Approximate Fekete points and discrete Leja points based on equal area partitions of the unit sphere

Transcription

1 Approximate Fekete points and discrete Leja points based on equal area partitions of the unit sphere Paul Leopardi Mathematical Sciences Institute, Australian National University. For presentation at Computational Techniques and Applications Conference Joint work with Alvise Sommariva and Marco Vianello. Made possible by donations from 31 supporters via Pozible. 2 December 2014

2 Acknowledgements My collaborators: Alvise Sommariva and Marco Vianello. Rob Womersley. Program on Minimal Energy Point Sets, Lattices, and Designs at the Erwin Schrödinger International Institute for Mathematical Physics, Australian National University.

3 Pozible supporters $32+ SvA, AD, CF, OF, KM, JP, AHR, RR, ES, MT. $50+ Yvonne Barrett, Angela M. Fearon, Sally Greenaway, Dennis Pritchard, Susan Shaw, Bronny Wright. $64+ Naomi Cole. $100+ Russell family, Jonno Zilber. $128+ Jennifer Lanspeary, Vikram.... and others, who did not want acknowledgement.

4 Outline of talk The EQ spherical codes Approximately optimal interpolating sets Results for the EQ spherical codes

5 The partition EQ(2,33) on S 2 R 3 EQ partitions: Recursive Zonal Equal Area partitions of the sphere, EQ(d, N ) = S d, with EQ(d, N ) = N.

6 The spherical code EQP(2,33) on S 2 EQ codes: The Recursive Zonal Equal Area spherical codes, EQP(d, N ) S d, with EQP(d, N ) = N.

7 Equal-area partitions of S d R d An equal area partition of S d R d is a finite set P of Lebesgue measurable subsets of S d, such that R = S d, and for each R P, R P λ d (R) = λ d(s d ), P where λ d is the Lebesgue area measure on S d.

8 Diameter bounded sets of partitions The diameter of a region R R d+1 is defined by diam R := sup{ x y x, y R}. A set Ξ of partitions of S d R d+1 is diameter-bounded with diameter bound K R + if for all P Ξ, for each R P, diam R K P 1/d.

9 Key properties of the EQ partition of S d EQ(d, N ) is the recursive zonal equal area partition of S d into N regions. The set of partitions EQ(d) := {EQ(d, N ) N N + }. The EQ partition satisfies: Theorem 1 For d 1, N 1, EQ(d, N ) is an equal-area partition. Theorem 2 For d 1, EQ(d) is diameter-bounded.

10 Geometric properties For EQP(d, N ) Good: Centre points of regions of diameter = O(N 1/d ), Mesh norm (covering radius) = O(N 1/d ), Minimum distance and packing radius = Ω(N 1/d ). Bad: Mesh ratio = Ω( d), Packing density π d/2 2 d Γ(d/2+1) as N.

11 Approximation properties Not so bad? Normalized spherical cap discrepancy = O(N 1/d ), Ugly: Normalized s -energy I s ± O(N 1/d ) 0 < s < d 1 I s ± O(N 1/d log N ) s = d 1 E s = I s ± O(N s/d 1 ) d 1 < s < d O(log N ) s = d O(N s/d 1 ) s > d. Cannot be used for polynomial interpolation: proven for large enough N, conjectured for small N.

12 EQ(3,99) Steps 1 to 2 EQ(3,99) Steps 3 to 5 θ F,1 θ c I y 1 = θ F,2 V(θ c ) = V R = σ(s 3 )/99 I = V R 1/3 y 2 = y 3 = F F F F θ F,3 y 4 = θ F,4 θ F,5 EQ(3,99) Steps 6 to 7 θ 1 m 1 = 15 θ 2 m 2 = 34 m 3 = θ 3 m 4 = 15 θ 4 θ 5

13 Admissible meshes Definition 3 For compact D R d, and C(D) the space of continuous functions on D, given a sequence of finite dimensional subspaces P t (D) C(D), a P t -norming mesh is a sequence (Z t ) of finite subsets of D such that p c sup z Z t p(z) for all p P t. For a P t -admissible mesh, P t (D) is the space of polynomials of maximum degree t on D, and the cardinality Z t = O(t s ) for some s 1. (Calvi and Levenberg 2008, Vianello 2013)

14 Approximate Fekete points Given a P t -admissible mesh with n t := Z t d t := dim(p t (D)), points z 1,..., z nt Z t, and a basis {p 1,..., p dt } of P t (D), the approximate Fekete points of order t are a subset of Z t with cardinality d t, obtained from the Vandermonde matrix A t := [p i (z j )] via QR decomposition with column pivoting. They approximate the maximal determinant Fekete points by having a large Vandermonde determinant and a small Lebesgue constant of interpolation. (Sommariva and Vianello 2009)

15 Approximate Fekete points The approximate Fekete points z af are obtained from the points z and corresponding Vandermonde matrix A as dim = rows(a); y = zeros(dim, 1); y(1) = 1; [Q, R, P] = qr(a); w = P(:, 1:dim) * (R(:, 1:dim) \ (Q * y)); z af = z(:, abs(w) > tol); where w is the optimal quadrature weight vector and tol is a small tolerance. Here all elements of w are non-zero, but z af may have dimension less than dim. (Sommariva and Vianello 2009)

16 Discrete Leja points For the Discrete Leja points, LU decomposition with partial row pivoting is used instead. dim = rows(a); y = zeros(dim, 1); y(1) = 1; [L, U, p] = lu(a, vector ); w = zeros(n, 1); w(p(1:dim)) = L(1:dim, :) \ (U \ y); z dl = z(:, p(1:dim)); Here z dl has dimension dim, but some elements of w may be zero. (Bos, De Marchi, Sommariva and Vianello 2010)

17 Non-negative least squares points We can also use non-negative least squares instead of either QR or LU decomposition. dim = rows(a); y = zeros(dim, 1); y(1) = 1; w = lsqnonneg(a, y); z nn = z(:, abs(w) > tol); Here all elements of w are positive, but z nn may have dimension less than dim. (Sommariva and Vianello 2014)

18 The EQ codes form an admissible mesh Theorem 4 The EQ codes form a P t -admissible mesh. Proof. Any finite point set on the unit sphere S d with mesh norm at most (1 c)/t generates a norming set with constant c for P t. The EQ spherical codes have mesh norm at most C d N 1/d. Thus if N (C d /(1 c)) d t d, then EQP(d, N ) is a norming set with constant c for P t. (Jetter, Stöckler and Ward, 1998; L and Vianello 2014)

19 The Fekete points on the sphere S 2 The Fekete (maximal determinant) points on the sphere S 2 are the points that maximize the determinant of the Vandermonde-type matrix A t := [p t,i (x j )], where i, j {1,..., (t + 1) 2 }, the p t,i form an orthonormal basis of the spherical polynomials of degree at most t, and x j S 2. Rob Womersley has (approximately) computed these points up to degree t = 165, and their corresponding optimal quadrature weights, as well as the log of the determinant of the Gram matrix G t := A T t A t. (Sloan and Womersley 2004; Womersley 2007)

20 The search algorithm For each type of point set (approximate Fekete points, discrete Leja points, non-negative least squares points), for t from 1 to 15, I used Octave with the EQ codes EQP(2, N ) for N from (t + 1) 2 to (t + 1) 3 to find: The smallest N such that the matrix A has full rank, and the point set has all corresponding weights non-zero. The smallest N such that the matrix A has full rank, and the point set has all corresponding weights positive. The value of N such that the matrix A has full rank, the point set has all corresponding weights positive, and the Gram determinant is maximal.

21 Approximate Fekete points Log of Gram determinant Non-zero wt Positive wt Maximum det Womersley Degree t

22 Discrete Leja points Log of Gram determinant Non-zero wt Positive wt Maximum det Womersley Degree t

23 Non-negative least squares points Log of Gram determinant Positive wt Maximum det Womersley Degree t

24 Maximum determinant positive weight points Relative log of Gram determinant Degree t Approx Fekete Discrete Leja NNLS

25 Left to do Numerical examples for larger degree t. Searching can be done in parallel. Prove that for some T > 0, for all t > T, for sufficiently large N, the EQ codes EQP(2, N ) yield positive weights for each of the 3 types of points. Estimate the N required. Investigate properties (mesh norm, discrepancy, energy) of the resulting point sets.