A family of Non-Oscillatory, 6 points, Interpolatory subdivision scheme

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1 A family of Non-Oscillatory, 6 points, Interpolatory subdivision scheme Rosa Donat, Sergio López-Ureña, Maria Santágueda Universitat de València. Universitat Jaume I. Bernried, February 29 - March 4, 2016 R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

2 Interpolatory subdivision from Interpolatory techniques 1 Interpolatory subdivision from Interpolatory techniques 2 Nonlinear averages and Non-oscillatory schemes A different perspective 3 6-Point Nonlinear, Non-Oscillatory, schemes Polynomial reproduction. Difference schemes Convergence. Smoothness of limit functions Order of accuracy Stability 4 Conclusion R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

3 Interpolatory subdivision from Interpolatory techniques 1 Interpolatory subdivision from Interpolatory techniques 2 Nonlinear averages and Non-oscillatory schemes A different perspective 3 6-Point Nonlinear, Non-Oscillatory, schemes Polynomial reproduction. Difference schemes Convergence. Smoothness of limit functions Order of accuracy Stability 4 Conclusion R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

4 Interpolatory subdivision from Interpolatory techniques χ j χ j+1 nested grids. f j is known data χ j. I[x, ] is a piecewise polynomial interpolatory technique: Generation of new data associated to χ j+1, f j+1 i = I[x j+1 i,f j ], for x j+1 i χ l+1. From these reconstruction techniques, local rules derived. R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

5 Interpolatory subdivision from Interpolatory techniques In general, the use of piecewise polynomial Lagrange interpolation based on l left points and r right points in a dyadic refinement framework leads to Binary schemes: (S l,r f ) 2i = f i, (S l,r f ) 2i+1 = ψ(f i l,...,f i+r 1 ) = r 1 k= l al,r k f i+k. Deslauries-Dubuc subdivision schemes: Interpolatory subdivision schemes related to piecewise polynomial interpolation based on a centered stencil, l = r. R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

6 Interpolatory subdivision from Interpolatory techniques Lagrange Interpolation techniques do not preserve the shape properties of the original data to be refined, when the degree of the polinomial pieces is larger than 1. ENO-WENO, PPH subdivision. Nonlinear piecewise polynomial interpolatory techniques are used for avoid Gibbs-like behavior. [F. Arándiga. R.D ], [A.Cohen, N.Dyn, B. Matei 2003], [S. Amat, R.D., J. Liandrat, J. Trillo 2006] Power p schemes, Shape-preserving schemes: Nonlinear averages versus linear averages. [S. Amat, K. Dadourian, J. Liandrat 2011], [ F. Kuijt, R. Van Damme 1999 ] R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

7 Interpolatory subdivision from Interpolatory techniques Nonoscillatory behavior of Power p subdivision 1 f S 2,2 0.5 S H2 S H f S 2,2 1.1 S H2 1 S H R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

8 Nonlinear averages and Non-oscillatory schemes 1 Interpolatory subdivision from Interpolatory techniques 2 Nonlinear averages and Non-oscillatory schemes A different perspective 3 6-Point Nonlinear, Non-Oscillatory, schemes Polynomial reproduction. Difference schemes Convergence. Smoothness of limit functions Order of accuracy Stability 4 Conclusion R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

9 Nonlinear averages and Non-oscillatory schemes 4-point DD (S 2,2 f ) 2n+1 = 1 2 (f n + f n+1 ) 1 8 (1 2 2 f n f n ) Power p (S Hp f ) 2n+1 = 1 2 (f n + f n+1 ) 1 8 H p( 2 f n 1, 2 f n ) H p (x,y) = ( sgn(x) + sgn(y) x + y x y x + y p ) ave 1 2, 1 (x,y) = x y m ave 1 2, 1 (x,y) M, { 2 m m H p (x,y) min{m,pm} M ave 1 2, 1 (O(h r ), O(h s )) = O(h min(r,s) ), 2 H p (O(h r ), O(h s )) = O(h max(r,s) ), = min{ x, y }, = max( x, y ) r > 0,s > 0, R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

10 Nonlinear averages and Non-oscillatory schemes 1 f S 2,2 0.5 S H2 S H f i = F(x i ), (x j+1 x j = h, j) F(x) piecewise-smooth, θ (x m,x m+1 ) isolated discontinuity. 2 f j = O(h 2 ), j m 1,m, 2 f m 1 = O(1) = 2 f m. (S 2,2 f ) 2j+1 = (S 1,1 f ) 2j+1 + O(h 2 ) j m 1,...,m + 1 (S 2,2 f ) 2j+1 = (S 1,1 f ) 2j+1 + O(1) j = m 1,m,m + 1. R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

11 Nonlinear averages and Non-oscillatory schemes 1 f S 2,2 0.5 S H2 S H f i = F(x i ), (x j+1 x j = h, j) F(x) piecewise-smooth, θ (x m,x m+1 ) isolated discontinuity. 2 f j = O(h 2 ), j m 1,m, 2 f m 1 = O(1) = 2 f m. (S Hp f ) 2j+1 = (S 1,1 f ) 2j+1 + O(h 2 ) j m. R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

12 Nonlinear averages and Non-oscillatory schemes A different perspective 1 Interpolatory subdivision from Interpolatory techniques 2 Nonlinear averages and Non-oscillatory schemes A different perspective 3 6-Point Nonlinear, Non-Oscillatory, schemes Polynomial reproduction. Difference schemes Convergence. Smoothness of limit functions Order of accuracy Stability 4 Conclusion R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

13 Nonlinear averages and Non-oscillatory schemes A different perspective Neville s algorithm for Lagrange interpolation: { S 2,2 = 1 2 S 2, S (S 2,1 f ) 2n+1 = (S 1,1 f ) 2n f n 1, 1,2 (S 1,2 f ) 2n+1 = (S 1,1 f ) 2n f n, 1 f S 1,2 1 f S 2,1 0.5 S H2 0.5 S H n-1 S 2,1 n n+1 n+2 m H p (x,y) min{m,pm} { S 2,1 S Hp around θ S 1,2 R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

14 Nonlinear averages and Non-oscillatory schemes A different perspective Neville s algorithm for Lagrange interpolation: { S 2,2 = 1 2 S 2, S (S 2,1 f ) 2n+1 = (S 1,1 f ) 2n f n 1, 1,2 (S 1,2 f ) 2n+1 = (S 1,1 f ) 2n f n, 1 f S 1,2 1 f S 2,1 0.5 S H2 0.5 S H n-1 S 2,1 n n+1 S 1,2 n+2 m H p (x,y) min{m,pm} { S Hp S 1,2 around θ R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

15 Nonlinear averages and Non-oscillatory schemes A different perspective General Neville: S l,r = r 1/2 l + r 1 S l,r 1 + l 1/2 l + r 1 S l 1,r. For the 6-point DD linear scheme: S 3,3 = 1 2 S 2, S 3,2 = 1 2 (3 8 S 3, S 2,2) (3 8 S 1, S 2,2). (1) n 2 S 3,1 n 1 n n + 1 n + 2 S 2,2 S 1,3 n + 3 R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

16 Nonlinear averages and Non-oscillatory schemes A different perspective General Neville: S l,r = r 1/2 l + r 1 S l,r 1 + l 1/2 l + r 1 S l 1,r. For the 6-point DD linear scheme: S 3,3 = 1 2 S 2, S 3,2 = 1 2 (3 8 S 3, S 2,2) (3 8 S 1, S 2,2). (2) S 3,1 n 1 n n 2 S 2,2 n + 1 n + 2 S 1,3 n + 3 R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

17 Nonlinear averages and Non-oscillatory schemes A different perspective General Neville: S l,r = r 1/2 l + r 1 S l,r 1 + l 1/2 l + r 1 S l 1,r. For the 6-point DD linear scheme: S 3,3 = 1 2 S 2, S 3,2 = 1 2 (3 8 S 3, S 2,2) (3 8 S 1, S 2,2). (3) n 2 S 3,1 n 1 n n + 1 n + 2 S 2,2 S 1,3 n + 3 R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

18 Nonlinear averages and Non-oscillatory schemes A different perspective General Neville: S l,r = r 1/2 l + r 1 S l,r 1 + l 1/2 l + r 1 S l 1,r. For the 6-point DD linear scheme: S 3,3 = 1 2 S 2, S 3,2 = 1 2 (3 8 S 3, S 2,2) (3 8 S 1, S 2,2). (4) n 2 S 3,1 n 1 n n + 1 n + 2 S 2,2 S 1,3 n + 3 R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

19 Nonlinear averages and Non-oscillatory schemes A different perspective Required: Nonlinear analogs of ave a,b (x,y) = ax + by Weighted-Power p mean. a > 0, b > 0 a + b = 1, p 1. W p,a,b (x,y) := sgn(x) + sgn(y) 2 ( ax + by 1 x y p (M + m α )(M + αm)p 1 ), M = max{ x, y }, m = min{ x, y }, α = max{a,b}/min{a,b}. Generalizes H p (x,y): W p, 1 2, 1 (x,y) = H p (x,y), 2 Non-oscillatory: 1 α m W p,a,b(x,y) pαm W p,a,b (O(h r ), O(h s )) = O(h max(r,s) ) R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

20 1 Interpolatory subdivision from Interpolatory techniques 2 Nonlinear averages and Non-oscillatory schemes A different perspective 3 6-Point Nonlinear, Non-Oscillatory, schemes Polynomial reproduction. Difference schemes Convergence. Smoothness of limit functions Order of accuracy Stability 4 Conclusion R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

21 (S 2,1 f ) 2n+1 = (S 1,1 f ) 2n f n 1, (S 1,2 f ) 2n+1 = (S 1,1 f ) 2n f n, S l,r = S 1,1 + L l,r 2, L l,r linear operator (L l,r f ) 2n = 0. S 3,3 = 1 2 (3 8 S 3, S 2,2) (3 8 S 1, S 2,2) S 3,3 = S 1,1 + ave 1 2, 1 [ave 3 2 8, 5 (L 1,3, L 2,2 ),ave 3 8 8, 5 (L 3,1, L 2,2 )] 2, 8 Non-linear non-oscillatory version: SHW q,p = S 1,1 + H q [W p, 3 8, 5 (L 1,3, L 2,2 ),W 8 p, 3 8, 5 (L 3,1, L 2,2 )] 2 ). 8 R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

22 (S 2,1 f ) 2n+1 = (S 1,1 f ) 2n f n 1, (S 1,2 f ) 2n+1 = (S 1,1 f ) 2n f n, S l,r = S 1,1 + L l,r 2, L l,r linear operator (L l,r f ) 2n = 0. S 3,3 = 3 8 (1 2 S 3, S 1,3) S 2,2 S 3,3 = S 1,1 + ave 3 8, 5 [ave 1 8 2, 1 (L 1,3, L 3,1 ), L 2,2 )] 2, 2 Non-linear non-oscillatory version: SWH p,q = S 1,1 + W p, 3 8, 5 [H q (L 1,3, L 3,1 ), L 2,2 )] 2, 8 R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

23 Thus, we consider the following two families of nonlinear schemes, SWH p,q = S 1,1 + W p, 3 8, 5 [H q (L 1,3, L 3,1 ), L 2,2 )] 2, 8 SHW q,p = S 1,1 + H q [W p, 3 8, 5 (L 1,3, L 2,2 ),W 8 p, 3 8, 5 (L 3,1, L 2,2 )] 2 ). 8 Nonlinear schemes of the form: f l (Z), (S N f ) n = (S 1,1 f ) n + F( 2 f ) n, n Z (5) where F : l (Z) l (Z) is a nonlinear operator. R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

24 1 f S 3, ,1 1,2 1,3 0 1 f S 3, ,1 2,2 2, f S 3, ,1 1,2 1 1,3 1.2 f S 3, ,1 2,2 1 2, R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

25 Polynomial reproduction. Difference schemes 1 Interpolatory subdivision from Interpolatory techniques 2 Nonlinear averages and Non-oscillatory schemes A different perspective 3 6-Point Nonlinear, Non-Oscillatory, schemes Polynomial reproduction. Difference schemes Convergence. Smoothness of limit functions Order of accuracy Stability 4 Conclusion R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

26 Polynomial reproduction. Difference schemes Proposition The schemes SWH p,q, SHW q,p reproduce exactly Π 3. Linear subdivision schemes: exact polynomial reproduction guarantees existence of difference schemes. S [l] l = l S. Nonlinear subdivision schemes: offset invariance guarantees existence of difference schemes. Definition (Oswald-Harizanov, 2010) A binary subdivision operator S is offset invariant (OSI) for Π k if for each f l (Z) and any polynomial P(x) Π m, m k there exists a polynomial, Q, of degree < m such that S(f + P Z ) = Sf + (P + Q) 2 1 Z R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

27 Polynomial reproduction. Difference schemes Proposition The schemes SWH p,q, SHW q,p are offset invariant for Π 1. There exist S [1] and S [2] for the new families of schemes. R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

28 Convergence. Smoothness of limit functions 1 Interpolatory subdivision from Interpolatory techniques 2 Nonlinear averages and Non-oscillatory schemes A different perspective 3 6-Point Nonlinear, Non-Oscillatory, schemes Polynomial reproduction. Difference schemes Convergence. Smoothness of limit functions Order of accuracy Stability 4 Conclusion R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

29 Convergence. Smoothness of limit functions S N = S 1,1 + F 2, (6) Theorem (C1+ C2 S N is uniformly convergent) C1. M > 0 : F(f ) M f, C2. L > 0, 0 < T < 1 : 2 S L N (f ) T 2 f. 2 S L N = (S [2] N )L 2 C2 C2 : L > 0, 0 < T < 1 : (S [2] N )L (f ) T f. Theorem SWH p,q and SHW q,p are uniformly convergent, for all p,q 1. SWH p,q, SHW q,p < 1 R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

30 Convergence. Smoothness of limit functions f SWH 2,2 S 3,3 S H f SWH 2,2 S 3,3 S H f 1,1 1,2 2,1 2,2 S 3, f 1,1 1,2 2,1 2,2 S 3, R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

31 Convergence. Smoothness of limit functions Numerical study of smoothness: Assume f (x) = S f 0 C r,l = [r], 0 β = r l < 1 f (l) (x k i+1 ) f (l) (x k i ) f (l) (x k+1 i+1 ) f (l) (x k+1 i ) Ch β k Ch β k+1 = 2 β. S interpolatory: f k i l+1 f k i l+1 f k+1 i = f (x k i ), f (l) (x k i ) l f k i /(h k ) l = 2 lk l f k i /h 0, ( 2 l+β l+1 f k ) l + β log 2 l+1 f k+1, ) ( l R l S = log 6 2, lk l7 := sup{ ( l+1 f k ) i : xi k [a,b]}. R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

32 Convergence. Smoothness of limit functions Left columns [a, b] = [ 0.1, 0.1]. Right columns [a, b] = [ 3, 3]. l R l S 3,3 R l SWH 1,2 R l SWH 2,1 R l SWH 2,2 R l S H Table: Coarse Gaussian data (x=0 not in initial grid). l R l S 3,3 R l SWH 1,2 R l SWH 2,1 R l SWH 2,2 R l S H Table: Coarse Gaussian data. (x=0 in initial grid). R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

33 Order of accuracy 1 Interpolatory subdivision from Interpolatory techniques 2 Nonlinear averages and Non-oscillatory schemes A different perspective 3 6-Point Nonlinear, Non-Oscillatory, schemes Polynomial reproduction. Difference schemes Convergence. Smoothness of limit functions Order of accuracy Stability 4 Conclusion R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

34 Order of accuracy f i = F(x i ), x i+1 x i = h, i F smooth, Proposition (r = 2p + 2) If 2 f n have the same sign for each n and F (x) > ρ > 0 S 2,2 f S Hp f = O(h r ) Proposition (r = min{2p + 2,3q + 2}) If, for each n, (L l,r f ) n have the same sign, and F (x) > ρ > 0, then S 3,3 f SWH p,q f = O(h r ) = S 3,3 f SHW q,p f R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

35 Order of accuracy Definition (approximation order em after one iteration r) Sf F 2 1 hz =O(h r ) Corollary (r = min{4,2p + 2}) If, for each n, (L l,r f ) n have the same sign, and F (x) > ρ > 0, then S Hp f F 2 1 hz = O(h r ) Corollary (r = min{6,2p + 2,3q + 2}) If, for each n, (L l,r f ) n have the same sign, and F (x) > ρ > 0, then SWH p,q f F 2 1 hz = O(h r ) = SHW q,p f F 2 1 hz. R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

36 Order of accuracy S a convergent subdivision scheme, f i = F(ih),i Z, F(x) smooth Definition (approximation order r) S f F = O(h r ) Proposition S convergent subdivision scheme s.t. Sf F 2 1 hz =O(h r ) then, if S stable S f F = O(h r ) R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

37 Order of accuracy S f 0 F L ([a,b]) = O(h r ): Numerical study of r E S (h) := max{ (S L f 0 ) n F(n2 L h), n2 L h [a,b]} S f 0 F L ([a,b]) Gaussian data: F(x) = e 2x2, h 0 = 0.1 L = 7 Left column :[a,b] = [ 0.4,0.4] ( F (x) > ρ > 0) Right column: [a,b] = [ 1, 0.3], (F (0.5) = 0). n E S3, e-6 2.6e e-8 4.0e e e e e-11 r n r n, the numerical order of accuracy= slope of the line obtained by linear regression of the data (log 2 (h n ),log 2 (E S (h n )), h n = h 0 /2 n, R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

38 Order of accuracy Numerical Evidence: The order of approximation of the new schemes coincides with the order of approximation after one interation. n E W(2,1) E W(2,2) E W(2,3) 0 1.7e-5 1.5e-5 6.3e-6 8.9e-6 6.3e-6 8.7e e-7 5.3e-7 1.0e-7 2.1e-7 1.0e-7 2.1e e-8 1.7e-8 1.7e-9 5.7e-9 1.7e-9 5.6e e e e e e e-10 r n r t r n = Numerical order of approximation (S S 7 ) r t =Theoretical order of approximation after one iteration. red r t : from Corollary (only F > ρ > 0). blue r t from direct Taylor expansions (if (L l,r 2 f ) i do not change sign). Similar conclusions for other data coming from smooth functions R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

39 Stability 1 Interpolatory subdivision from Interpolatory techniques 2 Nonlinear averages and Non-oscillatory schemes A different perspective 3 6-Point Nonlinear, Non-Oscillatory, schemes Polynomial reproduction. Difference schemes Convergence. Smoothness of limit functions Order of accuracy Stability 4 Conclusion R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

40 Stability Definition (Lipschitz Stability) S j f S j g C f g f,g l (Z) j 0 Numerical study of stability C j S (h) = sup 1 θ =1 h Sj (f 0 + hθ) S j (f 0 ), h > 0, (7) (sup is taken over a large number of perturbations θ, θ = 1. ) If S is stable, C j S C, j: any deviation with respect to this behavior is a sign of the unstability of the scheme. Numerical tests: f 0 = ( 1,0,1,1, 1, 1, 1,1,1), values of θ randomly chosen from the set { 1,0,1}. R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

41 Stability ,1 1,2 2,1 1, ,3 2,2 3, , Non-stable for p + q > 3! R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

42 Stability q\p q\p Table: Stability/accuracy persective on SWH p,q and SHW q,p. R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

43 Stability S N = S 1,1 + F 2 Stability follows from the contractivity of some power of the second difference scheme. Theoretical stability proofs in S. Harizanov, P. Oswald Stability of nonlinear subdivision and multiscale transforms Constr. Approx F. Arandiga, R.D., M. Santagueda The PCHIP Subdivision scheme JCAM, 2016 use the theory of Generalized Gradients of piecewise smooth Lipschitz functions/generalized Jacobians of nonlinear subdivision schemes defined by such functions. but... W p,a,b (x,y) admits Generalized Gradients SWH [2] p,q admits a Generalized Jacobian, R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

44 Stability Contractivity of S [2] : Numerical study T j S (h) sup 1 θ =1 h (S[2] ) j (f 0 + hθ) (S [2] ) j (f 0 ), h > 0, (8) ,1 1,2 2, R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

45 Conclusion 1 Interpolatory subdivision from Interpolatory techniques 2 Nonlinear averages and Non-oscillatory schemes A different perspective 3 6-Point Nonlinear, Non-Oscillatory, schemes Polynomial reproduction. Difference schemes Convergence. Smoothness of limit functions Order of accuracy Stability 4 Conclusion R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

46 Conclusion We have constructed two families of non-oscillatory subdivsion schemes that can be considered nonlinear/non-oscillatory versions of the 6-point Deslauries-Dubuc interpolatory subdivision scheme. Convergence (via second-difference scheme). Numerical study of regularity of limit function. Approximation properties. Order of approximation 5 (p = 2,q = 1). Order of approximation 6 for p 2,q 2 (Numerically) Stability. Some negative results. Some possibly possitive results hard to prove. R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

47 Conclusion THANKS FOR YOUR ATTENTION! R.D., S.L.-U., M.S. (UV, UJI) Non-Oscillatory 6-point schemes Bernried / 46

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