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What is cubature Archimedes and Sobolev High dimensions High degree Open problems

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38 Pages
English

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What is cubature? Archimedes and Sobolev High dimensions High degree Open problems Cubature from geometry and coding theory (Generalizations of Simpson's rule) arXiv:math/0402047 arXiv:math/0405366 arXiv:math/0408360 Greg Kuperberg UC Davis and Institut Fourier September 25, 2010

  • dimensions high degree

  • greg kuperberg

  • sobolev high

  • coding theory

  • higher dimension

  • oscillating weights

  • simpson's rule

  • open problems


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hWtasiucabuter?rAhcmidesenadoSbloveiHhgdmineisnosiHhgdgeereCubaturefromgeometryandcodingtheory(GeneralizationsofSimpson’srule)pOearXiv:math/0402047arXiv:math/0405366arXiv:math/0408360GregKuperbergUCDavisandInstitutFourierSeptember25,2010nrpboelsm
hWtasiucabuter?rAhcmidesenadoSbloveiHhgdmineisnosiHhgWhatisnumericalcubature?dgeereOGivenXRnanda(normalized)measureµonX,wewanttoestimate:ZXf(x~)dµf(F)=wkf(~pk).XkWewanttochooseFsothattheformupolynomials.ExampleSimpson’srule:alsixecatofr1Z0f(x)dx61f(0)+32f(21)+61f(1).osempenSimpson’sruleisexactiffR[x]3.Onesaysquadratureinonedimensionandcubatureinhigherdimensions.rpboelsm
hWtasiucabuter?rAhcmidesenadoSbloveiHhgdmineisnosiHhgWhatisnumericalcubature?dgeereOGivenXRnanda(normalized)measureµonX,wewanttoestimate:ZXf(x~)dµf(F)=wkf(~pk).XkWewanttochooseFsothattheformupolynomials.ExampleSimpson’srule:alsixecatofr1Z0f(x)dx61f(0)+32f(21)+61f(1).osempenSimpson’sruleisexactiffR[x]3.Onesaysquadratureinonedimensionandcubatureinhigherdimensions.rpboelsm