Polygonal Splines and Their Applications Motivation Polynomial for - PowerPoint PPT Presentation

Ming-Jun Lai Polygonal Splines and Their Applications Motivation Polynomial for Numerical Solution of PDE 1 Splines Spline Method for PDE GBC BB functions Ming-Jun Lai Polygonal Spline Space Department of Mathematics Hexahedral Spline Space University of Georgia Athens, GA. U.S.A. Solution of PDE Oct. 27, 2015 based on a joint work with Michael Floater of University of Oslo, Norway Atlanta, GA 1 This research is supported by National Science Foundation #DMS 1521537

Introduction Multivariate splines, usually defined on a triangulation in 2D, or a Ming-Jun Lai tetrahedral partition in 3D, or spherical surface, or a simplicial partition in R n , have been developed for 30 years and they are Motivation Polynomial extremely useful to various numerical applications such as Splines computer aided geometric design, numerical solutions of various Spline Method for PDE linear and nonlinear partial differential equations, scattered data GBC interpolation and fitting, image enhancements, spatial statistical BB functions analysis and data forecasting, and etc.. Polygonal Spline Space Hexahedral Spline Space Solution of PDE

Triangulated Splines for Applications Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE 2D spline interpolation 2 image data 2 M. -J. Lai and L. L. Schumaker, Domain decomposition method for scattered data fitting, SIAM J. Num. Anal. 47(2009) 911–928.

Triangulated Splines for Applications (II) Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE 3 3 M. -J. Lai and Meile, C. , Scattered data interpolation with nonnegative preservation using bivariate splines, Computer Aided Geometric Design, vol. 34 (2015) pp. 37–49

Triangulated Splines for Applications (III) Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE 4 4 Guo, W. H. and Lai, M. -J., Box Spline Wavelet Frames for Image Edge Analysis, SIAM Journal Imaging Sciences, vol. 6 (2013) pp. 1553–1578.

Triangulated Splines for Applications (IV) Ming-Jun Lai Motivation Polynomial Splines Spline Method for PDE GBC BB functions Polygonal Spline Space Hexahedral Spline Space Solution of PDE See 5 5 Lai, M. J., Shum, C. K., Baramidze, V. and Wenston, P ., Triangulated Spherical Splines for Geopotential Reconstruction, Journal of Geodesy, vol. 83 (2009) pp. 695–708.

What Are Multivariate Splines? Ming-Jun Lai Let ∆ be a triangulation of a domain Ω ⊂ R 2 . For integers d ≥ 1 , Motivation Polynomial − 1 ≤ r ≤ d define by Splines Spline Method for S r d (∆) = { s ∈ C r (Ω) , s | t ∈ P d , t ∈ ∆ } PDE GBC BB functions the spline space of smoothness r and degree d over ∆ . Polygonal Spline Space In general, let r = ( r 1 , · · · , r n ) with r i ≥ 0 be a vector of integers. Hexahedral Spline Space Define Solution of PDE S r d (∆) = { s ∈ C − 1 (Ω) , s | e i ∈ C r i , e i ∈ E } , where E is the collection of interior edges of △ . Each spline in S r d ( △ ) has variable smoothness. This can handle the situation of hanging nodes in a triangulation!

Definition of Spline Functions Let T = � ( x 1 , y 1 ) , ( x 2 , y 2 ) , ( x 3 , y 3 ) � . For any point ( x, y ) , let Ming-Jun Lai b 1 , b 2 , b 3 be the solution of Motivation x = b 1 x 1 + b 2 x 2 + b 3 x 3 Polynomial Splines y = b 1 y 1 + b 2 y 2 + b 3 y 3 Spline Method for 1 = b 1 + b 2 + b 3 . PDE GBC Fix a degree d > 0 . For i + j + k = d , let BB functions Polygonal Spline d ! Space 1 b j i ! j ! k ! b i 2 b k B ijk ( x, y ) = Hexahedral 3 Spline Space Solution of PDE which is called Bernstein-B´ ezier polynomials. For each T ∈ ∆ , let � c T S | T = ijk B ijk ( x, y ) . i + j + k = d We use s = ( c T ijk , i + j + k = d, T ∈ ∆) be the coefficient vector to denote a spline function in S − 1 d (∆) .

Evaluation and Derivatives Ming-Jun Lai Motivation We use the de Casteljau algorithm to evaluate a Bernstein-B´ ezier Polynomial Splines polynomial at any point inside the triangle. It is a simple and Spline Method for stable computation. PDE Let T = � v 1 , v 2 , v 3 � and S | T = � GBC i + j + k = d c ijk B ijk ( x, y ) . Then BB functions directional derivative Polygonal Spline � Space D v 2 − v 1 S | T = d ( c i,j +1 ,k − c i +1 ,j,k ) B ijk ( x, y ) . Hexahedral Spline Space i + j + k = d − 1 Solution of PDE Similar for D v 3 − v 1 S | T . D x and D y are linearly combinations of these two directional derivatives.

Smoothness Condition between Triangles Let T 1 and T 2 be two triangles in ∆ which share a common edge Ming-Jun Lai e . Then S ∈ C r ( T 1 ∪ T 2 ) if and only if the coefficients of c T 1 ijk and Motivation c T 2 ijk satisfy the following linear conditions. E.g., Polynomial Splines S ∈ C 0 ( T 1 ∪ T 2 ) iff c T 1 0 ,j,k = c T 2 j,k, 0 , j + k = d Spline Method for PDE S ∈ C 1 ( T 1 ∪ T 2 ) iff c T 1 1 ,j,k = b 1 c T 2 j +1 ,k, 0 + b 2 c T 2 j,k +1 , 0 + b 3 c T 2 GBC j,k, 1 BB functions for i + k = d − 1 and etc. (cf. [Farin, 86] and [de Boor, 87]). We Polygonal Spline code them by H c =0. Space Hexahedral Spline Space Solution of PDE

Integration Let s be a spline in S r d ( △ ) with s | T = � Ming-Jun Lai i + j + k = d c T ijk B i jk ( x, y ) , T ∈ △ . Then Motivation � � � A T Polynomial c T s ( x, y ) dxdy = � d +2 � ijk . Splines Ω Spline Method for T ∈△ 2 i + j + k = d PDE If p = � i + j + k = d a ijk B ijk ( x, y ) and q = � GBC i + j + k = d b ijk B ijk ( x, y ) BB functions over a triangle T , then Polygonal Spline Space � Hexahedral p ( x, y ) q ( x, y ) dxdy = a ⊤ M d b , Spline Space T Solution of PDE where a = ( a ijk , i + j + k = d ) ⊤ , b = ( b ijk , i + j + k = d ) ⊤ , M d is a symmetric matrix with known entries (cf. [Chui and Lai, 1992]). Similarly, we have (cf. [Awanou and Lai, 2005]) � p ( x, y ) q ( x, y ) r ( x, y ) dxdy = a ⊤ A d b ⊙ c . T

Spline Approximation Order We have (cf. [Lai and Schumaker’98] 6 ) Ming-Jun Lai Motivation Theorem Polynomial Splines Suppose that △ is a β -quasi-uniform triangulation of domain Spline Method for Ω ∈ R 2 and suppose that d ≥ 3 r + 2 . Fix 0 ≤ m ≤ d . Then for any PDE f in a Sobolev space W m +1 GBC (Ω) , there exists a quasi-interpolatory p BB functions spline Q f ∈ S r d ( △ ) such that Polygonal Spline Space � f − Q f � k,p, Ω ≤ C |△| m +1 − k | f | d +1 ,p, Ω , ∀ 0 ≤ k ≤ m + 1 , Hexahedral Spline Space Solution of PDE for a constant C > 0 independent of f , but dependent on β and d . See more detail in monograph 7 6 Lai, M. J. and Schumaker, L. L., Approximation Power of Bivariate Splines, Advances in Computational Mathematics, vol. 9 (1998) pp. 251–279. 7 M. -J. Lai and L. L. Schumaker, Spline Functions on Triangulations, Cambridge University Press, 2007.

The Weak Form of PDE’s The weak formulation for elliptic PDE’s reads : find u ∈ H k (Ω) Ming-Jun Lai which satisfies its boundary condition such that Motivation ∀ v ∈ H r a ( u, v ) = � f, v � , 0 (Ω) , (1) Polynomial Splines where a ( u, v ) is the bilinear form defined by Spline Method for PDE �  GBC   ∇ u · ∇ vdxdy, k = 1; BB functions � Ω a ( u, v ) = Polygonal Spline  Space  △ u △ vdxdy, k = 2 , Hexahedral Ω Spline Space � and � f, v � = Ω f ( x, y ) v ( x, y ) dxdy is the L 2 inner product of f and Solution of PDE v . Here H k (Ω) and H k 0 (Ω) are standard Sobolev spaces. Clearly, our model problem is the Euler-Lagrange equation associated with a minimization prpblem: u ∈ H k (Ω) E ( u ) , u satisfies given boundary conditions min (2) where the energy functional E ( u ) = 1 2 a ( u, u ) − � f, u � .

Our Spline Method [1] We write s ∈ S − 1 d ( △ ) in Ming-Jun Lai � Motivation c t i,j,k B t s ( x, y ) | t = i,j,k ( x, y ) , ( x, y ) ∈ t ∈ △ . Polynomial i + j + k = d t Splines Spline Method for PDE Let c = ( c t i,j,k , i + j + k = d t , t ∈ △ ) be the B-coefficient vector GBC associated with s . BB functions [2] We compute mass and stiffness matrices: Polygonal Spline Space E ( s ) = 1 Hexahedral 2 c ⊤ K c − c ⊤ M f Spline Space Solution of PDE where K and M are diagonally block matrices. [3] Since s ∈ S r d ( △ ) , we have the smoothness conditions H c = 0 . Also, the boundary condition can be written as B c = g . [4] FInally we solve the constrained minimization problem: min E ( s ) , subject to H c = 0 , B c = g .