Point sets of minimal energy Peter Grabner Institut fr Analysis und - PowerPoint PPT Presentation

Point sets of minimal energy Peter Grabner Institut für Analysis und Computational Number Theory Graz University of Technology October 16, 2013 Peter Grabner Point sets of minimal energy

Self-organisation by local interaction Self-organisation by local interaction is a far-reaching principle occurring in Biology, Chemistry, Physics. . . . It explains inter alia Peter Grabner Point sets of minimal energy

Self-organisation by local interaction Self-organisation by local interaction is a far-reaching principle occurring in Biology, Chemistry, Physics. . . . It explains inter alia the distribution of electrons on a surface (Thomson’s problem) the distribution of pollen grains (Tammes’ problem) the structure of polymers structure of ground states of particle systems viral morphology the arrangement of colloidal particles the structure of fullerenes Peter Grabner Point sets of minimal energy

Self-organisation by local interaction Self-organisation by local interaction is a far-reaching principle occurring in Biology, Chemistry, Physics. . . . It explains inter alia the distribution of electrons on a surface (Thomson’s problem) the distribution of pollen grains (Tammes’ problem) the structure of polymers structure of ground states of particle systems viral morphology the arrangement of colloidal particles the structure of fullerenes Idea: use this principle for generating Quasi-Monte Carlo point sets on manifolds, especially spheres. Peter Grabner Point sets of minimal energy

“Even” point distributions on the sphere The question of how to distribute N points “evenly” on the sphere has many important applications, such as Peter Grabner Point sets of minimal energy

“Even” point distributions on the sphere The question of how to distribute N points “evenly” on the sphere has many important applications, such as sampling functions on spheres Peter Grabner Point sets of minimal energy

“Even” point distributions on the sphere The question of how to distribute N points “evenly” on the sphere has many important applications, such as sampling functions on spheres integrating functions over spherical domains Peter Grabner Point sets of minimal energy

“Even” point distributions on the sphere The question of how to distribute N points “evenly” on the sphere has many important applications, such as sampling functions on spheres integrating functions over spherical domains solving PDEs by discretisation Peter Grabner Point sets of minimal energy

“Even” point distributions on the sphere The question of how to distribute N points “evenly” on the sphere has many important applications, such as sampling functions on spheres integrating functions over spherical domains solving PDEs by discretisation sampling spacial directions Peter Grabner Point sets of minimal energy

Pollen grains Peter Grabner Point sets of minimal energy

Potential function of a point distribution Peter Grabner Point sets of minimal energy

Quantify evenness For every point set X N = { x 1 , . . . , x N } of distinct points, we assign several qualitative measures that describe aspects of even distribution. Peter Grabner Point sets of minimal energy

Quantify evenness For every point set X N = { x 1 , . . . , x N } of distinct points, we assign several qualitative measures that describe aspects of even distribution. Then we can try to minimise or maximise these measures for given N . Peter Grabner Point sets of minimal energy

Combinatorial measures discrepancy � N � 1 � � � D N ( X N ) = sup χ C ( x n ) − σ ( C ) � � N � � C � � n = 1 Peter Grabner Point sets of minimal energy

Combinatorial measures discrepancy � N � 1 � � � D N ( X N ) = sup χ C ( x n ) − σ ( C ) � � N � � C � � n = 1 dispersion δ N ( X N ) = sup x ∈ S d min k | x − x k | Peter Grabner Point sets of minimal energy

Combinatorial measures discrepancy � N � 1 � � � D N ( X N ) = sup χ C ( x n ) − σ ( C ) � � N � � C � � n = 1 dispersion δ N ( X N ) = sup x ∈ S d min k | x − x k | separation ∆ N ( X N ) = min i � = j | x i − x j | Peter Grabner Point sets of minimal energy

Analytic measures error in numerical integration � N � � � � � I N ( f , X N ) = f ( x n ) − S d f ( x ) d σ d ( x ) � � � � � � n = 1 Peter Grabner Point sets of minimal energy

Analytic measures error in numerical integration � N � � � � � I N ( f , X N ) = f ( x n ) − S d f ( x ) d σ d ( x ) � � � � � � n = 1 Worst-case error for integration in a normed space H : I N ( X N , H ) = sup I N ( f , X N )) , f ∈ H � f � = 1 Peter Grabner Point sets of minimal energy

Analytic measures error in numerical integration � N � � � � � I N ( f , X N ) = f ( x n ) − S d f ( x ) d σ d ( x ) � � � � � � n = 1 Worst-case error for integration in a normed space H : I N ( X N , H ) = sup I N ( f , X N )) , f ∈ H � f � = 1 (generalised) energy: N N � � ˜ E g ( X N ) = g ( � x i , x j � ) = g ( � x i − x j � ) , i , j = 1 i , j = 1 i � = j i � = j where g denotes a positive definite function. Peter Grabner Point sets of minimal energy

Other concepts designs: N 1 � � f ( x n ) = S d f ( x ) d σ ( x ) N n = 1 for all polynomials of degree ≤ t . Peter Grabner Point sets of minimal energy

Other concepts designs: N 1 � � f ( x n ) = S d f ( x ) d σ ( x ) N n = 1 for all polynomials of degree ≤ t . L 2 -discrepancy: 2 � π � N � � 1 � � � χ C ( x , t ) ( x n ) − σ d ( C ( x , t )) d σ d ( x ) dt � � � N � S d 0 � � n = 1 Peter Grabner Point sets of minimal energy

Discrepancy Discrepancy is the most classical measure for the difference of two distributions � N � 1 � � � D N ( X N ) = sup χ C ( x n ) − σ ( C ) � . � � � N � C � n = 1 Peter Grabner Point sets of minimal energy

Discrepancy Discrepancy is the most classical measure for the difference of two distributions � N � 1 � � � D N ( X N ) = sup χ C ( x n ) − σ ( C ) � . � � � N � C � n = 1 It is rather difficult to compute explicitly, even for moderate values of N . Peter Grabner Point sets of minimal energy

Estimates for discrepancy Thus estimates for D N ( X N ) are of interest (PG 1991, X.-J. Li & J. Vaaler, 1999), Erdős-Turán type inequality   Z ( d , k ) M � N �  1 1 1 � � � � � D C N ( X N ) ≤ C d M + Y k , j ( x n ) � �  k N � � � � k = 1 j = 1 n = 1 Peter Grabner Point sets of minimal energy

Estimates for discrepancy Thus estimates for D N ( X N ) are of interest (PG 1991, X.-J. Li & J. Vaaler, 1999), Erdős-Turán type inequality   Z ( d , k ) M � N �  1 1 1 � � � � � D C N ( X N ) ≤ C d M + Y k , j ( x n ) � �  k N � � � � k = 1 j = 1 n = 1 F. J. Narcowich, X. Sun, J. D. Ward, and Z. Wu (2010), LeVeque-type inequality: 1 � 2   Z ( d ,ℓ ) d + 2 ∞ � N 1 � � � ℓ − ( d + 1 ) D ( X N ) ≤ B ( d ) Y ℓ, m ( x n ) .   N m = 1 n = 1 ℓ = 0 Peter Grabner Point sets of minimal energy

Irregularities On the other hand the theory of irregularities of distributions developed by K. F. Roth, W. Schmidt, J. Beck, W. Chen, . . . gives a lower bound D N ( X N ) ≥ CN − 1 2 − 1 2 d . Peter Grabner Point sets of minimal energy

Irregularities On the other hand the theory of irregularities of distributions developed by K. F. Roth, W. Schmidt, J. Beck, W. Chen, . . . gives a lower bound D N ( X N ) ≥ CN − 1 2 − 1 2 d . This is essentially best possible. Namely, for every N there exists a point set X N such that DN ( X N ) ≤ CN − 1 2 − 1 2 d log N . The construction of this point set is probabilistic. Peter Grabner Point sets of minimal energy

Energy Depending on the behaviour of the positive definite function g at 1, the corresponding energy-functionals are categorised Peter Grabner Point sets of minimal energy

Energy Depending on the behaviour of the positive definite function g at 1, the corresponding energy-functionals are categorised if g ( 1 ) exists, the energy functional N � E g ( X N ) = g ( � x i , x j � ) i , j = 1 is called non-singular Peter Grabner Point sets of minimal energy

Energy Depending on the behaviour of the positive definite function g at 1, the corresponding energy-functionals are categorised if g ( 1 ) exists, the energy functional N � E g ( X N ) = g ( � x i , x j � ) i , j = 1 is called non-singular if g ( 1 − t ) = O ( t − s / 2 ) for 0 ≤ s < d the energy functional N � E g ( X N ) = g ( � x i , x j � ) i , j = 1 i � = j is called singular . Peter Grabner Point sets of minimal energy