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An Adaptive Algorithm for the Approximate Calculation of Multiple Integrals
, 1991
"... An adaptive algorithm for numerical integration over hyperrectangular regions is described. The algorithm uses a globally adaptive subdivision strategy. Several precautions are introduced in the error estimation in order to improve the reliability. In each dimension more than one integration rule ..."
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Cited by 59 (7 self)
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An adaptive algorithm for numerical integration over hyperrectangular regions is described. The algorithm uses a globally adaptive subdivision strategy. Several precautions are introduced in the error estimation in order to improve the reliability. In each dimension more than one integration rule is made available to the user. The algorithm has been structured to allow efficient implementation on shared memory parallel computers. Keywords: automatic integration, adaptive, cubature, multidimensional integration, fully symmetric rules, null rules, parallel algorithms. Published in ACM Trans. Math. Softw. 17 (1991),pp. 437451. y Supported by the Norwegian Research Council for Humanities and Sciences and STATOIL. z Supported by the Norwegian Research Council for Humanities and Sciences. x Supported by the Norwegian Marshall Fund. 1 1 Introduction Adaptive algorithms are now used widely for the numerical calculation of multiple integrals. These algorithms have been develop...
Doubly Adaptive Quadrature Routines based on Newton–Cotes rules
 Reports in Informatics 229, Dept. of Informatics, Univ. of
, 2002
"... In this paper we test two recently published Matlab codes, adaptsim and adaptlob, using both a Lyness–Kaganove test and a battery type of test. Furthermore we modify these two codes using sequences of null rules in the error estimator with the intention to increase the reliability for both codes. In ..."
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Cited by 6 (1 self)
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In this paper we test two recently published Matlab codes, adaptsim and adaptlob, using both a Lyness–Kaganove test and a battery type of test. Furthermore we modify these two codes using sequences of null rules in the error estimator with the intention to increase the reliability for both codes. In addition two new Matlab codes applying a locally and a globally adaptive strategy respectively are developed. These two new codes turn out to have very good properties both with respect to reliability and efficiency. Both algorithms are using sequences of null rules in their local error estimators. These error estimators allow us both to test if we are in the region of asymptotic behavior and thus increase reliability and to take advantage of the degree of precision of the basic quadrature rule. The new codes compare favorably to the two recently published adaptive codes both when we use a Lyness–Kaganove testing technique and by using a battery test.
On Numerical Solutions to OneDimensional Integration Problems with Applications to Linear Light Sources
 ACM Transactions on Graphics
"... Many key problems in computer graphics require the computation of integrals. Due to the nature of the integrand and of the domain of integration, these integrals seldom can be computed analytically. As a result, numerical techniques are used to find approximate solutions to these problems. While the ..."
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Cited by 5 (0 self)
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Many key problems in computer graphics require the computation of integrals. Due to the nature of the integrand and of the domain of integration, these integrals seldom can be computed analytically. As a result, numerical techniques are used to find approximate solutions to these problems. While the numerical analysis literature offers many integration techniques, the choice of which method to use for specific computer graphic problems is a difficult one. This choice must be driven by the numerical efficiency of the method, and ultimately, by its visual impact on the computed image. In this paper, we begin to address these issues by methodically analyzing deterministic and stochastic numerical techniques and their application to the type of onedimensional problems that occur in computer graphics, especially in the context of linear light source integration. In addition to traditional methods such as GaussLegendre quadratures, we also examine Voronoi diagrambased sampling, jittered quadratures, random offset quadratures, weighted Monte Carlo, and a newly introduced method of compounding known as a difficulty driven compound quadrature. We compare the effectiveness of these methods using a threepronged approach. First, we compare the frequency domain characteristics of all the methods using periodograms. Next, applying ideas found in the numerical analysis literature, we examine the numerical and visual performance profiles of these methods for seven different oneparameter problem families. We then present results from the application of the methods for the example of linear light sources. Finally, we summarize the relative effectiveness of the methods surveyed, showing the potential power of difficultydriven compound quadratures.
Overturned internal capillarygravity waves
, 2015
"... A vortex sheet formulation of irrotational, incompressible Euler flow is used to compute periodic traveling waves at the interface between two constantdensity, twodimensional fluids, including waves with overturned crests. Branches of traveling waves are computed via numerical continuation, which ..."
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A vortex sheet formulation of irrotational, incompressible Euler flow is used to compute periodic traveling waves at the interface between two constantdensity, twodimensional fluids, including waves with overturned crests. Branches of traveling waves are computed via numerical continuation, which are jointly continuous in the physical parameters: Bond number, Atwood number and mean shear. Global branches are computed, for various choices of parameters, illustrating the termination criteria of the global bifurcation theorem of [1]. Small amplitude asymptotics of traveling waves are calculated. The role of the wave’s second harmonic in determining crest/trough shape is discussed. Waves of extremal displacement and those approaching selfintersection are computed. Surfaces in parameter space indicating where traveling waves exist are computed; these surfaces themselves have both overturned and selfintersecting boundaries. 1
Doubly Adaptive Quadrature Routines based on NewtonCote Rules
 Reports in Informatics 229, Dept. of Informatics, Univ. of
, 2002
"... In this paper we test two recently published Matlab codes, adaptsim and adaptlob, using both a LynessKaganove test and a battery type of test. Furthermore we modify these two codes using sequences of null rules in the error estimator with the intention to increase the reliability for both codes. ..."
Abstract
 Add to MetaCart
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In this paper we test two recently published Matlab codes, adaptsim and adaptlob, using both a LynessKaganove test and a battery type of test. Furthermore we modify these two codes using sequences of null rules in the error estimator with the intention to increase the reliability for both codes. In addition two new Matlab codes applying a locally and a globally adaptive strategy respectively are developed. These two new codes turn out to have very good properties both with respect to reliability and e#ciency.