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

Greg Kuperberg

Description

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

Sujets

• Greg Kuperberg
• Coding theory
• N-dimensional space
• Simpson's rule
• Open problem

Informations

• Publié par: pefav
• Nombre de lectures: 11
• Langue: English

Extrait

hWtasiucabuter?rAhcmidesenadoSbloveiHhgdmineisnosiHhgdgeere Cubaturefromgeometryandcodingtheory (GeneralizationsofSimpson’srule) pOe arXiv:math/0402047arXiv:math/0405366arXiv:math/0408360 GregKuperberg UCDavisandInstitutFourier September25,2010 nrpboelsm

hWtasiucabuter?rAhcmidesenadoSbloveiHhgdmineisnosiHhg Whatisnumericalcubature? dgeereO GivenX ⊆ R n anda(normalized)measureµonX,wewantto estimate: Z X f(x ~ )dµ ≈ f( F )=w k f( ~ p k ). X k Wewanttochoose F sothattheformu polynomials. Example Simpson’srule: alsixecatofr 1 Z 0 f(x)dx ≈ 6 1 f(0)+ 3 2 f( 2 1 )+ 6 1 f(1). osem pen Simpson’sruleisexactiff ∈ R [x] ≤ 3 .Onesaysquadratureinone dimensionandcubatureinhigherdimensions. rpboelsm

hWtasiucabuter?rAhcmidesenadoSbloveiHhgdmineisnosiHhg Whatisnumericalcubature? dgeereO GivenX ⊆ R n anda(normalized)measureµonX,wewantto estimate: Z X f(x ~ )dµ ≈ f( F )=w k f( ~ p k ). X k Wewanttochoose F sothattheformu polynomials. Example Simpson’srule: alsixecatofr 1 Z 0 f(x)dx ≈ 6 1 f(0)+ 3 2 f( 2 1 )+ 6 1 f(1). osem pen Simpson’sruleisexactiff ∈ R [x] ≤ 3 .Onesaysquadratureinone dimensionandcubatureinhigherdimensions. rpboelsm