Local Versus Global Strategies for Adaptive Quadrature

Malcolm, Michael A.; Simpson, R. B.

doi:10.1145/355637.355640

Cited by 73 publications

(40 citation statements)

References 10 publications

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“…The built-in numerical integration routine of Mathematica performs a recursive procedure called Global Adaptive Integration [21]. On each recursive call, it analyzes the behavior of the integrand function within the integration domain and automatically selects an integration rule to apply.…”

Section: Rq2: How Qcoral Compares With Mathematica and Volcomp?mentioning

confidence: 99%

Compositional solution space quantification for probabilistic software analysis

et al. 2014

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Probabilistic software analysis aims at quantifying how likely a target event is to occur during program execution. Current approaches rely on symbolic execution to identify the conditions to reach the target event and try to quantify the fraction of the input domain satisfying these conditions. Precise quantification is usually limited to linear constraints, while only approximate solutions can be provided in general through statistical approaches. However, statistical approaches may fail to converge to an acceptable accuracy within a reasonable time.We present a compositional statistical approach for the efficient quantification of solution spaces for arbitrarily complex constraints over bounded floating-point domains. The approach leverages interval constraint propagation to improve the accuracy of the estimation by focusing the sampling on the regions of the input domain containing the sought solutions. Preliminary experiments show significant improvement on previous approaches both in results accuracy and analysis time.

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Section: Rq2: How Qcoral Compares With Mathematica and Volcomp?mentioning

confidence: 99%

Compositional solution space quantification for probabilistic software analysis

et al. 2014

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“…As real numbers are identified to degenerated intervals, the interval arithmetic actually generalizes the real arithmetic, and mixed operations like 1 + [1, 2] = [2,3] are interpreted using (4).…”

Section: Interval Extensionsmentioning

confidence: 99%

“…Such adaptive strategies are known to be of critical importance also in the context of non verified quadrature, see e.g [2]. …”

mentioning

confidence: 99%

Convergence analysis and adaptive strategy for the certified quadrature over a set defined by inequalities

Goldsztejn

Cruz

Carvalho

2014

Journal of Computational and Applied Mathematics

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a b s t r a c tThis paper investigates the sufficient conditions for the asymptotic convergence of a generic branch and prune algorithm dedicated to the verified quadrature of a function in several variables. Quadrature over domains defined by inequalities, and adaptive meshing strategies are in the scope of this analysis. The framework is instantiated using certified quadrature methods based on Taylor models (i.e. Taylor approximations with rigorously bounded remainder), and reported experiments confirmed the analysis. They also show that the performances of the instantiated algorithm are comparable with current methods for certified quadrature.

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“…We have computed integrals over the triangle with vertices (0, 0), (1, 0), (0, 1) of the following functions: tricub: an ALGOL 60 code, as listed in Hemker [3], which uses an adaptive strategy of a "local type" (in the sence of Malcolm and Simpson [4]). …”

Section: Comparison With Other Algorithmsmentioning

confidence: 99%

Algorithm 34 An algorithm for the automatic integration over a triangle

Haegemans

1977

Computing

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--ZusammenfassungAlgorithm 34. An Algorithm for the Automatic Integration Over a Triangle. Numerical integration over an arbitrary triangular domain is considered. An adaptive algorithm with round-off error guard is described. A FORTRAN IV implementation is given and numerical results are reported. Algorithmus 34. Aigorithmus far das automatisehe Integrieren Uber Dreiecke. Es wird ein adaptiverAlgorithmus und ein FORTRAN IV Programm zur automatischen Berechnung yon Integralen fiber einem beliebigen Dreiecksbereich vorgestellt, und an numerischen Beispielen demonstriert.

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Local Versus Global Strategies for Adaptive Quadrature

Cited by 73 publications

References 10 publications

Compositional solution space quantification for probabilistic software analysis

Compositional solution space quantification for probabilistic software analysis

Convergence analysis and adaptive strategy for the certified quadrature over a set defined by inequalities

Algorithm 34 An algorithm for the automatic integration over a triangle

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