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] :=

• Ω

w(x)f(x)dx where Ω ⊂ Rs and w(x) ≥ 0, ∀x ∈ Rs. Search an approximation for I[f] I[f] ≃ Q[f] :=

N

• i=1

wif(y(i)) with wi ∈ R and y(i) ∈ Rs. 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

Ps = the vector space of all polynomials in s variables. Ps

d = the subspace of Ps of polynomials with degree ≤ d.

The cubature formula Q contains 2N + 1 parameters : the number N, the points y(i) the weights wi. 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 ∈ Ps

d

and ∃g ∈ Ps

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? Suppose a cubature formula has less than dim Ps

k =

s+k

k

• points.

⇒ there exists a f ∈ Ps

k that vanishes at the points

⇒ Q[f 2] = 0. I is a positive functional ⇒ I[f 2] > 0. ⇒ Q does not have degree 2k

• r

a cubature formula of degree d has its number of points N bounded below by Nmin = dim Ps

⌊ d

2 ⌋ =

s + ⌊ d

2⌋

n

• .

Used by Radon (1948).

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion

Radon’s approach: degree d → M = dim P2

d = (d+1)(d+2) 2

equations. 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

• btained.

d = 1 → N = 1 = Nmin: trivial d = 2 → N = 2 < Nmin d = 4 → N = 5 < Nmin d = 5 → N = 7 > Nmin: 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: degree d → M = dim P2

d = (d+1)(d+2) 2

equations. 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

• btained.

d = 1 → N = 1 = Nmin: trivial d = 2 → N = 2 < Nmin d = 4 → N = 5 < Nmin d = 5 → N = 7 > Nmin: 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 Nmin 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 C2 and C3 . 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 Ps such that if f1, f2 ∈ A and g1, g2 ∈ Ps, then f1g1 + f2g2 ∈ 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 ∈ Ps of degree d for which I[fg] = 0, ∀g ∈ Ps

d−1 is called an

• rthogonal polynomial for I.

Definition f ∈ Ps is d-orthogonal for I if I[fg] = 0 for all g that satisfy fg ∈ Ps

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 Ps

d − dim Ps d−1) unique orthogonal

polynomials of degree d of the form Pa1,a2,...,an := xa1

1 xa2 2 . . . xan n + Q

with ai ∈ N, s

i=1 ai = d and Q ∈ Ps 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 Ps

d − dim Ps d−1) unique orthogonal

polynomials of degree d of the form Pa1,a2,...,an := xa1

1 xa2 2 . . . xan n + Q

with ai ∈ N, s

i=1 ai = d and Q ∈ Ps 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. {f1, f2, . . . , ft} ⊂ A is an H-basis for A if for all f ∈ A polynomials g1, g2, . . . , gt exist such that f =

t

• i=1

figi and deg(figi) ≤ deg(f), i = 1, 2, . . . , t. 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 {f1, f2, . . . , ft} is an H-basis of A and if the set of common zeros of f1, f2, . . . , ft is finite and nonempty, then the following statements are equivalent :

1

There is a cubature formula of degree d for the integral I which has as points the common zeros of f1, f2, . . . , ft. These zeros may be multiple, leading to the use of function derivatives in the cubature formula.

2

fi is d-orthogonal for I, i = 1, 2, . . . , t. 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 = 2k − 1 ˜ Nmin = k(k + 1) 2 + k 2

• = dim P2

k−1 +

k 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 yλ(x − y)µ(1 − x)ν (Berens and Schmid) Cubature formulae with ˜ Nmin points do not always exist. But ˜ Nmin 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

−1

(1 − x2)α(1 − y2)αf(x, y)dxdy for α = ±0.5 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 ˜ Nmin points → Explicit expression for the points and weights

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion

Möller’s bound is known to be sharp for . . .

1

−1

1

−1

(1 − x2)α(1 − y2)αf(x, y)dxdy for α = ±0.5 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 ˜ Nmin points → Explicit expression for the points and weights

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion

Application for this special integral and cubature formulae

Chebyshev series → Clenshaw-Curtis integration

Chebyshev polynomials ˆ T0(x) := 1 and ˆ Th(x) := √ 2 cos (h arccos (x)). product Chebyshev polynomials ˆ Th of degree |h| := s

r=1 |hr|

defined as ˆ Th(x) := s

r=1 ˆ

Thr(xr). These are orthogonal on Cs := [−1, 1]s with respect to the scalar product with weight function ω(x) := π−s s

r=1(1 − x2 r)− 1

2 :

f1, f2ω :=

• Cs

f1(x) f2(x) ω(x) dx.

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion

An approximation of a function f can be obtained by Ps

n[f] :=

• h,|h|≤n

αh ˆ Th where αh := f, ˆ Thω. In practice, instead of evaluating the integrals from the continuous scalar product, one uses a cubature rule with nodes xℓ and corresponding weights wℓ αh ≈

• ℓ

wℓ f(xℓ) ˆ Th(xℓ)

• g(xℓ)

.

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion

Chebyshev lattices

Definition A s-dimensional rank-k Chebyshev lattice is described by the s-dimensional non-zero integer generating vectors z1, . . . , zk and z∆, positive integer denominators d1, . . . , dk and d∆

χ :=

• cos
• π

ℓ1 z1 d1 + · · · + ℓk zk dk + z∆ d∆

• , with ℓ1, . . . , ℓk ∈ Z
• .

Definition A Chebyshev lattice rule is a cubature rule with nodes xℓ from a Chebyshev lattice and weights wℓ defined as wℓ = ˜ wℓ ˜ W , where ˜ wℓ =

s

• r=1

1

2

φ(|xℓ,r|=1) and ˜ W =

• ℓ

˜ wℓ.

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion

Known 2D formulae fit into new framework

Many known cubature formulae for [−1, 1]2 with Chebyshev weight function fit into the Chebyshev lattice format. k = 2 (full rank) k = 1 s = 2 Morrow & Patterson (1978) Verlinden, C., Roose, Haegemans (1988) Padua (2005)

• C. & Schmid (1989)

Padua (2005) ≡ Caliari, De Marchi, Vianello (2005) + Bos, Caliari, De Marchi, Vianello, Xu (2006)

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion

Also for higher dimensions

and it allows to construct new cubature formulae k = s (full rank) k ≤ s − 1 s = 3 Noskov (1991) Godzina (1994) Poppe & C. (2011) Marchi, Vianello & Xu (2009) s = 4, 5 Godzina (1994) Poppe & C. (2011) s Godzina (1994)

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion

Conclusion

Chebyshev lattices provide a framework for many existing cubature formulae (minimal and non-minimal) an alternative way for construction of cubature formulae, not tight to lower bounds a direct route to extending Clenshaw-Curtis integration to 2 and more variables

easy representation of points, and FFT based implementations

→ extension of Chebfun (Trefethen et al.) to 2 and more variables

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion

Conclusion

Chebyshev lattices provide a framework for many existing cubature formulae (minimal and non-minimal) an alternative way for construction of cubature formulae, not tight to lower bounds a direct route to extending Clenshaw-Curtis integration to 2 and more variables

easy representation of points, and FFT based implementations

→ extension of Chebfun (Trefethen et al.) to 2 and more variables

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion

References:

• R. Cools & K. Poppe, Chebyshev lattices, a unifying framework

for cubature with the Chebyshev weight function, BIT Numerical Mathematics 51, 275–288 (2011)

• K. Poppe & R. Cools, In search of good Chebyshev lattices,

Proceedings of the Monte Carlo and Quasi-Monte Carlo Conference 2010, to appear (2011)

• K. Poppe & R. Cools, CHEBINT: a Matlab/Octave toolbox for

multivariate integration and interpolation based on Chebyshev approximation over a hypercube, in preparation

Introduction Polynomials–Ideals–Minimal cubature formulae Chebyshev series Chebyshev lattices Conclusion

Happy 70th birthday Claude & Sebastiano !