Relevant bibliographies by topics / Numerical integration. Integrals

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Numerical integration. Integrals'

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Published: 4 June 2021
Last updated: 10 February 2022

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Journal articles on the topic "Numerical integration. Integrals"

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Safouhi, Hassan. "A Generalized Technique in Numerical Integration." EPJ Web of Conferences 173 (2018): 01011. http://dx.doi.org/10.1051/epjconf/201817301011.

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Integration by parts is one of the most popular techniques in the analysis of integrals and is one of the simplest methods to generate asymptotic expansions of integral representations. The product of the technique is usually a divergent series formed from evaluating boundary terms; however, sometimes the remaining integral is also evaluated. Due to the successive differentiation and anti-differentiation required to form the series or the remaining integral, the technique is difficult to apply to problems more complicated than the simplest. In this contribution, we explore a generalized and formalized integration by parts to create equivalent representations to some challenging integrals.As a demonstrative archetype, we examine Bessel integrals, Fresnel integrals and Airy functions.
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Zadiraka, V. K., L. V. Luts, and I. V. Shvidchenko. "Optimal Numerical Integration." Cybernetics and Computer Technologies, no. 4 (December 31, 2020): 47–64. http://dx.doi.org/10.34229/2707-451x.20.4.4.

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Introduction. In many applied problems, such as statistical data processing, digital filtering, computed tomography, pattern recognition, and many others, there is a need for numerical integration, moreover, with a given (often quite high) accuracy. Classical quadrature formulas cannot always provide the required accuracy, since, as a rule, they do not take into account the oscillation of the integrand. In this regard, the development of methods for constructing optimal in accuracy (and close to them) quadrature formulas for the integration of rapidly oscillating functions is rather important and topical problem of computational mathematics. The purpose of the article is to use the example of constructing optimal in accuracy (and close to them) quadrature formulas for calculating integrals for integrands of various degrees of smoothness and for oscillating factors of different types and constructing a priori estimates of their total error, as well as applying to them of the theory of testing the quality of algorithms-programs to create a theory of optimal numerical integration. Results. The optimal in accuracy (and close to them) quadrature formulas for calculating the Fourier transform, wavelet transforms, and Bessel transform were constructed both in the classical formulation of the problem and for interpolation classes of functions corresponding to the case when the information operator about the integrand is given by a fixed table of its values. The paper considers a passive pure minimax strategy for solving the problem. Within the framework of this strategy, we used the method of “caps” by N. S. Bakhvalov and the method of boundary functions developed at the V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine. Great attention is paid to the quality of the error estimates and the methods to obtain them. The article describes some aspects of the theory of algorithms-programs testing and presents the results of testing the constructed quadrature formulas for calculating integrals of rapidly oscillating functions and estimates of their characteristics. The problem of determining the ranges of admissible values of control parameters of programs for calculating integrals with the required accuracy, as well as their best values for integration with the minimum possible error, is considered for programs calculating a priori estimates of characteristics. Conclusions. The results obtained make it possible to create a theory of optimal integration, which makes it possible to reasonably choose and efficiently use computational resources to find the value of the integral with a given accuracy or with the minimum possible error. Keywords: quadrature formula, optimal algorithm, interpolation class, rapidly oscillating function, quality testing.
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Kurihara, Y., and T. Kaneko. "Numerical contour integration for loop integrals." Computer Physics Communications 174, no. 7 (April 2006): 530–39. http://dx.doi.org/10.1016/j.cpc.2005.05.009.

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Hu, Jin-Xiu, Hai-Feng Peng, and Xiao-Wei Gao. "Numerical Evaluation of Arbitrary Singular Domain Integrals Using Third-Degree B-Spline Basis Functions." Mathematical Problems in Engineering 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/284106.

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A new approach is presented for the numerical evaluation of arbitrary singular domain integrals. In this method, singular domain integrals are transformed into a boundary integral and a radial integral which contains singularities by using the radial integration method. The analytical elimination of singularities condensed in the radial integral formulas can be accomplished by expressing the nonsingular part of the integration kernels as a series of cubic B-spline basis functions of the distancerand using the intrinsic features of the radial integral. In the proposed method, singularities involved in the domain integrals are explicitly transformed to the boundary integrals, so no singularities exist at internal points. A few numerical examples are provided to verify the correctness and robustness of the presented method.
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Lv, Jun, Guoyu Sheng, Xiaowei Gao, and Hongwu Zhang. "Numerical Integration Approach Based on Radial Integration Method for General 3D Polyhedral Finite Elements." International Journal of Computational Methods 12, no. 05 (October 2015): 1550026. http://dx.doi.org/10.1142/s0219876215500267.

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We construct an efficient quadrature method for the integration of the Galerkin weak form over general 3D polyhedral elements based on the radial integration method (RIM). The basic idea of the proposed method is to convert the polyhedral domain integrals to contour plane integrals of the element by utilizing the RIM which can be used for accurate evaluation of various complicated domain integrals. The quadrature construction scheme for irregular polyhedral elements involves the treatment of the nonpolynomial shape functions as well as the arbitrary geometry shape of the elements. In this approach, the volume integrals for polyhedral elements with triangular or quadrilateral faces are evaluated by transforming them into face integrals using RIM. For those polyhedral elements with irregular polygons, RIM is again used to convert the face integrals into line integrals. As a result, the volume integration of Galerkin weak form over the polyhedral elements can be easily carried out by a number of line integrals along the edges of the polyhedron. Some benchmark numerical examples including the patch tests are utilized to demonstrate the accuracy and convenience of the proposed method.
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Lindh, Roland, Per-Åke Malmqvist, and Laura Gagliardi. "Molecular integrals by numerical quadrature. I. Radial integration." Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chimica Acta) 106, no. 3 (July 1, 2001): 178–87. http://dx.doi.org/10.1007/s002140100263.

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Stolle, H. W., and R. Strauss. "On the numerical integration of certain singular integrals." Computing 48, no. 2 (June 1992): 177–89. http://dx.doi.org/10.1007/bf02310532.

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Ubale, P. V. "Numerical Solution of Boole’s Rule in Numerical Integration by Using General Quadrature Formula." Bulletin of Society for Mathematical Services and Standards 2 (June 2012): 1–4. http://dx.doi.org/10.18052/www.scipress.com/bsmass.2.1.

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We have seen that definite integrals arise in many different areas and that the fundamental theorem of calculus is a powerful tool for evaluating definite integrals. This paper describes classical quadrature method for the numerical solution of Boole’s rule in numerical integration.
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Brzeziński, D. W., and P. Ostalczyk. "High-accuracy numerical integration methods for fractional order derivatives and integrals computations." Bulletin of the Polish Academy of Sciences Technical Sciences 62, no. 4 (December 1, 2014): 723–33. http://dx.doi.org/10.2478/bpasts-2014-0078.

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Abstract In this paper the authors present highly accurate and remarkably efficient computational methods for fractional order derivatives and integrals applying Riemann-Liouville and Caputo formulae: the Gauss-Jacobi Quadrature with adopted weight function, the Double Exponential Formula, applying two arbitrary precision and exact rounding mathematical libraries (GNU GMP and GNU MPFR). Example fractional order derivatives and integrals of some elementary functions are calculated. Resulting accuracy is compared with accuracy achieved by applying widely known methods of numerical integration. Finally, presented methods are applied to solve Abel’s Integral equation (in Appendix).
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Sharma, M. D., and S. Nain. "Numerical evaluation of inverse integral transforms: Dynamic response of elastic materials." International Journal of Engineering, Science and Technology 12, no. 2 (June 1, 2020): 29–34. http://dx.doi.org/10.4314/ijest.v12i2.4.

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This study discusses the use of numerical integration in evaluating the improper integrals appearing as inverse integral transforms of non-analytic functions. These transforms appear while studying the response of various sources in an elastic medium through integral transform method. In these studies, the inverse Fourier transforms are solved numerically without bothering about the singularities and branch points in the corresponding integrands. References on numerical integration cited in relevant papers do not support such an evaluation but suggest contrary. Approximation of inverse Laplace transform integral into a series is used without following the essential restrictions and assumptions. Volume of the published papers using these dubious procedures has reached to an alarming level. The discussion presented aims to draw the attention of researchers as well as journals so as to stop this menace at the earliest possible.Keywords: Inverse Fourier transforms, inverse Laplace transforms, Romberg integration, improper integral, elastic waves
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Dissertations / Theses on the topic "Numerical integration. Integrals"

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Thompson, Jeremy Stewart. "High speed numerical integration of Fermi Dirac integrals." Thesis, Monterey, Calif. : Springfield, Va. : Naval Postgraduate School ; Available from National Technical Information Service, 1996. http://handle.dtic.mil/100.2/ADA311805.

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Webster, Jonathan Robert. "Methods of numerical integration for rapidly oscillatory integrals." Thesis, Loughborough University, 1999. https://dspace.lboro.ac.uk/2134/13776.

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This thesis is concerned with the evaluation of rapidly oscillatory integrals, that is integrals in which the integrand has numerous local maxima and minima over the range of integration. Three numerical integration rules are presented. The first is suitable for computing rapidly oscillatory integrals with trigonometric oscillations of the form f(x) exp(irq(x)). The method is demonstrated, empirically, to be convergent and numerically stable as the order of the formula is increased. For other forms of oscillatory behaviour, a second approach based on Lagrange's identity is presented. The technique is suitable for any oscillatory weight function, provided that it satisfies an ordinary linear differential equation of order m :2:: 1. The method is shown to encompass Bessel oscillations, trigonometric oscillations and Fresnel oscillations, and products of these terms. Examples are included which illustrate the efficiency of the method in practical applications. Finally, integrals where the integrand is singular and oscillatory are considered. An extended Clenshaw-Curtis formula is developed for Fourier integrals which exhibit algebraic and logarithmic singularities. An efficient algorithm is presented for the practical implementation of the method.
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Sinescu, Vasile. "Construction of lattice rules for multiple integration based on a weighted discrepancy." The University of Waikato, 2008. http://hdl.handle.net/10289/2542.

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High-dimensional integrals arise in a variety of areas, including quantum physics, the physics and chemistry of molecules, statistical mechanics and more recently, in financial applications. In order to approximate multidimensional integrals, one may use Monte Carlo methods in which the quadrature points are generated randomly or quasi-Monte Carlo methods, in which points are generated deterministically. One particular class of quasi-Monte Carlo methods for multivariate integration is represented by lattice rules. Lattice rules constructed throughout this thesis allow good approximations to integrals of functions belonging to certain weighted function spaces. These function spaces were proposed as an explanation as to why integrals in many variables appear to be successfully approximated although the standard theory indicates that the number of quadrature points required for reasonable accuracy would be astronomical because of the large number of variables. The purpose of this thesis is to contribute to theoretical results regarding the construction of lattice rules for multiple integration. We consider both lattice rules for integrals over the unit cube and lattice rules suitable for integrals over Euclidean space. The research reported throughout the thesis is devoted to finding the generating vector required to produce lattice rules that have what is termed a low weighted discrepancy . In simple terms, the discrepancy is a measure of the uniformity of the distribution of the quadrature points or in other settings, a worst-case error. One of the assumptions used in these weighted function spaces is that variables are arranged in the decreasing order of their importance and the assignment of weights in this situation results in so-called product weights . In other applications it is rather the importance of group of variables that matters. This situation is modelled by using function spaces in which the weights are general . In the weighted settings mentioned above, the quality of the lattice rules is assessed by the weighted discrepancy mentioned earlier. Under appropriate conditions on the weights, the lattice rules constructed here produce a convergence rate of the error that ranges from O(n−1/2) to the (believed) optimal O(n−1+δ) for any δ gt 0, with the involved constant independent of the dimension.
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Meszmer, Peter. "Hierarchische Integration und der Strahlungstransport in streuenden Medien." Doctoral thesis, Universitätsbibliothek Leipzig, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-98584.

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Der Strahlungstransport stellt eine von drei Arten des Wärmetransports zwischen Gebieten unterschiedlicher Temperatur dar. Eine der einfachsten Formen bildet der Strahlungstransport im Vakuum, ein Vorgang, der im kosmischen Umfeld, beispielsweise bei der Energieübertragung von einem Stern auf seine Planeten, beobachtbar ist. Hierbei ist es hinreichend, sich auf die Betrachtung von Oberflächen zu beschränken. Strahlungstransport kann jedoch auch in semitransparenten Medien, wie biologischem Gewebe oder Glas, beobachtet werden. Das Medium, in dem der Strahlungstransport erfolgt, wirkt sich durch Vorgänge wie Absorption, Emission, Reflexion oder Streuung auf den Strahlungstransport aus. Für die Modellierung des Strahlungstransports in einem solchen Umfeld können verschiedene Modelle, darunter das Strahlenmodell, genutzt werden. Dieses Modell beschreibt den Wärmetransport anhand einer skalaren Größe, die Strahlungsintensität genannt wird. Betrachtet wird die Strahlungsintensität in diesem Modell entlang eines Strahls in eine vorgegebene Richtung. Die mathematische Darstellung des Strahlenmodells des Strahlungstransports in partizipierenden Medien führt auf eine richtungsabhängige Integro-Differentialgleichung. Ist die Richtungsabhängigkeit nicht von Interesse, so kann der Übergang zu einer winkelintegrierten Form erfolgen. Dieser Übergang führt schließlich auf ein System schwach singulärer fredholmscher Integralgleichungen zweiter Art. Dieses charakterisiert nun nicht mehr die erwähnte Strahlungsintensität, sondern beschreibt die sogenannte Einstrahlung sowie den Strahlungsfluss. Das System singulärer Integralgleichungen kann mittels eines Galerkin-Ansatzes numerisch gelöst werden. Geht man von einer hinreichenden Glattheit des Randes aus, kann die Kompaktheit des Operators der Integralgleichungen gezeigt werden. Dies wiederum erlaubt Rückschlüsse auf die Existenz und Eindeutigkeit einer Lösung. Ein Augenmerk bei der Ermittlung der Galerkin-Näherung ist auf die Bestimmung der singulären Integrale der Galerkin-Diskretisierung zu richten. Für die Bestimmung multidimensionaler, singulärer Integrale stellt die Arbeit das Verfahren der hierarchischen Integration vor. Basierend auf einer Zerlegung des Integrationsgebietes, erfolgt die Beschreibung singulärer Integrale durch ein Gleichungssystem, dessen rechte Seite nur von regulären Integralen abhängig ist. Können diese regulären Integrale sowie die Lösung des Gleichungssystems exakt bestimmt werden, so sind auch die singulären Integrale exakt bestimmt. Bei einer numerischen Bestimmung der regulären Integrale ist die Fehlerordnung ausschlaggebend für den Fehler der singulären Integrale. Als Integrationsgebiete werden Hyperwürfel beliebiger Dimension sowie Simplizes bis einschließlich Dimension 3 als Integrationsgebiete betrachtet. Als Voraussetzungen an den Kern des Doppelintegrals sind nur die Eigenschaften der Translationsinvarianz sowie der Homogenität zu richten. Kann ein nicht translationsinvarianter oder nicht homogener Kern eines Integrals in Summanden zerlegt werden, die selbst translationsinvariant und homogen sind, ist auch die Bestimmung solcher Integrale möglich. Darüber hinaus stellt die Arbeit Verbindungen zu dem Begriff des Hadamard partie finie her. Auf diese Weise lässt sich das Verfahren der hierarchischen Integration für beliebige Dimensionen und beliebige Singularitätsordnungen anwenden. Die Strahlungstransportgleichung ist im Allgemeinen mittels eines Galerkin-Ansatzes lösbar, führt jedoch auf eine voll besetzte Systemmatrix. Numerische Beispiele beleuchten daher Methoden der Matrixkompression mittels hierarchischer Matrizen sowie der direkten Erzeugung schwach besetzter Matrizen über regulären Gittern und Gittern mit hängenden Knoten und skizziert Ansätze zur Parallelisierung auf entsprechenden Computersystemen.
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Johnson, Tomas. "Computer-aided Computation of Abelian integrals and Robust Normal Forms." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-107519.

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This PhD thesis consists of a summary and seven papers, where various applications of auto-validated computations are studied. In the first paper we describe a rigorous method to determine unknown parameters in a system of ordinary differential equations from measured data with known bounds on the noise of the measurements. Papers II, III, IV, and V are concerned with Abelian integrals. In Paper II, we construct an auto-validated algorithm to compute Abelian integrals. In Paper III we investigate, via an example, how one can use this algorithm to determine the possible configurations of limit cycles that can bifurcate from a given Hamiltonian vector field. In Paper IV we construct an example of a perturbation of degree five of a Hamiltonian vector field of degree five, with 27 limit cycles, and in Paper V we construct an example of a perturbation of degree seven of a Hamiltonian vector field of degree seven, with 53 limit cycles. These are new lower bounds for the maximum number of limit cycles that can bifurcate from a Hamiltonian vector field for those degrees. In Papers VI, and VII, we study a certain kind of normal form for real hyperbolic saddles, which is numerically robust. In Paper VI we describe an algorithm how to automatically compute these normal forms in the planar case. In Paper VII we use the properties of the normal form to compute local invariant manifolds in a neighbourhood of the saddle.
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Tamayo, Palau José María. "Multilevel adaptive cross approximation and direct evaluation method for fast and accurate discretization of electromagnetic integral equations." Doctoral thesis, Universitat Politècnica de Catalunya, 2011. http://hdl.handle.net/10803/6952.

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El Método de los Momentos (MoM) ha sido ampliamente utilizado en las últimas décadas para la discretización y la solución de las formulaciones de ecuación integral que aparecen en muchos problemas electromagnéticos de antenas y dispersión. Las más utilizadas de dichas formulaciones son la Ecuación Integral de Campo Eléctrico (EFIE), la Ecuación Integral de Campo Magnético (MFIE) y la Ecuación Integral de Campo Combinada (CFIE), que no es más que una combinación lineal de las dos anteriores.
Las formulaciones MFIE y CFIE son válidas únicamente para objetos cerrados y necesitan tratar la integración de núcleos con singularidades de orden superior al de la EFIE. La falta de técnicas eficientes y precisas para el cálculo de dichas integrales singulares a llevado a imprecisiones en los resultados. Consecuentemente, su uso se ha visto restringido a propósitos puramente académicos, incluso cuando tienen una velocidad de convergencia muy superior cuando son resuelto iterativamente, debido a su excelente número de condicionamiento.
En general, la principal desventaja del MoM es el alto coste de su construcción, almacenamiento y solución teniendo en cuenta que es inevitablemente un sistema denso, que crece con el tamaño eléctrico del objeto a analizar. Por tanto, un gran número de métodos han sido desarrollados para su compresión y solución. Sin embargo, muchos de ellos son absolutamente dependientes del núcleo de la ecuación integral, necesitando de una reformulación completa para cada núcleo, en caso de que sea posible.
Esta tesis presenta nuevos enfoques o métodos para acelerar y incrementar la precisión de ecuaciones integrales discretizadas con el Método de los Momentos (MoM) en electromagnetismo computacional.
En primer lugar, un nuevo método iterativo rápido, el Multilevel Adaptive Cross Approximation (MLACA), ha sido desarrollado para acelerar la solución del sistema lineal del MoM. En la búsqueda por un esquema de propósito general, el MLACA es un método independiente del núcleo de la ecuación integral y es puramente algebraico. Mejora simultáneamente la eficiencia y la compresión con respecto a su versión mono-nivel, el ACA, ya existente. Por tanto, representa una excelente alternativa para la solución del sistema del MoM de problemas electromagnéticos de gran escala.
En segundo lugar, el Direct Evaluation Method, que ha provado ser la referencia principal en términos de eficiencia y precisión, es extendido para superar el cálculo del desafío que suponen las integrales hiper-singulares 4-D que aparecen en la formulación de Ecuación Integral de Campo Magnético (MFIE) así como en la de Ecuación Integral de Campo Combinada (CFIE). La máxima precisión asequible -precisión de máquina se obtiene en un tiempo más que razonable, sobrepasando a cualquier otra técnica existente en la bibliografía.
En tercer lugar, las integrales hiper-singulares mencionadas anteriormente se convierten en casi-singulares cuando los elementos discretizados están muy próximo pero sin llegar a tocarse. Se muestra como las reglas de integración tradicionales tampoco convergen adecuadamente en este caso y se propone una posible solución, basada en reglas de integración más sofisticadas, como la Double Exponential y la Gauss-Laguerre.
Finalmente, un esfuerzo en facilitar el uso de cualquier programa de simulación de antenas basado en el MoM ha llevado al desarrollo de un modelo matemático general de un puerto de excitación en el espacio discretizado. Con este nuevo modelo, ya no es necesaria la adaptación de los lados del mallado al puerto en cuestión.
The Method of Moments (MoM) has been widely used during the last decades for the discretization and the solution of integral equation formulations appearing in several electromagnetic antenna and scattering problems. The most utilized of these formulations are the Electric Field Integral Equation (EFIE), the Magnetic Field Integral Equation (MFIE) and the Combined Field Integral Equation (CFIE), which is a linear combination of the other two.
The MFIE and CFIE formulations are only valid for closed objects and need to deal with the integration of singular kernels with singularities of higher order than the EFIE. The lack of efficient and accurate techniques for the computation of these singular integrals has led to inaccuracies in the results. Consequently, their use has been mainly restricted to academic purposes, even having a much better convergence rate when solved iteratively, due to their excellent conditioning number.
In general, the main drawback of the MoM is the costly construction, storage and solution considering the unavoidable dense linear system, which grows with the electrical size of the object to analyze. Consequently, a wide range of fast methods have been developed for its compression and solution. Most of them, though, are absolutely dependent on the kernel of the integral equation, claiming for a complete re-formulation, if possible, for each new kernel.
This thesis dissertation presents new approaches to accelerate or increase the accuracy of integral equations discretized by the Method of Moments (MoM) in computational electromagnetics.
Firstly, a novel fast iterative solver, the Multilevel Adaptive Cross Approximation (MLACA), has been developed for accelerating the solution of the MoM linear system. In the quest for a general-purpose scheme, the MLACA is a method independent of the kernel of the integral equation and is purely algebraic. It improves both efficiency and compression rate with respect to the previously existing single-level version, the ACA. Therefore, it represents an excellent alternative for the solution of the MoM system of large-scale electromagnetic problems.
Secondly, the direct evaluation method, which has proved to be the main reference in terms of efficiency and accuracy, is extended to overcome the computation of the challenging 4-D hyper-singular integrals arising in the Magnetic Field Integral Equation (MFIE) and Combined Field Integral Equation (CFIE) formulations. The maximum affordable accuracy --machine precision-- is obtained in a more than reasonable computation time, surpassing any other existing technique in the literature.
Thirdly, the aforementioned hyper-singular integrals become near-singular when the discretized elements are very closely placed but not touching. It is shown how traditional integration rules fail to converge also in this case, and a possible solution based on more sophisticated integration rules, like the Double Exponential and the Gauss-Laguerre, is proposed.
Finally, an effort to facilitate the usability of any antenna simulation software based on the MoM has led to the development of a general mathematical model of an excitation port in the discretized space. With this new model, it is no longer necessary to adapt the mesh edges to the port.
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Kraus, Michal. "Paralelní výpočetní architektury založené na numerické integraci." Doctoral thesis, Vysoké učení technické v Brně. Fakulta informačních technologií, 2013. http://www.nusl.cz/ntk/nusl-261227.

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This thesis deals with continuous system simulation. The systems can be described by system of differential equations or block diagram. Differential equations are usually solved by numerical methods that are integrated into simulation software such as Matlab, Maple or TKSL. Taylor series method has been used for numerical solutions of differential equations. The presented method has been proved to be both very accurate and fast and also procesed in parallel systems. The aim of the thesis is to design, implement and compare a few versions of the parallel system.
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Alsallami, Shami Ali M. "Discrete integrable systems and geometric numerical integration." Thesis, University of Leeds, 2018. http://etheses.whiterose.ac.uk/22291/.

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This thesis deals with discrete integrable systems theory and modified Hamiltonian equations in the field of geometric numerical integration. Modified Hamiltonians are used to show that symplectic schemes for Hamiltonian systems are accurate over long times. However, for nonlinear systems the series defining the modified Hamiltonian equation usually diverges. The first part of the thesis demonstrates that there are nonlinear systems where the modified Hamiltonian has a closed-form expression and hence converges. These systems arise from the theory of discrete integrable systems. Specifically, they arise as reductions of a lattice version of the Korteweg-de Vries (KdV) partial differential equation. We present cases of one and two degrees of freedom symplectic mappings, for which the modified Hamiltonian equations can be computed as a closed form expression using techniques of action-angle variables, separation of variables and finite-gap integration. These modified Hamiltonians are also given as power series in the time step by Yoshida's method based on the Baker-Campbell-Hausdorff series. Another example is a system of an implicit dependence on the time step, which is obtained by dimensional reduction of a lattice version of the modified KdV equation. The second part of the thesis contains a different class of discrete-time system, namely the Boussinesq type, which can be considered as a higher-order counterpart of the KdV type. The development and analysis of this class by means of the B{\"a}cklund transformation, staircase reductions and Dubrovin equations forms one of the major parts of the thesis. First, we present a new derivation of the main equation, which is a nine-point lattice Boussinesq equation, from the B{\"a}cklund transformation for the continuous Boussinesq equation. Second, we focus on periodic reductions of the lattice equation and derive all necessary ingredients of the corresponding finite-dimensional models. Using the corresponding monodromy matrix and applying techniques from Lax pair and $r$-matrix structure analysis to the Boussinesq mappings, we study the dynamics in terms of the so-called Dubrovin equations for the separated variables.
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Lastdrager, Boris. "Numerical time integration on sparse grids." [S.l. : Amsterdam : s.n.] ; Universiteit van Amsterdam [Host], 2002. http://dare.uva.nl/document/64526.

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Mikulka, Jiří. "Numerické výpočty určitých integrálů." Master's thesis, Vysoké učení technické v Brně. Fakulta informačních technologií, 2014. http://www.nusl.cz/ntk/nusl-236141.

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The application of the finite integral of multiple variable functions is penetrating into more and more industries and science disciplines. The demands placed on solutions to these problems (such as high accuracy or high speed) are often quite contradictory. Therefore, it is not always possible to apply analytical approaches to these problems; numerical methods provide a suitable alternative. However, the ever-growing complexity of these problems places too high a demand on many of these numerical methods, and so neither of these methods are useful for solving such problems. The goal of this thesis is to design and implement a new numerical method that provides highly accurate and very fast computation of finite integrals of multiple variable functions. This new method combines pre-existing approaches in the field of numerical mathematics.
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Books on the topic "Numerical integration. Integrals"

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Nicolas Bourbaki. Integration. Berlin: Springer, 2004.

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Nicolas Bourbaki. Integration. Berlin: Springer, 2004.

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Introduction to integration. Oxford: Clarendon Press, 1997.

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The general theory of integration. Oxford: Clarendon Press, 1991.

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Whitney, Hassler. Geometric integration theory. Mineola, N.Y: Dover Publications, 2005.

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Thompson, Jeremy Stewart. High speed numerical integration of Fermi Dirac integrals. Monterey, Calif: Naval Postgraduate School, 1996.

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Milʹshteĭn, G. N. Numerical integration of stochastic differential equations. Dordrecht: Kluwer Academic Publishers, 1995.

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Kythe, Prem K. Handbook of computational methods for integration. Boca Raton: Chapman & Hall/CRC, 2005.

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Measure and integration for use. Oxford: Clarendon Press, 1985.

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Lastdrager, Boris. Numerical time integration on sparse grids. [S.l: s.n.], 2002.

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Book chapters on the topic "Numerical integration. Integrals"

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Bialecki, Bernard. "Sinc Quadratures for Cauchy Principal Value Integrals." In Numerical Integration, 81–92. Dordrecht: Springer Netherlands, 1992. http://dx.doi.org/10.1007/978-94-011-2646-5_7.

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Sidi, Avram. "Computation of Oscillatory Infinite Integrals by Extrapolation Methods." In Numerical Integration, 349–51. Dordrecht: Springer Netherlands, 1992. http://dx.doi.org/10.1007/978-94-011-2646-5_29.

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Lyness, J. N. "Some Quadrature Rules for Finite Trigonometric and Related Integrals." In Numerical Integration, 17–33. Dordrecht: Springer Netherlands, 1987. http://dx.doi.org/10.1007/978-94-009-3889-2_2.

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Genz, Alan. "The Numerical Evaluation of Multiple Integrals on Parallel Computers." In Numerical Integration, 219–29. Dordrecht: Springer Netherlands, 1987. http://dx.doi.org/10.1007/978-94-009-3889-2_23.

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Hsu, L. C., and Y. S. Zhou. "Approximate Computation of Strongly Oscillatory Integrals with Compound Precision." In Numerical Integration, 91–101. Dordrecht: Springer Netherlands, 1987. http://dx.doi.org/10.1007/978-94-009-3889-2_7.

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Schwab, Christoph, and Wolfgang L. Wendland. "Numerical Integration of Singular and Hypersingular Integrals in Boundary Element Methods." In Numerical Integration, 203–18. Dordrecht: Springer Netherlands, 1992. http://dx.doi.org/10.1007/978-94-011-2646-5_16.

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Berntsen, Jarle. "On the Numerical Calculation of Multidimensional Integrals Appearing in the Theory of Underwater Acoustics." In Numerical Integration, 249–65. Dordrecht: Springer Netherlands, 1992. http://dx.doi.org/10.1007/978-94-011-2646-5_19.

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Hunter, D. B. "The Numerical Evaluation of Definite Integrals Affected by Singularities Near the Interval of Integration." In Numerical Integration, 111–20. Dordrecht: Springer Netherlands, 1992. http://dx.doi.org/10.1007/978-94-011-2646-5_9.

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Ioakimidis, Nikolaos I. "Application of Computer Algebra Software to the Derivation of Numerical Integration Rules for Singular and Hypersingular Integrals." In Numerical Integration, 121–31. Dordrecht: Springer Netherlands, 1992. http://dx.doi.org/10.1007/978-94-011-2646-5_10.

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Berens, H., and H. J. Schmid. "On the Number of Nodes of Odd Degree Cubature Formulae for Integrals with Jacobi Weights on a Simplex." In Numerical Integration, 37–44. Dordrecht: Springer Netherlands, 1992. http://dx.doi.org/10.1007/978-94-011-2646-5_3.

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Conference papers on the topic "Numerical integration. Integrals"

1

Chaloupka, Jan, Jiří Kunovský, Václav Šátek, Petr Veigend, and Alžbeta Martinkovičová. "Numerical Integration of Multiple Integrals using Taylor Polynomial." In 5th International Conference on Simulation and Modeling Methodologies, Technologies and Applications. SCITEPRESS - Science and and Technology Publications, 2015. http://dx.doi.org/10.5220/0005539701630171.

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Volskiy, Vladimir, Guy A. E. Vandenbosch, Athanasios G. Polimeridis, Juan R. Mosig, and Ruzica Golubovic Niciforovic. "KUL and EPFL cooperation on numerical integration of Sommerfeld integrals." In 2012 6th European Conference on Antennas and Propagation (EuCAP). IEEE, 2012. http://dx.doi.org/10.1109/eucap.2012.6206566.

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Volmer, Julia, Andreas Ammon, Alan Genz, Tobias Hartung, Karl Jansen, and Hernan Leövey. "Applying recursive numerical integration techniques for solving high dimensional integrals." In 34th annual International Symposium on Lattice Field Theory. Trieste, Italy: Sissa Medialab, 2017. http://dx.doi.org/10.22323/1.256.0335.

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Usovitsch, Johann, Ievgen Dubovyk, and Tord Riemann. "MBnumerics: Numerical integration of Mellin-Barnes integrals in physical regions." In Loops and Legs in Quantum Field Theory. Trieste, Italy: Sissa Medialab, 2018. http://dx.doi.org/10.22323/1.303.0046.

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Liao, Wen-I., and Tsung-Jen Teng. "On Evaluation of Lamb’s Integrals for Seismic Waves in a Three-Dimension Elastic Half-Space." In ASME 2005 Pressure Vessels and Piping Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/pvp2005-71448.

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The analytical method such as the boundary integral (element) method or the series expansion method is usually used to solve the wave scattering problem. In these methods, the singular Green’s function should be determined firstly; the main difficulty of the use of the Lamb’s singular solutions in integral form to represent the diffracted fields is the numerical implementation for the evaluation of those improper integrals. The integrands of these integrals are highly irregular and oscillatory. In this paper, a technique is proposed to calculate the integral in wave-number domain based on the method of steepest descent. After replacing the original integration path by steepest decent path, the wave-number integral results in a Gauss-Hermite type quadrature, so the oscillating characteristics of the original integrand can be removed and results a non-oscillating integrand, it is very helpful in computing efficiency.
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Helluy, Philippe, Sylvain Maire, and Patrice Ravel. "New Higher Order Numeric Quadratures for Regular or Singular Functions on an Interval: Applications for the Helmholtz Integral Equation." In ASME 1999 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1999. http://dx.doi.org/10.1115/detc99/vib-8119.

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Abstract A high order integration method is presented for regular or singular integrands over an integral. This method appears to be very useful to compute the integrals of the green function in the numerical resolution of boundary integral equations.
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Gluza, Janusz, Ievgen Dubovyk, Tord Riemann, and Johann Usovitsch. "Numerical integration of massive two-loop Mellin-Barnes integrals in Minkowskian regions." In Loops and Legs in Quantum Field Theory. Trieste, Italy: Sissa Medialab, 2016. http://dx.doi.org/10.22323/1.260.0034.

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Bora, Jugma N. "Analytical Evaluation of the Integrals Appearing in the Boundary Element Method for Some Problems in Mechanics." In ASME 1991 International Computers in Engineering Conference and Exposition. American Society of Mechanical Engineers, 1991. http://dx.doi.org/10.1115/cie1991-0102.

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Abstract The boundary integral equation method or simply the Boundary Element Method (BEM) is now considered to be a powerful tool for solving problems in mechanics. A large number of the line and area integrals appearing for two-dimensional problems, in the BEM, can be represented as linear combinations of four singular functions. These integrals that are generated are products of the approximating polynomials and one or more of the four singular functions. These integrals can be evaluated numerically or analytically. The advantage of numerical integration is that the shape of the boundary can be of any complexity. The big disadvantage is that another source of error, besides the polynomial interpolation, is added into the process due to the approximation of the integrand for numerical integration. The singular nature of the fundamental solutions further exacerbates this disadvantage. In analytical integration, the form of the boundary must be assumed. The usual representation is a sum of straight-line segments. Currently, analytical expressions are available only for a few formulations and they are valid only for zero- and first-order polynomials. In this paper analytical expressions of the integrals are obtained for approximating polynomials of any arbitrary order. The fundamental solutions of the Laplace, Biharmonic and the equation of Plane Elastostatics are considered. The validity and advantage of using these integrals are shown by some simple problems in mechanics that have theoretical solutions.
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Yan-Wen, Zhao, Zhao Qing-Guang, Luo Xi, Nie Zai-Ping, and Bi Hai-Yan. "Analysis of Numerical Integration Accuracy of Singular Integrals in Moment Method of TDEFIE." In The 2006 4th Asia-Pacific Conference on Environmental Electromagnetics. IEEE, 2006. http://dx.doi.org/10.1109/ceem.2006.257977.

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Volskiy, Vladimir, Guy A. E. Vandenbosch, Ruzica Golubovic Niciforovic, Athanasios G. Polimeridis, and Juan R. Mosig. "Numerical integration of Sommerfeld integrals based on singularity extraction techniques and double exponential-type quadrature formulas." In 2012 6th European Conference on Antennas and Propagation (EuCAP). IEEE, 2012. http://dx.doi.org/10.1109/eucap.2012.6205950.

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