Construction of wavelets. Rob Stevenson Korteweg-de Vries Institute for Mathematics University of Amsterdam

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1 Construction of wavelets Rob Stevenson Korteweg-de Vries Institute for Mathematics University of Amsterdam

2 Contents Stability of biorthogonal wavelets. Examples on IR, (0, 1), and (0, 1) n. General domains via domain decomposition. Wavelets in finite element spaces. 1/25

3 Ω = (0, 1) (sim. Ω = IR). Example: Haar wavelets (1910) V 0 V 1. j IN 0 V j = L 2 (Ω) Φ 0, Φ 1, Φ 2. Φ j = {φ j,k : k I j }, φ j,k = 2 j/2 φ(2 j k) (shift & dyadic dilation), diam supp φ j,k 2 j Φ 0, Ψ 1, Ψ 2. Ψ j = {ψ j,k : k J j } spans V j Ṽ j 1, ψ j,k = 2 j/2 ψ(2 j k) diam supp ψ j,k 2 j. Vanishing moment Ψ := Φ 0 j IN Ψ j = {ψ λ : λ }. λ = (j, k), λ = j. Orthonormal basis of L 2 (Ω). For s < 1 2, {2 λ s ψ λ : λ } Riesz basis for H s (Ω). For obtaining a Riesz basis: Orthogonality within and between levels can be relaxed. Smoothness and so range of stability can be increased. 2/25

4 Riesz bases and biorthogonality Recall, Ψ Riesz basis for H means F : l 2 ( ) H : c c Ψ F : H l 2 ( ) : g g(ψ) b.i., b.i.. From F 1 bounded, u c λ H. Denote it as ψ λ, i.e., c λ = ψ λ (u). Then Ψ := { ψ λ : λ } H and Ψ(Ψ) := [ ψ λ (ψ µ )] λ,µ = Id. Set F : l 2 ( ) H : c c Ψ F : H l 2 ( ) : u Ψ(u) Then FF = Id = F F. So Ψ R.b. H Ψ R.b. H. [ Via Riesz repr., alternatively Ψ R.b. H and Ψ, Ψ H = Id. ] Knowing biorthogonal Ψ and Ψ, b.i. of F can be proved by showing boundedness of F and F, i.e. boundedness of Ψ, Ψ H and Ψ, Ψ H, via strengthened CS ineq.. 3/25

5 Thm 1. Let Stability biorthogonal wavelets V 0 V 1 L 2 (Ω), Ṽ 0 Ṽ1 L 2 (Ω) (MRAs) s.t. inf sup 0 ṽ j Ṽj 0 v j V j For some 0 < γ < d, let inf j IN 0 ṽ j, v j L2(Ω) ṽ j L2 (Ω) v j L2 (Ω) > 0 and sim. with (V j ) j (Ṽj) j inf v v j L2 (Ω) 2 jd v H v j V d (Ω) (v H d (Ω)) (Jackson estimate), and j v j H s (Ω) 2 js v j L2 (Ω) (v j V j, s [0, γ)) (Bernstein estimate), where for a Hilbert space H d (Ω) L 2 (Ω) with dense embedding, H s (Ω) := [L 2 (Ω), H d (Ω)] s/d (s [0, d]). 4/25

6 Stability biorthogonal wavelets Let similar estimates be valid at the dual side with ((V j ) j, d, γ, H s (Ω)) reading as ((Ṽj) j, d, γ, H s (Ω)). Assume that for sufficiently small s > 0, H s (Ω) = H s (Ω). Then, with Φ 0 = {φ 0,k : k I 0 } basis for V 0 (scaling functions) and Ψ j = {ψ j,k : k J j } uniform L 2 (Ω)-Riesz bases for W j := V j Ṽ L 2 (Ω) j 1 (wavelets), for s ( γ, γ) Φ 0 j IN 2 sj Ψ j is a Riesz basis for H s (Ω), where H s (Ω) := ( H s (Ω)) for s < 0. For the (unique) collections Φ0 = { φ 0,k : k I 0 } Ṽ0 with Φ 0, Φ 0 L2 (Ω) = Id, Ψj = { ψ j,k : k J j } W j := Ṽj V L 2 (Ω) j 1 Ψ j, Ψ j L2 (Ω) = Id, and s ( γ, γ), Φ 0 j IN 2 sj Ψj is a Riesz basis for H s (Ω), where H s (Ω) := (H s (Ω)) for s < 0. with 5/25

7 inf-inf-sup conditions eq. to Stability biorthogonal wavelets unif. bounded Q j : L 2 (Ω) L 2 (Ω) with IQ j = V j, I(Id Q j ) = Ṽ j (biorthogonal projector), or unif. L 2 (Ω)-Riesz bases Φ j, Φj for V j, Ṽ j s.t. Φ j, Φ j 1 L 2 (Ω) is uniformly bounded. Two ways of appl. Thm.: 1): You have unif. L 2 (Ω)-Riesz coll. Ψ j, Ψ j, Φ j, Φ j s.t. Ψ j Ψ k when j k, Φ j, Φ j L2 (Ω) = Id, Ψ j, Ψ j L2 (Ω) = Id, and (V j :=) span Φ j = span Φ 0 Ψ 1... Ψ j, (Ṽj :=) span Φ j = span Φ 0 Ψ 1... Ψ j. Standard setting: Shift & dyadic dilation case on IR with 1 (or a few) generating primal and dual mother scaling function(s) and wavelet(s). Then inf-inf-sup is valid, and Ψ j, Ψ j unif. L 2 (Ω) Riesz for V j Ṽ L 2 (Ω) j 1, Ṽ j V L 2 (Ω) j 1. To verify J & B estimates. 2): You specify MRA s (V j ) j, (Ṽj) j that satisfy J & B. To verify inf-infsup and to construct unif. L 2 (Ω)-Riesz bases Ψ j for W j = V j Ṽ L 2 (Ω) j 1, preferably local ones. For some applications, needed that resulting bases Ψ j for Ṽj V L2 (Ω) j 1 6/25 are also local. Not needed for awgm.

8 Proof of Thm. (sketch) W j = I(Q j Q j 1 ) (Q 1 := 0), Wj = I(Q j Q j 1 ). l 2,s (Q) := {(w j ) j : w j W j : (w j ) j l2,s (Q) := j 4sj w j 2 L 2 (Ω) < }, l 2,s (Q ) :=... To prove G : l 2,s (Q) H s : (w j ) j j G : l 2,s (Q ) H s : ( w j ) j j w j b.i. (s ( γ, γ)) w j b.i. (s ( γ, γ)) Boundedness of G from w j, w k H s 2 ε j k (2 sj w j L2 (Ω))(2 sk w k L2 (Ω)) for ε s.t. s ± ε ( γ, γ). Sim. for G. [ j k: w j, w k H s w j H s+ε w k H s ε (for s ε < 0 < s+ε use H t = H t for t > 0 small). For t = s ± ε > 0, use B at primal side. Otherwise, w i H t = sup v w i,(q i Q i 1 )v v H t and J at dual side. ] 7/25

9 Proof of Thm. (sketch) Since (H s ) = H s and l 2,s (Q) = l 2, s (Q ), boundedness of G or G gives that of G : H s l 2, s (Q ) : u ((Q j Q j 1)u) j G : H s l 2, s (Q) : u ((Q j Q j 1 )u) j (s ( γ, γ)), (s ( γ, γ)). Now use G G = Id = G G and G G = Id = GG to conclude boundedness of (G ) 1 and ( G ) 1. 8/25

10 Examples in shift & dyadic dilation setting i.e. application of first type. Orthogonal wavelets from the Daubechies family of orders d(= d) = 2 and 6. d = 1 is Haar. Smoothness, and so γ, increases with d. Riesz for s < γ : Daubechies 4 tap wavelet scaling function wavelet function Functions implicitly defined as solution of a refinement equation. file:///users/robstevenson/desktop/daubechies4-functions.webarchive Page 1 of 1 9/25

11 Examples in shift & dyadic dilation setting Cohen-Daubecies-Feauveau family Primals are B-splines. Duals are implicitly defined as solution of a ref. eq Figure 1: Primal and dual scaling function of orders d = d = Figure 2: Corresponding primal and dual wavelet 10/25

12 Daubecies-Cohen-Feauveau family Figure 3: Primal and dual scaling function of orders d = 3, d = Figure 4: Corresponding primal and dual wavelet γ = d 1 2, and γ increases linearly with d d, d + d even. Primals Riesz for s ( γ, γ), duals for s ( γ, γ). 11/25

13 Examples in shift & dyadic dilation setting Donovan-Geronimo-Hardin piecewise orthogonal multi-wavelets Piecewise polynomial orthogonal scaling functions (and) wavelets. Multiple generators. Here continuous ones (i.e. γ = 3 2 ) of order d = 2. Riesz for s < γ Figure 5: Scaling functions Figure 6: Corresponding wavelets (Figures CDF and DGH wavelets by Sebastian Kestler, Uni Ulm) 12/25

14 Towards more interesting domains for num. appl. Step 1: Adaptation to (0, 1): Several constructions that yield on each level a few boundary adapted scaling functions and wavelets. Needed to maintain simultaneously biorthogonality and polynomial reproduction of orders d and d. Boundary conditions can be incorporated at primal and/or dual side, so H d can read as H d (0, 1), H d 0(0, 1) or, say H d (0, 1) H 1 0(0, 1). Sim. for H d. Recent constructions yield condition numbers on (0, 1) that are only slightly worse than those on IR. Step 2: Wavelets on (0, 1) n. Let (V j ) j, (Ṽj) j MRA s on (0, 1), orders d, d, regularity γ, γ. Wj = V j Ṽ j 1, Wj = Ṽj Vj 1. Then (V j V j ) j, (Ṽj Ṽj) j MRA s on (0, 1), orders d, d, regularity γ, γ. V j V j (Ṽj 1 Ṽj 1) = W j V j 1 V j 1 W j W j W j, so primal wavs ψ j,k φ j 1,k, φ j 1,k ψ j,k, φ j 1,k φ j 1,k. Sim. at dual side. 13/25

15 Towards more interesting domains for num. appl. Step 3: DD First approach: Composite wavelets. Connect primal and dual MRAs by identifying DOFs on the interfaces. Construct bases (wavelets) for the biorthogonal complements. New types of wavelets near the interfaces. 14/25

16 Towards more interesting domains for num. appl. Figure 7: Primal (left) and corresponding dual (right) wavelets built from CDF (2, 2). 15/25

17 Second approach (DD): Using extension operators. Let Ω = Ω 1 Ω 2, Ω 1 Ω 2 =. Let R i restriction of functions on Ω to Ω i E i extension of functions on Ω i to Ω η i extension by zero of functions on Ω i to Ω Towards more interesting domains for num. appl. From R i E i = Id (= R i η i ), and R i η j = 0, [ ] R 1 : L R 2 (I E 1 R 1 ) 0 (Ω) L 0 (Ω 1 ) L 0 (Ω 2 ) are each others inverse. [E 1 η 2 ] : L 0 (Ω 1 ) L 0 (Ω 2 ) L 0 (Ω) Note if black is C k, and E is C k, then blue extended with zero is C k. 16/25

18 Using extension operators For m IN, if E 1 B(H m (Ω 1 ), H m (Ω)), then [E 1 η 2 ] B(H m (Ω 1 ) H0, Ω m 1 Ω 2 (Ω 2 ), H m (Ω)) [ ] R 1 B(H m (Ω), H m (Ω R 2 (I E 1 R 1 ) 1 ) H0, Ω m 1 Ω 2 (Ω 2 )). So for Ψ 1 and Ψ 2 Riesz bases of H m (Ω 1 ) and H m 0, Ω 1 Ω 2 (Ω 2 ), E 1 Ψ 1 η 2 Ψ 2 is Riesz basis for H m (Ω). Construction can be applied recursively in the multi-subdomain case. If Ω 1, Ω 2 are hypercubes that share a face, E 1 can be univariate extension. Hestenes extensions can be applied, e.g., reflection. Latter suitable for construction of Riesz bases for H s (Ω) for s ( 1 2, 3 2 ). Only basis functions that don t vanish at interface have to be extended. Kind of basis functions irrelevant. 17/25

19 Construction of a basis for the biorthogonal complement (second appl. of Thm.) Let (V j ) j, (Ṽj) j MRAs that satisfy J & B and inf-inf-sup. [(e.g., Φ j, Φ j 1 L 2 (Ω) unif. bounded for unif. L 2(Ω)-Riesz bases Φ j, Φ j for V j, Ṽj).] Thm 2. Let { Φj 1 }, prop. sc., be unif. L 2 (Ω)-Rsz bases for {Ṽj 1 Θ j Ξ j V j with Θ j, Φ j 1 L2 (Ω) scaled, an invertible diagonal matrix. Then, properly Ψ j := Ξ j Ξ j, Φ j 1 L2 (Ω) Θ j, Φ j 1 1 L 2 (Ω) Θ j is unif. L 2 (Ω)-Riesz basis for V j Ṽ L 2 (Ω) j 1. [ Indeed, Ψ j, Φ j 1 L2 (Ω) = 0 ] Note Ψ j unif. local when Ξ j, Θ j, Φ j 1 unif. local. [ If additionally, span Θ j = V j 1, i.e. Θ j = Φ j 1 (so classical wavelet setting) ( severe ), and both transformations Φ j Φ j 1 Ξ j unif. local ( mild ), then resulting Ψ j also unif. local. ] }, 18/25

20 Application: (Pre)wavelets on polytopes Let (V j ) j seq. of cont. piecewise f.e. spaces on Ω corr. to dyadic refs. ( red-refinements ) of initial conforming subdivision T 0 into n-simplices. Then J & B with d = 2, γ = 3 2. Let Φ j 1 (= Φ j 1 ) = {φ j 1,x : x I j 1 } nodal basis V j 1 (bdr. vertices may or may not be included depending on bdr. conds.), being prop. sc. a unif. L 2 (Ω)-Riesz basis. Set Ξ j = {φ j,x : x I j \I j 1 } and Θ j = {φ j,x 2 (n+1) φ j 1,x : x I j 1 }. Then, Θ j Ξ j is, properly scaled, unif. L 2 (Ω)-Riesz bases for V j, and Θ j, Φ j 1 L2 (Ω) is an invertible diagonal matrix. Figure 8: Nonzero coefficients Ξ j, Φ j 1 L2 (Ω). Riesz for s < 3 2. [ Orthogonality between levels, but not inside levels. ] 19/25

21 Wavelets on polytopes Generalizations possible. E.g. take Ṽj continuous piecewise quadratics w.r.t T j, V j continuous piecewise linears w.r.t. T j+1. inf-inf-sup satisfied. J & B with d = 2, γ = 3 2, d = 3, γ = 3 2. (Primal) wavelets Riesz for s < 3 2. Three vanishing moments. 20/25

22 Three-point hierarchical basis For (Ṽj) j = (V j ) j, recall construction unif. L 2 (Ω)-stable basis for orth. complement V j V L 2 (Ω) j 1. (wavconstr) This two-level result equally applies with, L2 (Ω) reading as some scalar product, j on V j. Again, let (V j ) j seq. of cont. piecewise f.e. spaces on Ω corr. to dyadic ref. of initial conforming subd. T 0 into n-simplices. Take u, v j = x I j w j,x u(x)v(x), where w j,x = {T T j :T x} vol(t ) n+1. Take Ξ j = {φ j,x : x I j \ I j 1 }, Θ j = {φ j,x : x I j 1 }. Then Θ j Ξ j is, properly scaled, unif. j Riesz basis for V j, and Θ j, Φ j 1 j is invertible diagonal matrix. So Ψ j := Ξ j Ξ j, Φ j 1 j Θ j, Φ j 1 1 j Θ j is unif. j Riesz basis for V j V j j 1. 21/25

23 Figure 9: Nonzero coefficients Ξ j, Φ j 1 j. So, by definition of Ξ j and Θ j, each wavelet is a linear combi. of only three nodal basis functions (any n). Multi-level stability? Inner product changes from level to level. So space decomposition isn t orthogonal. No information about dual side. Stability L 2 (Ω)-orthogonal decomposition + perturbation arguments gives 0.29 n = 1 that, properly scaled, 3-pt hb is Riesz for s > 0.42 n = n = 3 Likely not sharp. In shift-invariant case, Riesz for H s (IR n ) for s ( , 3 2 ) for n {1, 2, 3} (sharp). The (one-point) hierarchical basis (Schauder basis) is Riesz for H s (Ω) for s > n 2 (sharp). 22/25

24 Conclusion Wavelet bases suitable for the application of awgm can be constructed on general domains. 23/25

25 References [1] N.G. Chegini, S. Dahlke, U. Friedrich, and R.P. Stevenson. Piecewise tensor product wavelet bases by extensions and approximation rates. Technical report, KdV Institute for Mathematics, University of Amsterdam, To appear in Math. Comp. [2] A. Cohen, I. Daubechies, and J.C. Feauveau. Biorthogonal bases of compactly supported wavelets. Comm. Pur. Appl. Math., 45: , [3] A. Cohen, I. Daubechies, and P. Vial. Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal., 1(1):54 81, [4] W. Dahmen. Stability of multiscale transformations. J. Fourier Anal. Appl., 2(4): , [5] W. Dahmen, A. Kunoth, and K. Urban. Biorthogonal spline-wavelets on the interval - Stability and moment conditions. Appl. Comp. Harm. Anal., 6: , [6] W. Dahmen and R. Schneider. Composite wavelet bases for operator equations. Math. Comp., 68: , [7] W. Dahmen and R. Schneider. Wavelets on manifolds I: Construction and domain decomposition. SIAM J. Math. Anal., 31: , [8] W. Dahmen and R.P. Stevenson. Element-by-element construction of wavelets satisfying stability and moment conditions. SIAM J. Numer. Anal., 37(1): , [9] I. Daubechies. Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math., 41: , [10] T.J. Dijkema. Adaptive tensor product wavelet methods for solving PDEs. PhD thesis, Utrecht University, /25 [11] G.C. Donovan, J.S. Geronimo, and D.P. Hardin. Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets. SIAM J. Math. Anal.,

26 [14] M. Primbs. New stable biorthogonal spline-wavelets on the interval. Results Math., 57(1-2): , [15] R.P. Stevenson. Piecewise linear (pre-)wavelets on non-uniform meshes. In Multigrid methods V (Stuttgart, 1996), volume 3 of Lect. Notes Comput. Sci. Eng., pages Springer, Berlin, [16] R.P. Stevenson. Stable three-point wavelet bases on general meshes. Numer. Math., 80: , /25