Spherical designs and QMC design sequences Valuable tools for - PowerPoint PPT Presentation

Spherical designs and QMC design sequences – Valuable tools for numerical analysis Ian H. Sloan University of New South Wales, Sydney, Australia Joint work with J Brauchart, J Dick, EB Saff, YG Wang and R Womersley

QMC? QMC = Quasi-Monte Carlo Quasi-Monte Carlo rules are integration rules in which all weights are equal. An important special case: 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 Here d σ d (x) is normalised measure on S d . So X N is a spherical t -design if the QMC (i.e. 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 : Given s > d/ 2 , there exists C ( s, d ) > 0 such that for Theorem. every spherical t -design X N on S d there holds � � � � � � 1 � ≤ C ( s, d ) � � f (x) − � f � H s . S d f (x) d σ d (x) � � � t s N x ∈ X N The result holds more generally, for all positive-weight rules that are exact for polynomials of degree up to t .

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 .

A constant × t d is 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).

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 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 .

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? There are many. Here’s one: Theorem. (J Brauchart, EB Saff, IH Sloan, R Womersley, Math Comp, 2014) N that maximize the sum of pairwise A sequence of N -point sets X ∗ 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. 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 strength of the sequence of QMC designs ( 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 .

Worst case error � N � Let Q [ X N ]( f ) := 1 f (x j ) ≈ S d f (x) d σ d (x) . N j =1 Then the worst case error in H s ( S d ) is defined by: wce( Q [ X N ]; H s ( S d )) �� � � � � � � � : f ∈ H s ( S d ) , � f � H s ≤ 1 := sup � Q [ X N ]( f ) − I ( f ) � � , QMC designs defined in terms of WCE: A sequence ( X N ) of N point configurations on S d is a sequence of QMC designs for H s iff there exists c ( s, d ) > 0 such that wce( Q [ X N ]; H s ( S d )) ≤ c ( s, d ) N s/d .

Reproducing kernel for H s ( S d ) , s > d/ 2 Z ( d,ℓ ) ∞ � � a ( s ) K ( s ) (x , y)= Y ℓ,k (x) Y ℓ,k (y) ℓ ℓ =0 k =1 ∞ � a ( s ) Z ( d, ℓ ) P ( d ) = (x · y) , ℓ ℓ ℓ =0 where P ( d ) Legendre polynomial associated with S d . ℓ It is a zonal kernel : i.e. K ( s ) (x , y) = K ( s ) (x · y) . The reproducing kernel properties are easily verified: K ( s ) ( · , x) ∈ H s ( S d ) , x ∈ S d , ( f, K ( s ) ( · , x)) H s = f (x) , x ∈ S d , f ∈ H s ( S d ) .

WCE in terms of the reproducing kernel Recall: For a ( s ) ≍ (1 + ℓ ) − 2 s , ℓ Z ( d,ℓ ) ∞ ∞ a ( s ) a ( s ) Z ( d, ℓ ) P ( d ) K ( s ) (x · y) := � � � Y ℓ,k (x) Y ℓ,k (y) = (x · y) ℓ ℓ ℓ ℓ =0 k =1 ℓ =0 is a reproducing kernel for H s ( S d ) . Theorem N N � � � � 2 = 1 wce( Q [ X N ]; H s ( S d )) K ( s ) (x j · x i ) . N 2 j =1 i =1 Here ∞ � a ( s ) Z ( d, ℓ ) P ( d ) (x · y) = K ( s ) (x · y) − a ( s ) K ( s ) (x · y) := 0 . ℓ ℓ ℓ =1

Optimal QMC designs Recall: K ( s ) (x · y) := K ( s ) (x · y) − a ( s ) 0 . Let X ∗ N = { x ∗ 1 , · · · , x ∗ N } , for N = 2 , 3 , 4 , . . . be a sequence of minimizers of N N � � K ( s ) (x j · x i ) , j =1 i =1 Theorem. (op cit) For all s.d/ 2 there exists c ( s, d ) > 0 such that for all N ≥ 2 N ]; H s ( S d )) ≤ c ( s, d ) wce( Q [ X ∗ N s/d . Consequently, ( X ∗ N ) is a sequence of QMC designs for H s ( S d ) . Proof: Spherical designs with N ≍ t d exist, and satisfy the bounds in the theorem for all s > d/ 2 . The minimizer for a particular s > d/ 2 must be at least as good.

QMC designs from distance kernels � � Let V ( S d ) := S d | x − y | d σ d (x) d σ d (y) . It can be shown that S d K ( d +1) / 2 (x , y) := 2 V ( S d ) − | x − y | is a reproducing kernel for H ( d +1) / 2 ( S d ) , and that correspondingly K ( d +1) / 2 (x , y) = V ( S d ) − | x − y | . Thus   1 / 2 N N � � 1  V ( S d ) −  wce( Q [ X N ]; H ( d +1) / 2 ( S d )) = | x j − x i | N 2 j =1 i =1 is the corresponding WCE in H ( d +1) / 2 ( S d ) . Corollary. A sequence of N -point sets X ∗ N that maximize the sum of Euclidean distances is a sequence of QMC designs for H ( d +1) / 2 ( S d ) .

Generalized distance kernels In the same way, a kernel for Sobolev spaces with s ∈ ( d/ 2 , d/ 2 + 1) is given by (Hubbert & Baxter 2012, Brauchart & Womersley 201?) K ( s ) gd (x , y) := 2 V 2 s − d ( S d ) − | x − y | 2 s − d where � � S d | x − y | 2 s − d d σ d (x) d σ d (y) . V 2 s − d := V 2 s − d ( S d ) := S d