QMC Designs on the Sphere Ian H. Sloan University of New South - PowerPoint PPT Presentation

QMC Designs on the Sphere Ian H. Sloan University of New South Wales, Sydney, Australia Uniform distribution and QMC Methods, RICAM 2013 Joint with J Brauchart, EB Saff and R Womersley

QMC designs Definition. A sequence of point sets ( X N ) ⊂ S d with N → ∞ is a sequence of QMC designs for the Sobolev space H s ( S d ) , for some s > d/ 2 , if there exists c ( s, d ) > 0 , such that for all f ∈ H s ( S d ) � � � � � � 1 � ≤ c ( s, d ) � � f (x) − S d f (x) d σ d (x) N s/d � f � H s . � � � N x ∈ X N Here d σ d (x) is normalised measure on S d .

QMC designs Definition. A sequence of point sets ( X N ) ⊂ S d with N → ∞ is a sequence of QMC designs for the Sobolev space H s ( S d ) , for some s > d/ 2 , if there exists c ( s, d ) > 0 , such that for all f ∈ H s ( S d ) � � � � � � 1 � ≤ c ( s, d ) � � f (x) − S d f (x) d σ d (x) N s/d � f � H s . � � � N x ∈ X N Here d σ d (x) is normalised measure on S d . This is the optimal rate of convergence in H s ( S d ) K Hesse and IHS 2005 for d = 2 , Hesse 2006 for general d .

QMC designs Definition. A sequence of point sets ( X N ) ⊂ S d with N → ∞ is a sequence of QMC designs for the Sobolev space H s ( S d ) , for some s > d/ 2 , if there exists c ( s, d ) > 0 , such that for all f ∈ H s ( S d ) � � � � � � 1 � ≤ c ( s, d ) � � f (x) − S d f (x) d σ d (x) N s/d � f � H s . � � � N x ∈ X N Here d σ d (x) is normalised measure on S d . This is the optimal rate of convergence in H s ( S d ) K Hesse and IHS 2005 for d = 2 , Hesse 2006 for general d . The idea grew from properties of spherical designs .

Spherical designs Definition : A spherical t -design on S d ⊂ R d +1 is a set X N := { x 1 , . . . , x N } ⊂ S d such that � N � 1 p (x j ) = p ( x ) d σ d (x) ∀ p ∈ P t . N j =1 S d

Spherical designs Definition : A spherical t -design on S d ⊂ R d +1 is a set X N := { x 1 , . . . , x N } ⊂ S d such that � N � 1 p (x j ) = p ( x ) d σ d (x) ∀ p ∈ P t . N j =1 S d So X N is a spherical t -design if the equal weight cubature rule with these points integrates exactly all polynomials of degree ≤ t .

A spherical 50 -design 2 51 2 points. This is a Womersley spherical 50 -design with 1302 ≈ 1

Spherical designs are good for integration Spherical designs are tools for numerical integration . The following theorem (Hesse & IHS, 2005, 2006) shows a good rate of convergence for sufficiently smooth functions f : Theorem. Given s > d/ 2 , there exists C ( s, d ) > 0 such that for every spherical t -design X N on S d there holds � � � � � � 1 � ≤ C ( s, d ) � � f (x) − S d f (x) d σ d (x) � f � H s . � � � t s N x ∈ X N

How many points for a spherical t -design? It is known (Seymour & Zaslavsky, 1984) that for every t ≥ 1 (and for every dimension of the sphere) there always exists a spherical design. But how many points does a spherical t -design need? There is no possible upper bound because ...

How many points for a spherical t -design? It is known (Seymour & Zaslavsky, 1984) that for every t ≥ 1 (and for every dimension of the sphere) there always exists a spherical design. But how many points does a spherical t -design need? There is no possible upper bound because ... Delsarte, Goethals, Seidel (1977) established lower bounds of exact order t d : � � � �  d + t/ 2 − 1 d + t/ 2   + if t is even ,  d d � � N ≥  d + ⌊ t/ 2 ⌋   if t is odd , d

How many points for a spherical t -design? It is known (Seymour & Zaslavsky, 1984) that for every t ≥ 1 (and for every dimension of the sphere) there always exists a spherical design. But how many points does a spherical t -design need? There is no possible upper bound because ... Delsarte, Goethals, Seidel (1977) established lower bounds of exact order t d : � � � �  d + t/ 2 − 1 d + t/ 2   + if t is even ,  d d � � N ≥  d + ⌊ t/ 2 ⌋   if t is odd , d Yudin (1997) established larger lower bounds, still of exact order t d .

Is (constant × t d ) enough points? It has long been conjectured that c d t d points is enough, for some c d > 0 , but until very recently there was no proof. Recently Bondarenko, Radchenko and Viazovska (Annals of Mathematics, 2013) proved this important existence result.

Is (constant × t d ) enough points? It has long been conjectured that c d t d points is enough, for some c d > 0 , but until very recently there was no proof. Recently Bondarenko, Radchenko and Viazovska (Annals of Mathematics, 2013) proved this important existence result. But what is the constant? The Bondarenko et al. constant is huge.

Is (constant × t d ) enough points? It has long been conjectured that c d t d points is enough, for some c d > 0 , but until very recently there was no proof. Recently Bondarenko, Radchenko and Viazovska (Annals of Mathematics, 2013) proved this important existence result. But what is the constant? The Bondarenko et al. constant is huge. For S 2 Chen, Frommer and Lang (2011) proved that ( t + 1) 2 points is enough for all t up to 100 . For S 2 , we believe that ( t + 1) 2 points is enough for all t . 2 ( t + 1) 2 seems to be enough (R. Womersley, private Even N ≈ 1 communication).

Efficient spher. designs are QMC designs Clearly, every sequence of spherical t -designs is a sequence of QMC designs for H s ( S d ) , for all s > d/ 2 , iff N ≍ t d as t → ∞ , since Theorem. Given s > d/ 2 , there exists C ( s, d ) > 0 such that for every spherical t -design X N on S d there holds � � � � � � 1 � ≤ C ( s, d ) � � f (x) − S d f (x) d σ d (x) � f � H s . � � � t s N x ∈ X N If N ≍ t d this gives � � � � � � 1 � ≤ c ( s, d ) � � f (x) − S d f (x) d σ d (x) N s/d � f � H s . � � � N x ∈ X N

Are there other QMC designs? We think there are many. Here’s one that is certain: Theorem. (J Brauchart, EB Saff, IH Sloan, R Womersley, Math Comp, to appear) A sequence of N -point sets X ∗ N that maximize the sum of pairwise Euclidean distances is a sequence of QMC designs for H ( d +1) / 2 ( S d ) .

Are there other QMC designs? We think there are many. Here’s one that is certain: Theorem. (J Brauchart, EB Saff, IH Sloan, R Womersley, Math Comp, to appear) A sequence of N -point sets X ∗ N that maximize the sum of pairwise Euclidean distances is a sequence of QMC designs for H ( d +1) / 2 ( S d ) . Thus for S 2 the points that maximize the sum of Euclidean distances form a sequence of QMC designs for H 3 / 2 .

Are there other QMC designs? We think there are many. Here’s one that is certain: Theorem. (J Brauchart, EB Saff, IH Sloan, R Womersley, Math Comp, to appear) A sequence of N -point sets X ∗ N that maximize the sum of pairwise Euclidean distances is a sequence of QMC designs for H ( d +1) / 2 ( S d ) . Thus for S 2 the points that maximize the sum of Euclidean distances form a sequence of QMC designs for H 3 / 2 . To prove this and other things we need some machinery. But first:

The nested property of QMC designs Theorem. (Brauchart, Saff, IHS, Womersley, op. cit.) Given s > d/ 2 , let ( X N ) be a sequence of QMC designs for H s ( S d ) . Then ( X N ) is a sequence of QMC designs for all coarser H s ′ ( S d ) , i.e. for all s ′ satisfying d/ 2 < s ′ ≤ s . This result isn’t trivial – for the smaller set H s ( S d ) we demand faster convergence.

The nested property of QMC designs Theorem. (Brauchart, Saff, IHS, Womersley, op. cit.) Given s > d/ 2 , let ( X N ) be a sequence of QMC designs for H s ( S d ) . Then ( X N ) is a sequence of QMC designs for all coarser H s ′ ( S d ) , i.e. for all s ′ satisfying d/ 2 < s ′ ≤ s . This result isn’t trivial – for the smaller set H s ( S d ) we demand faster convergence. So there is some upper bound on the admissible values of s : s ∗ := sup { s : ( X N ) is a sequence of QMC designs for H s } . We call s ∗ the QMC index of the sequence ( X N ) .

Generic QMC designs If s ∗ = + ∞ , we say the sequence of QMC designs is “generic”. Every sequence of spherical t -designs with N ≍ t d as t → ∞ is a generic sequence of QMC designs. We don’t know if there are other interesting examples.

The Sobolev space H s ( S d ) With λ ℓ := ℓ ( ℓ + d − 1) ( λ ℓ is the ℓ th eigenvalue of − ∆ ∗ d ) , � � (1 + λ ℓ ) s � � Z ( d,ℓ ) ∞ � � 2 � � � � H s ( S d ) = f ∈ L 2 ( S d ) : f ℓ,k < ∞ � . ℓ =0 k =1 Thus H 0 ( S d ) = L 2 ( S d ) . Here Laplace-Fourier coefficients � � f ℓ,k = ( f, Y ℓ,k ) L 2 ( S d ) = S d f (x) Y ℓ,k (x) d σ d (x) . Y ℓ,k for k = 1 , . . . , Z ( d, ℓ ) is an orthonormal set of spherical harmonics of degree ℓ : ∆ ∗ d Y ℓ,k = − λ ℓ Y ℓ,k .

Norms for H s ( S d ) It is useful to allow also other equivalent norms for H s ( S d ) : Let ( a ( s ) ) ℓ ≥ 0 satisfy ℓ ≍ (1 + λ ℓ ) − s ≍ (1 + ℓ ) − 2 s . a ( s ) ℓ Inner product and norm for f, g ∈ H s ( S d ) Z ( d,ℓ ) ∞ � � 1 � ( f, g ) H s := f ℓ,k � g ℓ,k , a ( s ) ℓ =0 k =1 ℓ � � f � H s := ( f, f ) H s . It is easily seen that H s ( S d ) is embedded in C ( S d ) iff s > d/ 2 .