Quarkonial frames of wavelet type - Stability, approximation and compression properties

Transcription

1 Quarkonial frames of wavelet type - Stability, approximation and compression properties Stephan Dahlke 1 Peter Oswald 2 Thorsten Raasch 3 ESI Workshop Wavelet methods in scientific computing Vienna, November 12-16, 212

2 Outline 1 Motivation 2 Quarkonial frames 3 Quarklet frames 4 Conclusion

3 Basic setting numerical solution of operator equations F (u) = g, u X basis-oriented approach with dictionary Ψ = {ψ λ : λ J } u v Ψ = λ J v λ ψ λ, # supp v small stability properties of the dictionary, e.g., Ψ frame span Ψ = X, u X inf u l2(j ) u=u Ψ very fast decay of best approximations σ N (u) = inf u v X Ce αnβ, α, β > v=v Ψ:# supp v N

4 Examples Fourier basis Ψ = {e i k, : k Z n } on (, 1) n, consider the Gevrey space for α, β, s { G α,β,s := u(x) = u k e i k,x : } e 2α k β (1 + k 2s ) u k 2 < k Z n k Z n e.g., β 1 G α,β,s consists of analytic functions hp finite element frames span Ψ = j= γ= S γ,k (Ω, T j ), S γ,k (Ω, T ) := {ψ H k (Ω) : ψ T P γ, T T } e.g., Ω = (, 1), u analytic with algebraic singularity at x = geometrically graded meshes T j with σ N (u) Ce α N, α >

5 Another example: Quarkonial frames Triebel 1997ff.: quarkonial/subatomic decompositions combines Weierstraß approach for analytic functions... f (x) = c γ,m (x x m ) γ η m (x), η m (x) = 1, m=1 γ N n m=1 x Ω... with wavelet philosophy (dilation and translation) f (x) = γ N n j= k Z n c γ,j,k ϕ γ (2 j x k), ϕ γ (x) = g γ (x)ϕ(x), x Ω, k Z n ϕ(x k) = 1, characterization of function spaces g γ P γ enrichment function similar exponential approximation spaces as hp-fem

6 Relation of quarkonial frames with other approaches close relation with partition of unity methods (PUM) ϕ θ (x) = 1, x Ω = Ω θ, θ Θ θ Θ ϕ γ,θ (x) := g γ,θ (x)ϕ θ (x), supp ϕ θ Ω θ, g γ,θ V θ enrichment function see Babuška/Melenk 1996ff. Dahmen/Dekel/Petrushev 27, and their multilevel modification, see Schweitzer 23 stable space splittings, Oswald, Griebel, Yserentant, s ff. fusion frames, Casazza/Kutyniok et al. 24ff.

7 Triebel s stability result Theorem (Triebel 1997ff.) Consider the system Ψ := { w γ,j,k ϕ γ,j,k : γ N n, j N, k Z n} with ϕ Bŝ,, ϕ( k) = 1, ϕ γ(x) = x γ ϕ(x), ϕ γ,j,k (x) = ϕ γ(2 j x k), k Z n (*) w γ,j,k max { 2 js ϕ γ,j,k B s p,p, ϕ γ,j,k Lp } C2 ρ γ, ρ >. Then for < p, σ p := max{, n( 1 p 1)} < s < ŝ, each u Bs p,p(r n ) has an unconditionally S - and L max{1,p} convergent expansion u = u Ψ with u B s p,p inf u=u Ψ u b s p,p, u p b s p,p := γ N n j= k Z n 2 jsp u γ,j,k p. (generalization to non-shiftinvariant case and bounded domains possible)

8 Improving Triebel s result Note that condition (*) means that w γ,j,k 2 ρ γ 2 jn/p, ρ > which is not desirable from the viewpoint of numerical applications. Improvement: Theorem (Dahlke/Oswald/R. 211) In the shift-invariant case with polynomial enrichment, and under certain approximation and smoothness properties of the subspaces V r,j := clos Lp span{ϕ γ,j,k : γ r, k Z n }, r, j, algebraic decay w γ,j,k γ β 2 jn/p suffices, for some β >.

9 Some details By approximation and smoothness properties we mean Jackson and (in particular) Bernstein inequalities for variable degree r: (J) u = uj r, uj r V r,j, j= ( ) 1/p 2 jsp uj r p L p CJr,s u B s p,p j= (B) u r j B t p,p CB r,t 2 jt u r j Lp, u j r V r,j, r r, j, σ p < t < ŝ p J r,s scales like r s, using results on polynomial approximation. B r,t scales like r αt, α 2, which is essentially due to Markov-type inequalities for algebraic polynomials q Lp(I ) C I 1 r 2 q Lp(I ), q P r. B r,t determines the asymptotics of w γ,j,k, r = γ. The proof uses the machinery of stable subspace splitting.

10 Some quarkonial frames for H s (R), s > Let us look at the following special cases: partition function ϕ given by B-spline N d, d N first option: enrichment by monomials g r (x) = x r second option: enrichment by orthogonal polynomials g r (x) = L r (x)

11 Special case I: monomial B-spline quarks ϕ = N m, ϕ r (x) = (x/m) r ϕ(x) m = 2, r = m = 2, r = 1 m = 2, r = 2 Lemma It holds that ϕ r Lp (r + 1) m 1+1/p, r (m 1) 2, 1 p. Theorem (Dahlke/Oswald/R. 212) It holds that for all u V r,j, ω m (u, t) Lp C min{1, (r + 1) 2 2 j t} m 1+1/p u Lp u H t C(r + 1) 2t 2 jt u L2, t < m 1/2

12 Special case II: quarks based on orthogonal polynomials Let ϕ(x) := N 1 (x) = χ [,1) (x) and consider the Legendre wavelet basis (see Alpert 1993, Razzaghi/Yousefi 2) ϕ r (x) := 2L r (2x 1)ϕ(x), L r r-th Legendre polynomial Ψ L := { ϕ r ( k) : r, k Z } r = r = 1 r = 2 r = 3 Alpert 1993: orthonormal multiwavelet basis of length q N with generator Φ := (ϕ r ) r q, Φ(x) = k Z A kφ(2x k).

13 Special case II: quarks based on orthogonal polynomials Concerning stability: use that ϕ r L2 = 1 for all r... Theorem (Dahlke/Oswald/R. 212) It holds that for all u V r,j, ω m (u, t) L2 C min{1, (r + 1) 2 2 j t} 1/2 u L2 u H t C(r + 1) 2t 2 jt u L2, t < 1/2.

14 Quarklets Issue: quarkonial systems are unstable in L 2 Way out: do some stabilization introduce (more) vanishing moments one out of many options: wavelet-type modification, e.g., Haar-like ψ r (x) := ϕ r (2x) ϕ r (2x 1), r let s call them quarklets...

15 Quarklets: moment conditions Legendre quarklets := Legendre quarks with Haar-type modification ψ r (x) := ϕ r (2x) ϕ r (2x 1), ϕ r (x) := 2L r (2x 1)ϕ(x), r Lemma It holds that ψ r ϕ q ( k) for all q r, k Z. In particular, ψ r has r + 1 vanishing moments, ψ r P r. This also works for monomial quarks and for other wavelet masks with m discrete vanishing moments ψ r (x) := k Z b k ϕ r (2x k), r Lemma In this case, ψ p has m vanishing moments, like ψ = ψ.

16 Example: monomial CDF-quarklets ϕ = N 2, {b k } k Z mask of Cohen/Daubechies/Feauveau spline wavelets with 2 vanishing moments r = r = 1 r = 2

17 Quarklets: compression properties q t + 1 vanishing moments induce compression estimates of the form u, ψ r,j,k L2 C2 j(t+1/2) ψ r L1 u W t (L (supp ψ r,j,k )) via the well-known technique u, ψr,j,k L2 = inf g P t u g, ψr,j,k L2 inf g P t u g L (supp ψ r,j,k ) ψ r,j,k L1 and Whitney-type estimates. Note that the constant in Whitney s theorem is independent (!) of r. Moreover, can hope that ψ r L1 decays algebraically in r.

18 Quarklets: stability in L 2 (R) We obtain decay estimates for L 2 inner products ψ r,j,k, ψ r,j,k L 2, which imply the L 2 stability of quarklet systems: Theorem (Dahlke/Oswald/R. 212) Let ψ r,j,k := { ψ r (2 j k), j, for all r, j 1, k Z. ϕ r ( k), j = 1 There exist weights w r with algebraic decay in r, such that is a frame for L 2 (R). {w r 2 j/2 ψ r,j,k : r, j 1, k Z}

19 Quarklets: stability in H s (R), s < Let ϕ = N m and let {b k } k Z be the mask of a CDF wavelet with m vanishing moments. For the aforementioned weights w r, consider the L 2 (R) quarklet frame Ψ := {w r 2 j/2 ψ r,j,k : r, j 1, k Z} dual CDF wavelet basis 1 w ΨCDF is a dual frame when rescaled, Ψ CDF gets stable in H s (R), s < š the rescaled quarklet system Ψ is a frame for H s (R), š < s < ŝ

20 Concluding Remarks We have seen: subatomic decomposition of function spaces quarkonial frames of piecewise polynomial functions stability in H s (R), < s < ŝ wavelet-like modification yields quarklets vanishing moments stability in H s (R), š < s < ŝ Work in progress: quark(let)s on bounded domains approximation spaces numerical tests... Thank you for you attention!