Orthogonality of hat functions in Sobolev spaces

1 Orthogonality of hat functions in Sobolev spaces Ulrich Reif Technische Universität Darmstadt A Strobl, September 18, 27

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3 Outline: Recap: quasi interpolation Recap: orthogonality of uniform B-splines Orthogonality of hat functions in IR d

4 Quasi interpolation Consider a univariate spline space S of order n with B-splines B n j. A quasi interpolant Λ of order ν is a linear operator f Λf = j (λ j f) B n j S given by functionals λ j, with Locality: λ j f depends only on f restricted to supp Bj n. Boundedness: sup j λ j < Polynomial precision: Λp = p for all p IP ν, or ideally p IP n.

5 Schoenberg: With µ j denoting the Greville abscissae, Λ j f = j f(µ j )B n j is a QI of second order, f Λf = O(h 2 ). 4 2 2 1 2 3 4 5

6 Schoenberg: With µ j denoting the Greville abscissae, Λ j f = j f(µ j )B n j is a QI of second order, f Λf = O(h 2 ). 4 2 2 1 2 3 4 5

7 Schoenberg: With µ j denoting the Greville abscissae, Λ j f = j f(µ j )B n j is a QI of second order, f Λf = O(h 2 ). 4 2 2 1 2 3 4 5

8 Schoenberg: With µ j denoting the Greville abscissae, Λ j f = j f(µ j )B 2 j is a QI of second order, f Λf = O(h 2 ). This is already optimal for the piecewise linear case n = 2, 4 2 2 1 2 3 4 5

9 de Boor-Fix: For certain coefficients ψ j,k, Λ j f = j (λ j f)b n j, λ j f = n 1 k= ψ j,k D k f(µ j ), is a QI of maximal order n, f Λf = O(h n ). Example: quadratic splines with integer knots λ j f = f(j + 3/2) 1 8 f (j + 3/2)

1 3 2.5 2 1.5 1.5.35.3.25.2.15.1.5.5.1 de Boor-Fix 1.5.5 1.15 1.5.5 1

1 3 2.5 2 1.5 1.5.35.3.25.2.15.1.5.5.1 de Boor-Fix 1.5.5 1.15 1.5.5 1 3 2.5 2 1.5 1.5.35.3.25.2.15.1.5.5.1 Least Squares 1.5.5 1.15 1.5.5 1

11 3 2.5 2.1.5 Interpolation 1.5 1.5.5.1 1.5.5 1 1.5.5 1

11 3 2.5 2.1.5 Interpolation 1.5 1.5.5.1 1.5.5 1 1.5.5 1 3 2.5 2.1.5 Least Squares 1.5 1.5.5.1 1.5.5 1 1.5.5 1

12 Quasi interpolation vs. Least squares local rules for coefficients asymptotically optimal approximation order h n error not minimal wrt. norm solve global system asymptotically optimal approximation order h n error minimal wrt. norm

12 Quasi interpolation vs. Least squares local rules for coefficients asymptotically optimal approximation order h n error not minimal wrt. norm solve global system asymptotically optimal approximation order h n error minimal wrt. norm How to combine the advantages?

For polynomials: adapt the basis to the inner product. 13

13 For polynomials: adapt the basis to the inner product. For splines: adapt the inner product to the basis.

13 For polynomials: adapt the basis to the inner product. For splines: adapt the inner product to the basis. For positive weights ω := [ω,..., ω m ], (f, g) ω := m µ= ω µ µ f, µ g defines the Sobolev space H m ω (IR). The induced norm ω is equivalent to the standard norm with weights ω = = ω m = 1.

13 For polynomials: adapt the basis to the inner product. For splines: adapt the inner product to the basis. For positive weights ω := [ω,..., ω m ], (f, g) ω := m µ= ω µ µ f, µ g defines the Sobolev space H m ω (IR). The induced norm ω is equivalent to the standard norm with weights ω = = ω m = 1. Since B n j Hn 1 (IR), one can try to determine ω such that (B n j, B n k ) ω = δ j,k.

14 For uniform B-splines, this yields n conditions for the n weights in ω. 1 8 6 4 2 1 1 2 3 4 5 (B 3, B 3 ) ω = 1 (B 3, B 3 1) ω = (B 3, B 3 2) ω =

14 For uniform B-splines, this yields n conditions for the n weights in ω. 1 8 6 4 2 1 1 2 3 4 5 (B 3, B 3 ) ω = 1 (B 3, B 3 1) ω = (B 3, B 3 2) ω = Example: Quadratic B-splines with integer knots are orthonormal wrt. (f, g) ω(3) = f, g + f, g /4 + f, g /3.

15 More particular cases: n ω (n) ω 1 (n) ω 2 (n) ω 3 (n) ω 4 (n) ω 5 (n) 1 1 2 1 1 6 3 1 1 4 1 3 4 1 1 3 7 12 1 14 5 1 5 12 13 144 41 324 1 63 6 1 1 2 31 24 139 648 479 1512 1 2772 Example: Quadratic B-splines with integer knots are orthonormal wrt. (f, g) ω(3) = f, g + f, g /4 + f, g /3.

16 Under mild regularity assumptions, Fourier series and transforms ˆϕ(y) := ϕ(x) exp( ixy) dx IR ϕ(y) := j Z ϕ(j) exp( ijy) are related by the Poisson summation formula ϕ(y) = k Z ˆϕ(y + 2kπ),

16 Under mild regularity assumptions, Fourier series and transforms ˆϕ(y) := ϕ(x) exp( ixy) dx IR ϕ(y) := j Z ϕ(j) exp( ijy) are related by the Poisson summation formula ϕ(y) = k Z ˆϕ(y + 2kπ), With B n the centered B-spline of order n, let ϕ(x) := ( B n, B n ( x) ). ω Then orthonormality is equivalent to ϕ(j) = δ j, ϕ(y) 1 ϕ(y) 1 + O(y 2n ).

By the convolution property of B-splines, 17 ϕ(x) := (B n, B n ( x)) ω = ˆϕ(y) = n 1 m= n 1 m= ω m ( 1) m 2m B 2n (x) ω m y 2m sinc 2n (y/2)

By the convolution property of B-splines, 17 ϕ(x) := (B n, B n ( x)) ω = ˆϕ(y) = n 1 m= n 1 m= ω m ( 1) m 2m B 2n (x) ω m y 2m sinc 2n (y/2) By the Poisson summation formula, ϕ(y) = k Z ˆϕ(y + 2kπ) = n 1 m= k Z = ( n 1 m= ω m y 2m sinc 2n (y/2 + kπ) ω m y 2m)( sinc 2n (y/2) + O(y 2n ) )! = 1 + O(y 2n ).

18 Theorem [R. 95] The sequence {Bj n, j Z} of cardinal B-splines is orthonormal in (IR) for weights ω = ω(n) defined by H n 1 ω 1/sinc 2n (y/2) = n 1 m= ω m (n)y 2m + O(y 2n ).

18 Theorem [R. 95] The sequence {Bj n, j Z} of cardinal B-splines is orthonormal in (IR) for weights ω = ω(n) defined by H n 1 ω 1/sinc 2n (y/2) = n 1 m= ω m (n)y 2m + O(y 2n ). For knot spacing h, weights are scaled according to ω m (n, h) := h 2m 1 ω m (n).

18 Theorem [R. 95] The sequence {Bj n, j Z} of cardinal B-splines is orthonormal in (IR) for weights ω = ω(n) defined by H n 1 ω 1/sinc 2n (y/2) = n 1 m= ω m (n)y 2m + O(y 2n ). For knot spacing h, weights are scaled according to ω m (n, h) := h 2m 1 ω m (n). The solution g := Λ n hf to the least squares approximation problem of f in H n 1 ω(n,h) (IR) is given by the quasi interpolant Λn h with λ n hf := (B n h, f) ω(n,h).

19 3 2.5 2 1.5 1.5.35.3.25.2.15.1.5.5.1 de Boor-Fix 1.5.5 1.15 1.5.5 1 3 2.5 2 1.5 1.5.35.3.25.2.15.1.5.5.1 Orthogonal 1.5.5 1.15 1.5.5 1

2 3 2.5 2.1.5 Interpolation 1.5 1.5.5.1 1.5.5 1 1.5.5 1 3 2.5 2.1.5 Orthogonal 1.5 1.5.5.1 1.5.5 1 1.5.5 1

21 Generalizations: approximation on bounded intervals for n 1 non-uniform B-splines obvious for n = 2 tricky for n = 3 open for n 4 uniform tensor product B-splines open for unimodular box splines

21 Generalizations: approximation on bounded intervals for n 1 non-uniform B-splines obvious for n = 2 tricky for n = 3 open for n 4 uniform tensor product B-splines open for unimodular box splines new: hat functions on arbitrary triangulations in IR d

22 Hat functions: The triangulation of a set Ω IR d consists of simplices S k, k K, and vertices V i, i I. The hat functions B i : Ω IR are defined by S k B i (V j ) = δ i,j, i, j I B i S k IP 2 (S k ), k K.

23 Hat functions: The orthogonality conditions (B i, B j ) Ω = δ i,j, i, j I are globally coupled. A stronger, but local condition requires orthogonality on each simplex, S k (B i, B j ) k = δ i,j, i, j I, k K.

24 In IR 2 : For each triangle S k, there exist 3 non-vanishing hat functions, yielding 3 homogeneous conditions for orthogonality.

24 In IR 2 : For each triangle S k, there exist 3 non-vanishing hat functions, yielding 3 homogeneous conditions for orthogonality. The ansatz ( (f, g) k := ωk fg + f W k g t) S k ω k >, W k spd (2 2)-matrix, provides the appropriate number of degrees of freedom.

25 Theorem [App, R.] On the unit triangle S := {x IR 2 : x 1 + x 2 1}, the weights ω := 6, W := [ ] 1 1/2 1/2 1 yield orthonormality.

26 In IR d : For each simplex S k, there exist d + 1 non-vanishing hat functions, yielding d 2 + d 2 homogeneous conditions for orthogonality. The ansatz ( (f, g) k := ωk fg + f W k g t) S k ω k >, W k spd (d d)-matrix, provides the appropriate number of degrees of freedom.

27 Theorem [App, R.] On the unit simplex S := {x IR d : x 1 + + x d 1}, the weights ω := (d + 1)!, W := d! d + 2 d 1 1 1 1 d 1 1..... 1 1 1 d yield orthonormality.

28 For an arbitrary simplex S k, obtained from S by an affine map the weights yield orthogonality. ω k := S k = AS + a, ω det A, W k := AW At det A (A,a)

29 Orthogonality: Define the Sobolev type inner product (f, g) Ω := (f, g) k = ( ωk fg + f W k g t), k K k K S k then, with #B i the number of simplices in supp B i, (B i, B j ) Ω = δ i,j #B i.

29 Orthogonality: Define the Sobolev type inner product (f, g) Ω := (f, g) k = ( ωk fg + f W k g t), k K k K S k then, with #B i the number of simplices in supp B i, (B i, B j ) Ω = δ i,j #B i. Approximation: The solution g = Λf to the least squares approximation problem of f wrt. the inner product (, ) Ω is given by Λf = i (f, B i ) Ω #B i B i.

3 Discrete Variants: Denote vertices and edge midpoints of S k by P k = p 1 k. p m k and the quadratic polynomial interpolating the function f at P k by q k = Q k f. Then [f, g] Ω := k K(Q k f, Q k g) k is a discrete inner product with [B i, B j ] Ω = (Q k B i, Q k B j ) k = i, B j ) k = (B i, B j ) Ω. k K k K(B

31 Evaluation of [f, B i ] Ω is cheap, [f, B i ] k = (Q k f, B i ) k = V F k, F k = f(p k ), where V is independent of k. In the 2d-case, 1/15 4 11 11 2 7 2

32 Average savings in 2d: For similar accuracy wrt. max norm, interpolation requires 4% more triangles. L 2 -norm, interpolation requires 1% more triangles.

32 Average savings in 2d: For similar accuracy wrt. max norm, interpolation requires 4% more triangles. L 2 -norm, interpolation requires 1% more triangles. Applications: Computer graphics? FEM Nonlinear optimization...

33 Conclusion: Sobolev orthogonality available for uniform B-splines Sobolev orthogonality available for hat functions in IR d Cheaper than L 2 Better than QI