Generated sets as sampling schemes for hyperbolic cross trigonometric polynomials

Transcription

1 Generated sets as sampling schemes for hyperbolic cross trigonometric polynomials Lutz Kämmerer Chemnitz University of Technology joint work with S. Kunis and D. Potts supported by DFG-SPP / 16

2 Road map Introduction Hyperbolic cross discrete Fourier transform Lattices Generated sets Summary 2 / 16

3 Introduction T [0, 1), f C(T), = 2 n, n, x j = j/ ˆf k = f (x) e 2πikx dx T 1 1 f (x j ) e 2πikx j, j=0 discrete Fourier transform (DFT) f (x j ) = /2 k= /2+1 unitary up to a scaling factor complexity DFT: O ( 2) FFT: O ( log ) k = /2 + 1,..., /2, ˆf k e 2πikx j, j = 0,..., 1 3 / 16

4 Introduction discrete Fourier transform (d-dimensional) f (x) = ˆf k e 2πikx, k Ĝ d Ĝ d = { /2 + 1,..., /2}d Z d x = ( j1,..., j d ) T T d, j 1,..., j d {0,..., 1} unitary up to a scaling factor problem size Ĝ d = d, complexity DFT: O ( 2d) or O ( d+1) FFT: O ( d d log ) problem size and complexity increase strongly with dimension d 4 / 16

5 Hyperbolic cross evaluate trigonometric polynomial f (x) = aim: k H d,γ ˆf k e 2πikx, x X, X H d,γ n stable spatial discretisation & fast algorithm weighted hyperbolic cross (weighted Zaremba cross) H d,γ = {k Zd : d ( max 1, k ) s } γ s s=1 γ = (γ s ) s weight sequence with 1 γ 1 γ (a) = 32, γ = ( 1 2 ) s (b) = 32, γ = ( 1 2 s ) s 5 / 16

6 Lattice based HCFFT (LHCFFT) rank-1 lattice: z d, M x j = jz M mod 1; j = 0,..., M 1 f (x j ) = reformulation as 1-dim. DFT, f = Aˆf k H d,γ ˆf k e 2πikx j = M 1 l=0 k H d,γ kz l (mod M) ˆf k e2πi jkz M M 1 = l=0 jl 2πi ĝ l e M complexity O(M log M + H d,γ ) applying a 1-dim. FFT Aim: stable and unique reconstruction of ˆf k 6 / 16

7 Lattice based HCFFT (LHCFFT) Theorem (Kunis, Potts, K. 2012) Let, d, a sampling scheme X = {x j, j = 0,..., M 1}, and the Fourier matrix A = ( e 2πikx ) j. j=0,...,m 1;k H d,γ 1 X is arbitrary and A A = MI, then M γ 1 γ 2. 2 X is a rank-1 lattice, then A A = MI or A A is singular. cardinality M of a reconstruction lattice at least γ 1 γ 2 for comparison: H d,γ C d,γ(log ) d 1 complexity of a stable LHCFFT Ω( 2 log + H d,γ ) 7 / 16

8 LHCFFT - integration reconstruct Fourier coefficients exactly ˆf k = f (x)e 2πikx dx = ˆf h e 2πi(h k)x dx, k H d,γ T T h H d,γ integrate monomials exactly (also all linear combinations) e 2πilx, l {k Z d : k = h 1 h 2 : h 1, h 2 H d,γ } =: Hd,γ adapt results from numerical integration (C., K.,. 2010) Theorem (K. 2012) There exists { a rank-1 lattice of size ( M max 2 γ 1 + 1; max s=2,...,d H s,γ ) } Hs 1,γ 4 γ s + 4 that allows a stable and unique reconstruction of trigonometric polynomials with frequencies supported on H d,γ. 8 / 16

9 LHCFFT - cardinality of H d,γ Lemma (K. 2012) The cardinality of H d,γ is bounded above by C d,γ ( 2 (log ) d 2 + 2(log ) d 1 ). for comparison: necessary: M γ 1 γ 2 complexity Ω( 2 ) sufficient: M H d,γ complexity O(2 (log ) d 1 ) searching for z d (adapted CBC type algorithm) reducing M while retaining the desired properties 9 / 16

10 LHCFFT - examples = 4, γ = ( 1 2 ) s cardinality H d,γ 4 = 2d 2 + 2d + 1 reconstruction lattice of size M d 8 3 d(d 2 + 2) 4 [ ] M oversampling factor d 4 H d,γ 4 3 d M e.g H 100,γ 4 = 8, γ = ( 1 2 ) s, dimension d = 50 cardinality H 50,γ 8 = cardinality H 50,γ 8 > difficult to manage reconstruction lattice of size M = oversampling factor M H 50,γ Aim: simpler search strategy using continuous optimisation 10 / 16

11 LHCFFT - generalisation generated set: r R d, M x j = jr mod 1, j = 0,..., M 1 1 2/3 reformulation as 1-dim. DFT, f = Aˆf f (x j ) = k H d,γ ˆf k e 2πik x j = y Y k H d,γ kr y (mod 1) 0 0 1/3 2/3 1 ˆf k e2πijy = ĝ y e 2πijy y Y with Y = {k r mod 1; k H d,γ }; A = ( e 2πijk r) j=0,...,m 1;k H d,γ complexity O(M log M + log ɛ H d,γ ) using a 1-dim. FFT 1/3 Aim: stable and unique reconstruction of ˆf k 11 / 16

12 LHCFFT - generalisation matrix elements of the matrices A A 1 M (A A) k,k = 1 M 1 e 2πij(k k) r =: K M (y k y k ) M j=0 y k = k r mod 1, k H d,γ K M 1-periodic univariate Dirichlet kernel K M (y) 2My 1 for y [ 1 2, 1 ] 2 Lemma (Gershgorin circle theorem applied to M 1 A A) The intervall [1 δ, 1 + δ] contains all eigenvalues of M 1 A A, δ := max k H d,γ k H d,γ \{k} K M (y k y k ). 12 / 16

13 LHCFFT - generalisation y k1 y k10 y k20 y k30 y k assume 0 = y k1 < y k2 <... < y k H < 1 d,γ 1 y k H, for j = H d,γ d,γ define t j =, y kj+1 y kj, for 1 j < H d,γ π: permutation arranging 0 < t π(1) t π(2)... t π( H d,γ ) Lemma The maximum Gershgorin radius δ of M 1 A A is bounded above by 1 δ 1 2 Hd,γ ( j ) 1 t M π(l). j=1 l=1 13 / 16

14 LHCFFT - generalisation Corollary (K. 2012) For a fixed generating vector r, H d,γ, C > 1, and 1 2 C + 1 Hd,γ ( j ) 1 M(C) t C 1 π(l) j=1 l=1 (1) the generated set {jr mod 1 : j = 0,..., M(C) 1} guarantees cond 2 (A A) C. new approach: find minimisers r for (1) ( use function evaluations, complexity O simplex search method H d,γ ) (d + log Hd,γ ) 14 / 16

15 LHCFFT - generalisation r minimiser of (1) M(10) minimum of (1) cond 2(A A) 10 r minimiser of (1) M (10) = min l {2 l : δ 9 11 } cond 2(A A) 10 d H d,γ M(10) M (10) cardinalities of stable generated sets, γ = ( 1 2 ) s 15 / 16

16 Summary full grid FFT - suffers from curse of dimensionality hyperbolic crosses decreases problem size strongly condition number increases strongly for standard sparse grids rank-1 lattices perfectly stable spatial discretisation (A A = MI) fast for reasonable problem sizes (1-dimensional FFT) generated sets stable spatial discretisation (cond 2 (A A) very small) fast for reasonable problem sizes (1-dimensional FFT) K., Reconstructing hyperbolic cross trigonometric polynomials by sampling along generated sets, Preprint 2012, lkae 16 / 16