematically accurate and unambiguous description, and available field data are often very noisy, and hence it can be hard to understand how the results of computations should be interpreted from the ecological viewpoint. In this scientific context, computational ecology has to deal with a new paradigm: conventional issues of numerical modelling such as convergence and stability become less important than the qualitative analysis that can be provided with the help of computational techniques. We discuss this paradigm by considering computational challenges arising in several specific ecological applications.

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Bianca, Carlo. "Mathematical and computational modeling of biological systems: advances and perspectives." AIMS Biophysics 8, no. 4 (2021): 318–21. http://dx.doi.org/10.3934/biophy.2021025.

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<abstract> <p>The recent developments in the fields of mathematics and computer sciences have allowed a more accurate description of the dynamics of some biological systems. On the one hand new mathematical frameworks have been proposed and employed in order to gain a complete description of a biological system thus requiring the definition of complicated mathematical structures; on the other hand computational models have been proposed in order to give both a numerical solution of a mathematical model and to derive computation models based on cellular automata and agents. Experimental methods are developed and employed for a quantitative validation of the modeling approaches. This editorial article introduces the topic of this special issue which is devoted to the recent advances and future perspectives of the mathematical and computational frameworks proposed in biosciences.</p> </abstract>

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Eslami, M., and A. Neyrame. "Numerical methods." Computational Mathematics and Modeling 22, no. 1 (January 2011): 92–97. http://dx.doi.org/10.1007/s10598-011-9091-0.

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Sullivan, Jillian C. F. "Microcomputer-Assisted Mathematics: Polynomial Equations Revisited." Mathematics Teacher 79, no. 9 (December 1986): 732–37. http://dx.doi.org/10.5951/mt.79.9.0732.

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Although solving polynomial equations is important in mathematics, most high school students can solve only linear and quadratic equations. This is because the methods for solving cubic and quartic equations are difficult, and no general methods of solution are available for equations of degree higher than four. However, numerical methods can be used to approximate the real solutions of polynomial equations of any degree. Because they involve a great deal of computation they have not traditionally been taught in the schools. Now that most students have access to calculators and computers, this computational difficulty is easily overcome.

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Petrauskas, K., and R. Baronas. "Computational Modelling of Biosensors with an Outer Perforated Membrane." Nonlinear Analysis: Modelling and Control 14, no. 1 (January 20, 2009): 85–102. http://dx.doi.org/10.15388/na.2009.14.1.14532.

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This paper presents one-dimensional (1-D) and two-dimensional (2-D) in-space mathematical models for amperometric biosensors with an outer perforated membrane. The biosensor action was modelled by reaction-diffusion equations with a nonlinear term representing the Michaelis-Menten kinetics of an enzymatic reaction. The conditions at which the 1-D model can be applied to simulate the biosensor response accurately were investigated numerically. The accuracy of the biosensor response simulated by using 1-D model was evaluated by the response simulated with the corresponding 2-D model. A procedure for a numerical evaluation of the effective diffusion coefficient to be used in 1-D model was proposed. The numerically calculated effective diffusion coefficient was compared with the corresponding coefficients derived analytically. The numerical simulation was carried out using the finite difference technique.

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Seidel, Edward, and Wai-Mo Suen. "Numerical relativity as a tool for computational astrophysics." Journal of Computational and Applied Mathematics 109, no. 1-2 (September 1999): 493–525. http://dx.doi.org/10.1016/s0377-0427(99)00169-7.

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Lemeshevsky, Sergey, Almas Sherbaf, and Petr Vabishchevich. "The Contribution of Piotr Matus to Computational Mathematics." Computational Methods in Applied Mathematics 13, no. 4 (October 1, 2013): 363–67. http://dx.doi.org/10.1515/cmam-2013-0019.

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Ibrahim, Adel K., and Medhat A. Rakha. "Numerical computations of infinite products." Applied Mathematics and Computation 161, no. 1 (February 2005): 271–83. http://dx.doi.org/10.1016/j.amc.2003.12.027.

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Rowland, John H. "Computational Mathematics—An Introduction to Numerical Approximation (T. R. F. Nonweilerz)." SIAM Review 28, no. 3 (September 1986): 437–38. http://dx.doi.org/10.1137/1028145.

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Faragó, István, Krassimir Georgiev, Per Grove Thomsen, and Zahari Zlatev. "Numerical and computational issues related to applied mathematical modelling." Applied Mathematical Modelling 32, no. 8 (August 2008): 1475–76. http://dx.doi.org/10.1016/j.apm.2007.06.035.

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Matos, José A. O., and Paulo B. Vasconcelos. "Effectiveness of Floating-Point Precision on the Numerical Approximation by Spectral Methods." Mathematical and Computational Applications 26, no. 2 (May 26, 2021): 42. http://dx.doi.org/10.3390/mca26020042.

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With the fast advances in computational sciences, there is a need for more accurate computations, especially in large-scale solutions of differential problems and long-term simulations. Amid the many numerical approaches to solving differential problems, including both local and global methods, spectral methods can offer greater accuracy. The downside is that spectral methods often require high-order polynomial approximations, which brings numerical instability issues to the problem resolution. In particular, large condition numbers associated with the large operational matrices, prevent stable algorithms from working within machine precision. Software-based solutions that implement arbitrary precision arithmetic are available and should be explored to obtain higher accuracy when needed, even with the higher computing time cost associated. In this work, experimental results on the computation of approximate solutions of differential problems via spectral methods are detailed with recourse to quadruple precision arithmetic. Variable precision arithmetic was used in Tau Toolbox, a mathematical software package to solve integro-differential problems via the spectral Tau method.

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Le Bris, Claude. "Computational chemistry from the perspective of numerical analysis." Acta Numerica 14 (April 19, 2005): 363–444. http://dx.doi.org/10.1017/s096249290400025x.

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We present the field of computational chemistry from the standpoint of numerical analysis. We introduce the most commonly used models and comment on their applicability. We briefly outline the results of mathematical analysis and then mostly concentrate on the main issues raised by numerical simulations. A special emphasis is laid on recent results in numerical analysis, recent developments of new methods and challenging open issues.

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Smirnov, N. N., V. F. Nikitin, L. I. Stamov, V. A. Nerchenko, and V. V. Tyrenkova. "Numerical Simulations of Gaseous Detonation Propagation Using Different Supercomputing Architechtures." International Journal of Computational Methods 14, no. 04 (April 18, 2017): 1750038. http://dx.doi.org/10.1142/s0219876217500384.

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The aim of the present study is to calculate the process of detonation combustion of gas mixtures in engines. Development and verification of 3D transient mathematical model of chemically reacting gas mixture flows incorporating hydrogen was performed. Development of a computational model based on the mathematical one for parallel computing on supercomputers incorporating CPU and GPU units was carried out. Investigation of the influence of computational grid size on simulation precision and computational speed was performed. Investigation of calculation runtime acceleration was carried out subject to variable number of parallel threads on different architectures and implying different strategies of parallel computation.

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KALTENBACHER, MANFRED. "COMPUTATIONAL ACOUSTICS IN MULTI-FIELD PROBLEMS." Journal of Computational Acoustics 19, no. 01 (March 2011): 27–62. http://dx.doi.org/10.1142/s0218396x11004286.

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We present physical/mathematical models base on partial differential equations (PDEs) and efficient numerical simulation schemes based on the Finite Element (FE) method for multi-field problems, where the acoustic field is the field of main interest. Acoustics, the theory of sound, is an emerging scientific field including disciplines from physics over engineering to medical science. We concentrate on the following three topics: vibro-acoustics, aero-acoustics and high intensity focused ultrasound. For each topic, we discuss the physical/mathematical modeling, efficient numerical schemes and provide practical applications.

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Bragin, M. D., O. A. Kovyrkina, M. E. Ladonkina, V. V. Ostapenko, V. F. Tishkin, and N. A. Khandeeva. "Combined Numerical Schemes." Computational Mathematics and Mathematical Physics 62, no. 11 (November 2022): 1743–81. http://dx.doi.org/10.1134/s0965542522100025.

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SUZUKI, M. "MATHEMATICAL BASIS OF COMPUTATIONAL PHYSICS." International Journal of Modern Physics C 07, no. 03 (June 1996): 355–59. http://dx.doi.org/10.1142/s0129183196000296.

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The present paper explains some general basic formulas concerning quantum Monte Carlo simulations, symplectic integration and other numerical calculations. A generalization of the BCH formula is given with an application to the decomposition of exponential operators in the presence of small parameters.

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Bultheel, A. "Numerical methods." Journal of Computational and Applied Mathematics 24, no. 3 (December 1988): N2. http://dx.doi.org/10.1016/0377-0427(88)90305-6.

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Higdon, Robert L. "Numerical modelling of ocean circulation." Acta Numerica 15 (May 2006): 385–470. http://dx.doi.org/10.1017/s0962492906250013.

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Computational simulations of ocean circulation rely on the numerical solution of partial differential equations of fluid dynamics, as applied to a relatively thin layer of stratified fluid on a rotating globe. This paper describes some of the physical and mathematical properties of the solutions being sought, some of the issues that are encountered when the governing equations are solved numerically, and some of the numerical methods that are being used in this area.

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Guo, Yuling, and Jianguo Huang. "A Domain Decomposition Based Spectral Collocation Method for Lane-Emden Equations." Communications in Computational Physics 22, no. 2 (June 21, 2017): 542–71. http://dx.doi.org/10.4208/cicp.oa-2016-0181.

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AbstractA domain decomposition based spectral collocation method is proposed for numerically solving Lane-Emden equations, which are frequently encountered in mathematical physics and astrophysics. Compared with the existing methods, this method requires less computational cost and is particularly suitable for long-term computation. The related error estimates are also established, indicating the spectral accuracy of the method. The numerical performance and efficiency of the method are illustrated by several numerical experiments.

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Ruhe, Axel, M. G. Cox, and S. Hammarling. "Reliable Numerical Computation." Mathematics of Computation 59, no. 199 (July 1992): 298. http://dx.doi.org/10.2307/2152999.

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Hauenstein, Jonathan D., Jose Israel Rodriguez, and Frank Sottile. "Numerical Computation of Galois Groups." Foundations of Computational Mathematics 18, no. 4 (June 14, 2017): 867–90. http://dx.doi.org/10.1007/s10208-017-9356-x.

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Evangelista-Alvarado, Miguel Ángel, José Crispín Ruíz-Pantaleón, and Pablo Suárez-Serrato. "On computational Poisson geometry II: Numerical methods." Journal of Computational Dynamics 8, no. 3 (2021): 273. http://dx.doi.org/10.3934/jcd.2021012.

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<p style='text-indent:20px;'>We present twelve numerical methods for evaluation of objects and concepts from Poisson geometry. We describe how each method works with examples, and explain how it is executed in code. These include methods that evaluate Hamiltonian and modular vector fields, compute the image under the coboundary and trace operators, the Lie bracket of differential 1–forms, gauge transformations, and normal forms of Lie–Poisson structures on <inline-formula><tex-math id="M1">\begin{document}$ {\mathbf{R}^{{3}}} $\end{document}</tex-math></inline-formula>. The complexity of each of our methods is calculated, and we include experimental verifications on examples in dimensions two and three.</p>

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Kahaner, David K., and Carl-Erik Froberg. "Numerical Mathematics--Theory and Computer Applications." Mathematics of Computation 48, no. 178 (April 1987): 829. http://dx.doi.org/10.2307/2007845.

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I., E., David Kincaid, and Ward Cheney. "Numerical Analysis--Mathematics of Scientific Computing." Mathematics of Computation 59, no. 199 (July 1992): 297. http://dx.doi.org/10.2307/2152998.

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Yuan, Yirang, Qing Yang, Changfeng Li, and Tongjun Sun. "A Numerical Approximation Structured by Mixed Finite Element and Upwind Fractional Step Difference for Semiconductor Device with Heat Conduction and Its Numerical Analysis." Numerical Mathematics: Theory, Methods and Applications 10, no. 3 (June 20, 2017): 541–61. http://dx.doi.org/10.4208/nmtma.2017.y15013.

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AbstractA coupled mathematical system of four quasi-linear partial differential equations and the initial-boundary value conditions is presented to interpret transient behavior of three dimensional semiconductor device with heat conduction. The electric potential is defined by an elliptic equation, the electron and hole concentrations are determined by convection-dominated diffusion equations and the temperature is interpreted by a heat conduction equation. A mixed finite element approximation is used to get the electric field potential and one order of computational accuracy is improved. Two concentration equations and the heat conduction equation are solved by a fractional step scheme modified by a second-order upwind difference method, which can overcome numerical oscillation, dispersion and computational complexity. This changes the computation of a three dimensional problem into three successive computations of one-dimensional problem where the method of speedup is used and the computational work is greatly shortened. An optimal second-order error estimate in L2 norm is derived by prior estimate theory and other special techniques of partial differential equations. This type of parallel method is important in numerical analysis and is most valuable in numerical application of semiconductor device and it can successfully solve this international famous problem.

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Khoromskaia, V. "Computation of the Hartree-Fock Exchange by the Tensor-Structured Methods." Computational Methods in Applied Mathematics 10, no. 2 (2010): 204–18. http://dx.doi.org/10.2478/cmam-2010-0012.

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AbstractWe propose a novel numerical method for fast and accurate evaluation of the exchange part of the Fock operator in the Hartree-Fock equation which is a (nonlocal) integral operator. Usually, this challenging computational problem is solved by analytical evaluation of two-electron integrals using the “analytically separable” Galerkin basis functions, like Gaussians. Instead, we employ the agglomerated “grey-box” numerical computation of the corresponding six-dimensional integrals in the tensor-structured format which does not require analytical separability of the basis set. The point of our method is a low-rank tensor representation of arising functions and operators on an n×n×n Cartesian grid and the implementation of the corresponding multi-linear algebraic operations in the tensor product format. Linear scaling of the tensor operations, including the 3D convolution product, with respect to the one-dimension grid size n enables computations on huge 3D Cartesian grids thus providing the required high accuracy. The presented algorithm for evaluation of the exchange operator and a recent tensor method for the computation of the Coulomb matrix are the main building blocks in the numerical solution of the Hartree-Fock equation by the tensor-structured methods. These methods provide a new tool for algebraic optimization of the Galerkin basis in the case of large molecules.

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Sabin, Malcolm. "Numerical geometry of surfaces." Acta Numerica 3 (January 1994): 411–66. http://dx.doi.org/10.1017/s0962492900002476.

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The mathematical techniques used within Computer Aided Design software for the representation and calculation of surfaces of objects are described. First the main techniques for dealing with surfaces as computational objects are described, and then the methods for enquiring of such surfaces the properties required for their assessment and manufacture.

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Lavrischeva, Ekaterina Mikhailovna, and Igor Borisovich Petrov. "Modeling Technical and Mathematical Tasks of Applied Knowledge Areas on Computers." Proceedings of the Institute for System Programming of the RAS 32, no. 6 (2020): 167–82. http://dx.doi.org/10.15514/ispras-2020-32(6)-13.

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The paper considers modeling of technical problems and problems of applied mathematics, their algorithms and programming. The characteristics of the numerical modeling of technical problems and applied mathematics are given: physical and technical experiments, energy, ballistic and seismic methods of I.V. Kurchatov, starting with mathematical methods of the 17-20th centuries, the first computers and computers. The analysis of the first technical problems and problems of applied mathematics, their modeling, algorithmization and programming using the A.A. Lyapunov graph-schematic language, address language and programming languages is given. Numerical methods are presented, implemented under the guidance of A.A. Dorodnitsyn, A.A. Samarsky, O.M. Belotserkovsky and other scientists on modern supercomputers. Examples of mathematical modeling of the biological problem of eye treatment and the subject of «Computational geometry» on the Internet are given.

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Shcherban, V. "Arithmetic Table as an Integral Part of all Computational Mathematics." Bulletin of Science and Practice 6, no. 6 (June 15, 2020): 31–41. http://dx.doi.org/10.33619/2414-2948/55/04.

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The paper is devoted to studying the following issue as a statement. What do we know and what we don’t know about arithmetic tables. Perhaps there is no mathematical problem as naive or simple as finding a method for creating arithmetic tables. We confirm that the general method has not been found yet. This study provides nonterminal solution to this problem. Why? The presentation of arithmetic material in essence, plus some accompanying ideas, makes it possible to develop them further in the system. Materials and methods. The system looks like this: a numerical table as a Pascal's triangle and a symmetric polynomial in two or three variables. Some arithmetic properties of such tables will be found, studied and proved. All this was made possible only after successful decryption of the entire class of numeric tables of truncated triangles in the cryptographic system. Results. For example, the arithmetic properties of truncated Pascal’s triangle for finding all prime numbers have been found and presented, and then their formulas have been placed. In addition to elementary addition and subtraction tables, unlimited “comparison” tables of numbers are given and presented for the first time. Conclusions. For computer implementation of the objectives set, the rules of real actions that should exist for tables have been laid down. Only recurrent numeric series should be used for this purpose.

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Liu, Xue Ting. "The Judgement for Generalized Positive Definite Matrices in Signal Processing." Advanced Materials Research 121-122 (June 2010): 128–32. http://dx.doi.org/10.4028/www.scientific.net/amr.121-122.128.

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The generalized positive definite matrix is an active research field of special matrix, they have applied in computational mathematics, economics, physics, biology, applied mathematics, numerical computation, signal processing, coding theory, oil investigation in recent years, and so on. In this paper, motivated by [3], we give a simple and convenient judging methodwhich can be used to judge whether an nonnegative real matrix A is an generalized positive definite matrix or not.

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Bini, Dario, Khalide Jbilou, Marilena Mitrouli, and Lothar Reichel. "Numerical Analysis and Scientific Computation (NASCA18)." Journal of Computational and Applied Mathematics 373 (August 2020): 112612. http://dx.doi.org/10.1016/j.cam.2019.112612.

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KHANI, A., M. M. MOGHADAM, and S. SHAHMORAD. "Approximate Solution Of The System Of Non-Linear Volterra Integro-Differential Equations." Computational Methods in Applied Mathematics 8, no. 1 (2008): 77–85. http://dx.doi.org/10.2478/cmam-2008-0005.

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Abstract In this paper we develop a new method to find a numerical solution for the system of non-linear Volterra integro-differential equations (SNVE). To this end, we present our method based on the matrix form of SNVE. The corresponding unknown coefficients of our method have been determined by using the computational aspects of matrices. Finally the accuracy of the method has been verified by presenting some numerical computations.

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Abdulle, A., and Y. Bai. "Reduced-order modelling numerical homogenization." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 372, no. 2021 (August 6, 2014): 20130388. http://dx.doi.org/10.1098/rsta.2013.0388.

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A general framework to combine numerical homogenization and reduced-order modelling techniques for partial differential equations (PDEs) with multiple scales is described. Numerical homogenization methods are usually efficient to approximate the effective solution of PDEs with multiple scales. However, classical numerical homogenization techniques require the numerical solution of a large number of so-called microproblems to approximate the effective data at selected grid points of the computational domain. Such computations become particularly expensive for high-dimensional, time-dependent or nonlinear problems. In this paper, we explain how numerical homogenization method can benefit from reduced-order modelling techniques that allow one to identify offline and online computational procedures. The effective data are only computed accurately at a carefully selected number of grid points (offline stage) appropriately ‘interpolated’ in the online stage resulting in an online cost comparable to that of a single-scale solver. The methodology is presented for a class of PDEs with multiple scales, including elliptic, parabolic, wave and nonlinear problems. Numerical examples, including wave propagation in inhomogeneous media and solute transport in unsaturated porous media, illustrate the proposed method.

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Et.al, Ji-Sun Kang. "Computational Efficiency Examination of a Regional Numerical Weather Prediction Model using KISTI Supercomputer NURION." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 12, no. 6 (April 10, 2021): 743–49. http://dx.doi.org/10.17762/turcomat.v12i6.2088.

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For well-resolving extreme weather events, running numerical weather prediction model with high resolution in time and space is essential. We explore how efficiently such modeling could be, using NURION. We have examined one of community numerical weather prediction models, WRF, and KISTI’s 5th supercomputer NURION of national HPC. Scalability of the model has been tested at first, and we have compared the computational efficiency of hybrid openMP + MPI runs with pure MPI runs. In addition to those parallel computing experiments, we have tested a new storage layer called burst buffer to see whether it can accelerate frequent I/O. We found that there are significant differences between the computational environments for running WRF model. First of all, we have tested a sensitivity of computational efficiency to the number of cores per node. The sensitivity experiments certainly tell us that using all cores per node does not guarantee the best results, rather leaving several cores per node could give more stable and efficient computation. For the current experimental configuration of WRF, moreover, pure MPI runs gives much better computational performance than any hybrid openMP + MPI runs. Lastly, we have tested burst buffer storage layer that is expected to accelerate frequent I/O. However, our experiments show that its impact is not consistently positive. We clearly confirm the positive impact with relatively smaller problem size experiments while the impact was not seen with bigger problem experiments. Significant sensitivity to the different computational configurations shown this paper strongly suggests that HPC users should find out the best computing environment before massive use of their applications

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Riachy, Samer, Mamadou Mboup, and Jean-Pierre Richard. "Multivariate numerical differentiation." Journal of Computational and Applied Mathematics 236, no. 6 (October 2011): 1069–89. http://dx.doi.org/10.1016/j.cam.2011.07.031.

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Zotos, Kostas. "Improving numerical software." Applied Mathematics and Computation 192, no. 1 (September 2007): 247–51. http://dx.doi.org/10.1016/j.amc.2007.03.005.

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Journal articles on the topic 'Numerical and computational mathematics'

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