From optimal cubature formulae to Chebyshev lattices: a way towards - PowerPoint PPT Presentation

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion From optimal cubature formulae to Chebyshev lattices: a way towards generalised Clenshaw-Curtis quadrature Ronald Cools Koen Poppe � (ronald.cools|koen.poppe)@cs.kuleuven.be � Department of Computer Science, K.U.Leuven, Belgium International Conference on Scientific Computing SC2011 10–14 October 2011

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion Introduction to the problem Given is an integral � I [ f ] := w ( x ) f ( x ) dx Ω where Ω ⊂ R s and w ( x ) ≥ 0, ∀ x ∈ R s . Search an approximation for I [ f ] N � w i f ( y ( i ) ) I [ f ] ≃ Q [ f ] := i = 1 with w i ∈ R and y ( i ) ∈ R s . Webster: quadrature: the process of finding a square equal in area to a given area. cubature: the determination of cubic contents. If n = 1 then Q is called a quadrature formula. If n ≥ 2 then Q is called a cubature formula.

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion P s = the vector space of all polynomials in s variables. d = the subspace of P s of polynomials with degree ≤ d . P s The cubature formula Q contains 2 N + 1 parameters : the number N , the points y ( i ) the weights w i . Criterion to specify and classify cubature formulae: Definition A cubature formula Q for an integral I has degree d if Q [ f ] = I [ f ] , ∀ f ∈ P s d and ∃ g ∈ P s d + 1 such that Q [ g ] � = I [ g ] .

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion The quest for minimal formulae How many points are needed in a cubature formula to obtain a specified degree of precision? � s + k � Suppose a cubature formula has less than dim P s k = points. k ⇒ there exists a f ∈ P s k that vanishes at the points ⇒ Q [ f 2 ] = 0. I is a positive functional ⇒ I [ f 2 ] > 0. ⇒ Q does not have degree 2 k or a cubature formula of degree d has its number of points N bounded below by � s + ⌊ d � 2 ⌋ N min = dim P s 2 ⌋ = . ⌊ d n Used by Radon (1948).

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion Radon’s approach: d = ( d + 1 )( d + 2 ) degree d → M = dim P 2 equations. 2 Each point introduces 3 unknowns. If d � = multiple of 3, then M is a multiple of 3. In that case, if the number of points in the cubature formula is N = ( d + 1 )( d + 2 ) , 6 then a system with the same number of equations as unknowns is obtained. d = 1 → N = 1 = N min : trivial d = 2 → N = 2 < N min d = 4 → N = 5 < N min d = 5 → N = 7 > N min : formulae for square, circle and triangle whose points are common zeros of 3 orthogonal polynomials.

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion Radon’s approach: d = ( d + 1 )( d + 2 ) degree d → M = dim P 2 equations. 2 Each point introduces 3 unknowns. If d � = multiple of 3, then M is a multiple of 3. In that case, if the number of points in the cubature formula is N = ( d + 1 )( d + 2 ) , 6 then a system with the same number of equations as unknowns is obtained. d = 1 → N = 1 = N min : trivial d = 2 → N = 2 < N min d = 4 → N = 5 < N min d = 5 → N = 7 > N min : formulae for square, circle and triangle whose points are common zeros of 3 orthogonal polynomials.

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion Under what conditions is N min sharp? What is the highest lower bound for standard regions?

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion The first century in a nutshell 1877: J.C. Maxwell constructed the first cubature formulae. Fully symmetric, degree 7 for C 2 and C 3 . 1890: P. Appell : orthogonal polynomials ↔ cubature formulae 1948: J. Radon : cubature formula of degree 5 using the common zeros of 3 orthogonal polynomials of degree 3. 196*: A. Stroud and I.P. Mysovskih : orthogonal polynomials ↔ cubature formulae for n -dimensional regions. 196*-197*: Two groups of researchers: The first : attacked the system of nonlinear equations. → consistency conditions (P. Rabinowitz, N. Richter and F. Mantel) The second : used the relation between orthogonal polynomials and cubature formulae. (A. Stroud, I.P. Mysovskih , G. Rasputin , R. Franke, R. Piessens and A. Haegemans) → introduction of polynomial ideals in 1973 (H.M. Möller). introduction of real ideals in 1978 (H.J. Schmid).

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion Ideal theory is relevant Definition A polynomial ideal A is a subset of P s such that if f 1 , f 2 ∈ A and g 1 , g 2 ∈ P s , then f 1 g 1 + f 2 g 2 ∈ A . If a set of points is given, then the set of all polynomials that vanish at these points is an ideal.

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion Multivariate orthogonal polynomials Definition f ∈ P s of degree d for which I [ fg ] = 0, ∀ g ∈ P s d − 1 is called an orthogonal polynomial for I . Definition f ∈ P s is d-orthogonal for I if I [ fg ] = 0 for all g that satisfy fg ∈ P s d .

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion There exist more than one orthogonal polynomial for each degree: Choice 1: There exist ( dim P s d − dim P s d − 1 ) unique orthogonal polynomials of degree d of the form P a 1 , a 2 ,..., a n := x a 1 1 x a 2 2 . . . x a n n + Q with a i ∈ N , � s i = 1 a i = d and Q ∈ P s d − 1 . Choice 2: Orthonormal sequence − → reproducing kernel There exist several types of bases for A : H-basis (= Macaulay basis = canonical basis) − → theoretical importance G-basis (= Gröbner basis) − → practical importance Under mild restrictions, a G-basis is an H-basis.

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion There exist more than one orthogonal polynomial for each degree: Choice 1: There exist ( dim P s d − dim P s d − 1 ) unique orthogonal polynomials of degree d of the form P a 1 , a 2 ,..., a n := x a 1 1 x a 2 2 . . . x a n n + Q with a i ∈ N , � s i = 1 a i = d and Q ∈ P s d − 1 . Choice 2: Orthonormal sequence − → reproducing kernel There exist several types of bases for A : H-basis (= Macaulay basis = canonical basis) − → theoretical importance G-basis (= Gröbner basis) − → practical importance Under mild restrictions, a G-basis is an H-basis.

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion Definition Let A be a polynomial ideal. { f 1 , f 2 , . . . , f t } ⊂ A is an H-basis for A if for all f ∈ A polynomials g 1 , g 2 , . . . , g t exist such that t � f = f i g i and deg ( f i g i ) ≤ deg ( f ) , i = 1 , 2 , . . . , t . i = 1 Theorem (Möller 1973) For any polynomial ideal an H-basis exists.

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion H -bases are important because Theorem (Möller 1973) If { f 1 , f 2 , . . . , f t } is an H-basis of A and if the set of common zeros of f 1 , f 2 , . . . , f t is finite and nonempty, then the following statements are equivalent : There is a cubature formula of degree d for the integral I which 1 has as points the common zeros of f 1 , f 2 , . . . , f t . These zeros may be multiple, leading to the use of function derivatives in the cubature formula. f i is d-orthogonal for I , i = 1 , 2 , . . . , t. 2 Corollary Every polynomial of degree τ ≤ d that is zero in all the points of a cubature formula of degree d is orthogonal to all polynomials of degree ≤ d − τ .

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion Improved lower bound For degree m = 2 k − 1 � k � � k � N min = k ( k + 1 ) ˜ = dim P 2 + k − 1 + 2 2 2 1973 : for � with central symmetric weight function 1976 : for △ 1 ≤ k ≤ 6 ( m ≤ 11 ) (Möller) 1983 : for △ ∀ k ( Rasputin ) 1992 : for △ with weight function y λ ( x − y ) µ ( 1 − x ) ν (Berens and Schmid) Cubature formulae with ˜ N min points do not always exist. But ˜ N min is the best possible if one only takes into account the central symmetry.

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion Möller’s bound is known to be sharp for . . . � 1 � 1 ( 1 − x 2 ) α ( 1 − y 2 ) α f ( x , y ) dxdy for α = ± 0 . 5 − 1 − 1 More than one minimal formulae can exist! Explicit expression for the points are known, but not for the weights. (Morrow & Patterson 1978) (Schmid 1983) (Verlinden, C., Roose, Haegemans, 1988) (C. & Schmid 1989) Note that rules based on the so-called Padua points require more than ˜ N min points → Explicit expression for the points and weights