Integration on manifolds by mapped low-discrepancy points and greedy minimal k s -energy points 1 Stefano De Marchi Department of Mathematics - University of Padova IBC 2016 - Bedlewo (Poland) August 29- September 2, 2016 1 joint work with G. Elefante (Fribourg - CH)
Outline Motivations and aims 1 2 Preserving measure maps (on 2-manifolds) Minimal Riesz-energy points 3 Greedy minimal Riesz-energy points Numerical results 4 2 of 33
Motivations and aims 3 of 33
Facts Integrate functions on manifolds by QMC method: low-discrepancy points (Sobol, Hammersley, Fibonacci lattices,...) N � f ( x ) d x ≈ 1 � f ( x i ) , x i ∈ X N X i = 1 4 of 33
Facts Integrate functions on manifolds by QMC method: low-discrepancy points (Sobol, Hammersley, Fibonacci lattices,...) N � f ( x ) d x ≈ 1 � f ( x i ) , x i ∈ X N X i = 1 Low-discrepancy points are uniformly distributed w.r.t. Lebegsue measure d x 4 of 33
Facts Integrate functions on manifolds by QMC method: low-discrepancy points (Sobol, Hammersley, Fibonacci lattices,...) N � f ( x ) d x ≈ 1 � f ( x i ) , x i ∈ X N X i = 1 Low-discrepancy points are uniformly distributed w.r.t. Lebegsue measure d x Take � 1 f ( x ) d H d ( x ) H d ( M ) M where M is a d -dimensional manifold and H d is the d -dimensional Hausdorff measure. Here we can again use QMC 4 of 33
Facts Integrate functions on manifolds by QMC method: low-discrepancy points (Sobol, Hammersley, Fibonacci lattices,...) N � f ( x ) d x ≈ 1 � f ( x i ) , x i ∈ X N X i = 1 Low-discrepancy points are uniformly distributed w.r.t. Lebegsue measure d x Take � 1 f ( x ) d H d ( x ) H d ( M ) M where M is a d -dimensional manifold and H d is the d -dimensional Hausdorff measure. Here we can again use QMC Poppy-seed Bagel Theorem (PsB) [Hardin, Saff 2004]: “minimal Riesz s-energy points, under some assumptions, are uniformly distributed with respect to the Hausdorff measure H d ” 4 of 33
Aims Observe that from the (PsB)-Theorem, the Riesz s -energy points can be useful for a QMC method when using the Hausdorff measure 5 of 33
Aims Observe that from the (PsB)-Theorem, the Riesz s -energy points can be useful for a QMC method when using the Hausdorff measure Ideas Find a measure preserving map so that we can map low discrepancy poins to points nearly uniformly distributed wrt the Hausdorff measure on the manifold Extract approximate minimal Riesz s -energy points from a suitable discretization of the manifold Compare results 5 of 33
Preserving measure maps (on 2-manifolds) 6 of 33
Error bound on bounded domains and discrepancy Theorem (Zaremba 1970) Let B ⊆ [ 0 , 1 ) d be a convex d-dimensional subset and f a function with bounded variation V ( f ) on [ 0 , 1 ) d in the sense of Hardy and Krause. Then, for any points set P = { x 1 , . . . , x N } ⊆ [ 0 , 1 ) d , we have that � � � � � � N � � 1 � � � � f ( x i ) − f ( x ) d x ≤ ( V ( f ) + | f ( 1 ) | ) J N ( P ) , (1) � � � � N � � B � i = 1 � � � x i ∈ B where 1 = ( 1 , . . . , 1 ) . � ����� �� ����� � d where J N ( P ) is the isotropic discrepancy of the points set P defined as J N ( P ) = D N ( C ; P ) with C a family of all convex subsets of [ 0 , 1 ) d and D N ( C ; P ) the classical discrepancy of the set P . 7 of 33
Error bound on manifolds and discrepancy Theorem (Brandolini et al. JoC 2013) Let M be a smooth compact manifold with a normalized measure d x. Fix k = 1 , ϕ k : [ 0 , 1 ) d → M , and a smooth partition a family of local charts { ϕ k } K of unity { ψ k } K k = 1 subordinate to these charts. Then, there exists c > 0 depending only on the local charts ( not on the function f and the measure µ ), � � � � � � � f ( y ) d µ ( y ) � ≤ c D ( µ ) || f || W d , 1 ( M ) , (2) � � � � � M � � � where D ( µ ) = sup U ∈A � U d µ ( y ) � � , A is the collection of all intervals in M � and �� � 1 / p � � p ∂ α � � � � � � || f || W n , p ( M ) = ∂ x α ( ψ k ( ϕ k ( x )) f ( ϕ k ( x ))) d x . � � � � [ 0 , 1 ) d 1 ≤ k ≤ K | α |≤ n 8 of 33
Error bound on manifolds and discrepancy Theorem (Brandolini et al. JoC 2013) Let M be a smooth compact manifold with a normalized measure d x. Fix k = 1 , ϕ k : [ 0 , 1 ) d → M , and a smooth partition a family of local charts { ϕ k } K of unity { ψ k } K k = 1 subordinate to these charts. Then, there exists c > 0 depending only on the local charts ( not on the function f and the measure µ ), � � � � � � � f ( y ) d µ ( y ) � ≤ c D ( µ ) || f || W d , 1 ( M ) , (2) � � � � � M � � � where D ( µ ) = sup U ∈A � U d µ ( y ) � , A is the collection of all intervals in M � � and �� � 1 / p � � p ∂ α � � � � � � || f || W n , p ( M ) = ∂ x α ( ψ k ( ϕ k ( x )) f ( ϕ k ( x ))) d x . � � � � [ 0 , 1 ) d 1 ≤ k ≤ K | α |≤ n Notice: if d µ = 1 � x ∈ X N δ x − d x in (2), we have the analogue of the N Koksma-Hlawka inequality for manifolds 8 of 33
On M = S 2 It is not easy to compute an estimate of the error using the previous inequality If M = S 2 [Marques et al. 2013] observed that minimizing the spherical cap discrepancy (s.c.d.) is equivalent to minimize the w.c.e. (worst case error) � � � � � 1 f ( x ) − 1 � � � � S 2 f ( x ) d σ ( x ) � sup � , � � N 4 π � � f ∈H � x ∈ X N with H a normed function space ( C 0 are ok! polynomials → sperical design). By using the Stolarsky’s invariance principle [Stolarsky ‘73, Brauchard&Dick 2013], the w.c.e is propotional to the distance-based energy metric 1 / 2 4 3 − 1 � E N ( X N ) = | x i − x j | . N 2 x i , x j ∈ X N Then we can maximize the term � x i , x j ∈ X N | x i − x j | instead of the s.c.d. 9 of 33
On M = S 2 It is not easy to compute an estimate of the error using the previous inequality If M = S 2 [Marques et al. 2013] observed that minimizing the spherical cap discrepancy (s.c.d.) is equivalent to minimize the w.c.e. (worst case error) � � � � � 1 f ( x ) − 1 � � � � S 2 f ( x ) d σ ( x ) � sup � , � � N 4 π � � f ∈H � x ∈ X N with H a normed function space ( C 0 are ok! polynomials → sperical design). By using the Stolarsky’s invariance principle [Stolarsky ‘73, Brauchard&Dick 2013], the w.c.e is propotional to the distance-based energy metric 1 / 2 4 3 − 1 � E N ( X N ) = | x i − x j | . N 2 x i , x j ∈ X N Then we can maximize the term � x i , x j ∈ X N | x i − x j | instead of the s.c.d. On different manifolds? 9 of 33
Preserving measure maps on 2-manifolds Idea construct a sequence which is uniformly distributed w.r.t. Hausdorff measure on M , by a preserving measure map. 10 of 33
Preserving measure maps on 2-manifolds Idea construct a sequence which is uniformly distributed w.r.t. Hausdorff measure on M , by a preserving measure map. Letting S = ( X N ) N ≥ 1 uniformly distributed w.r.t. the Lebesgue measure λ on a rectangle U ⊂ R 2 , M a regular manifold of dimension 2 and Φ an invertible map from U to M . Definition Let us consider A ⊂ M . We define the measure µ Φ ( A ) as � µ Φ ( A ) := λ (Φ − 1 ( A )) = (3) d λ. Φ − 1 ( A ) 10 of 33
Preserving measure maps on 2-manifolds Idea construct a sequence which is uniformly distributed w.r.t. Hausdorff measure on M , by a preserving measure map. Letting S = ( X N ) N ≥ 1 uniformly distributed w.r.t. the Lebesgue measure λ on a rectangle U ⊂ R 2 , M a regular manifold of dimension 2 and Φ an invertible map from U to M . Definition Let us consider A ⊂ M . We define the measure µ Φ ( A ) as � µ Φ ( A ) := λ (Φ − 1 ( A )) = (3) d λ. Φ − 1 ( A ) ֒ → Hence the sequence of points Φ( S ) is uniformly distributed with respect to the measure µ Φ (by definition!). 10 of 33
Preserving measure maps on 2-manifolds (cont) 1 Take the measure H 2 on the manifold M which, by means of the area formula [Folland, p. 353] is of the type � g ( x ) d x , (4) U with g a density function (that depends on the parametrization Φ of M ). 11 of 33
Preserving measure maps on 2-manifolds (cont) 1 Take the measure H 2 on the manifold M which, by means of the area formula [Folland, p. 353] is of the type � g ( x ) d x , (4) U with g a density function (that depends on the parametrization Φ of M ). Consider the change of variables from another rectangle U ′ ⊂ R 2 2 U ′ Ψ : −→ U (5) x ′ �→ Ψ( x ′ ) = x , so that g (Ψ( x ′ )) | J Ψ( x ′ ) | = g ( x ) = 1 , (6) 11 of 33
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