Schumaker, L. L. (1988), Dual bases for spline spaces on cells, Comp. Aided Geom. Design, 5, pp. 277--284.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....method of passing through the vertices of 4, v is not semi singular of type 1 w.r.t. 4 i . In particular, if v = w for some 2 f0; kg. Moreover, by assumption v can be semi singular of type 2 w.r.t. 4 i only if v is singular. In this case, we have r = 3, and it follows from Theorem 3. 3 in [13] that the coefficient a [T3 ] 1;1;1 is uniquely determined. Otherwise, if r 4, then for some l 2 f1; r Gamma 1g one common edge of T n l and T n l 1 is non degenerate at v, and we can also proceed with our arguments. Since all relevant differentiability conditions at the edges with ....

....vertices v; v l Gamma1 ; v l and by p l 2 Pi 3 the polynomial pieces on T l , l = 1; 5, in the representation (1) We consider the set C 2 = fa [T l ] 3 Gamma Gammaoe;oe ; oe = 0; 3 Gamma ; 1; 3; l = 1; 5g. For C 1 splines, it follows from Theorem 3. 3 in [13] that each subset of C 2 that uniquely determines all coefficients of C 2 has cardinality 8 and contains the coefficients a [T l ] 1;0;2 ; l = 3; 4. By the proof of Theorem 4, the coefficients a [T1 ] 1;2 Gammaoe;oe ; oe = 0; 1; 2, and a [T2 ] 1;2 Gammaoe;oe ; oe = 1; 2, are uniquely ....

Schumaker, L.L., Dual bases for spline spaces on a cell, Comput. Aided

....admissible set for S r q ( if for every choice of coecients a( 2 IR; l; m, a unique spline s 2 S r q ( exists with these coecients in the representation (1) of s. We remark that the notion of admissible sets is closely related to the notion of minimally determining sets (cf. [3, 4, 31, 53, 54]) However, we need this notion for describing the interpolation sets in a uni ed way and for the argumentations in our proofs. We need the following simple lemma on the connection of admissible sets and the dimension of S r q ( Lemma 3.1 Let f 1 ; m g be an admissible set for S r ....

L.L. Schumaker, Dual bases for spline spaces on a cell, Comp. Aided Geom. Design 5 (1987) 277-284.

.... surface are discussed in [6] Even though we are working in IR 3 , because of the nature of homogeneous polynomials which are essentially bivariate functions the entire development is closely modelled after the analysis of the bivariate spaces of splines S r d ( Delta) carried out in [8, 15, 17]. 2. Homogeneous spline spaces. We begin by introducing some notation, closely following [4] Definition 1. Let fv1 ; v2 ; v3g be a set of linearly independent unit vectors in IR 3 . We call (3) T = fv 2 IR 3 : v = b1v1 b2v2 b3v3 with b i 0g the trihedron generated by fv1 ; v2 ; v3g. ....

....In view of these properties, we say that such splines have local support. The duality property (11) assures that the splines s for 2 are linearly independent, and since there are precisely #(G) of them, they form a basis for H 0 d (T ) To obtain analogous results for H r d (T ) we follow [8, 15, 17]. To get an upper bound on dimension, we construct a determining set Gamma ae such that if s 2 H r d (T ) 12) fls = 0 for all fl 2 Gamma implies s j 0: Then as shown in [8] dimH r d (T ) is bounded above by the cardinality of Gamma. We can get a lower bound for the dimension (and construct ....

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L. L. Schumaker, Dual bases for spline spaces on cells, Comput. Aided Geom. Design, 5 (1988), pp. 277--284.

....65D07 1. Introduction. Given a regular triangulation 4, let S r d (4) fs 2 C r( Omega Gamma : sj T 2 P d for all triangles T 2 4g; where P d is the space of polynomials of degree d, and Omega is the union of the triangles in 4. Such spline spaces have been heavily studied, cf. e.g. [1 14] and references therein. Of particular interest for applications are spline spaces that possess a basis where every spline is supported only on the star of a vertex. The star of a vertex is the set of triangles sharing that vertex. Using such bases in applications leads to sparse linear systems. ....

.... in rings R 0 (v) R 2 (v) with the B net of a spline in S 1 2 (4H ) The subset of points which are marked with a box in the figure form a minimal determining set for S 1 2 (4H ) This follows from the general theory of minimal determining sets for spline spaces on vertex stars given in [14], but can also easily be verified directly. For a spline s 2 V 1 4 (4H ) not all of these coefficients can be set independently, since the smoothness conditions coupled with the boundary conditions imply that certain coefficients in the second ring must be automatically zero. In particular, the ....

[Article contains additional citation context not shown here]

L.L. Schumaker, Dual bases for spline spaces on cells, Comput. Aided Geom. Design, 5 (1988), pp. 277--284.

....M r is a determining set for S r (TCT ) To show that M r is a minimal determining set, we show that its cardinality is equal to the dimension of S r (TCT ) It is easy to check that #M r is equal to the number in (4.6) Now consider the superspline space S 2m;5m 1 6m 1 (TCT ) By Theorem 2. 2 in [16], the dimension of this space is (46m 2 34m 6) 2. Our space S r (TCT ) is the subspace which satisfies the 2m 2 Gamma m special conditions (4.2) 4.5) and the supersmoothness C 4m (v i ) for i = 1; 2; 3. Enforcing this supersmoothness requires an additional 3(m 2 m) 2 conditions, ....

....and thus M r is a determining set. To show that M r is a minimal determining set, we now show that its cardinality is equal to the dimension of S r (TCT ) It is easy to check that #M r is equal to the number in (5.6) Now consider the superspline space S 2m 1;5m 2 6m 3 (TCT ) By Theorem 2. 2 in [16], the dimension of this space is (46m 2 68m 24) 2. Our space S r (TCT ) is the subspace which satisfies the 2m 2 m special conditions (5.2) 5.5) and the supersmoothness C 4m 1 (v i ) for i = 1; 2; 3. Enforcing this supersmoothness requires an additional 3(m 2 m) 2 conditions, and ....

Schumaker, L. L., Dual bases for spline spaces on cells, Comput. Aided Geom. Design 5 (1988), 277--284.

....; a 1 ;n g. Nevertheless, these conditions must be satisfied since the number of free parameters c , 2 Gamma k n Gamma k Gamma1 , used in the above computation on ring R k (v) is equal to dimS r; k (4 v ) Gamma dimS r; k Gamma1 (4 v ) 4(k Gamma r) n k (cf. Theorem 2. 2 of [31]) Thus, we are able to compute all coefficients c , 2 D k (v)n Gamma k , by applying Lemma 6.1 of [24] several times. By that lemma, the maximum of the computed coefficients is bounded by a constant K times the maximum of the set coefficients, where K depends only on d and the smallest angle ....

....r; d (4 v ) on D k (v) we need to supplement Gamma k Gamma1 with an appropriate subset of the domain points on the ring R k (v) Using the fact that v is not a singular vertex, it is easy to see that the number of edges attached to v with different slopes is at least three. Then Theorem 2. 2 of [31] implies m : dimS r; k (4 v ) Gamma dimS r; k Gamma1 (4 v ) 4(k Gamma r) 3:8) Thus, to get a minimal determining set Gamma k for S r; d (4 v ) on D k (v) we need to add to Gamma k Gamma1 exactly m points on the ring R k (v) To simplify the discussion of how to choose these m ....

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Schumaker, L. L., Dual bases for spline spaces on cells, Comput. Aided Geom. Design 5 (1988), 277--284. 24

....in (4.2) supported on T is r 2 2 Gammar X j=1 (r j 1) 2 2 = dimP ; which shows (1.2) and proves our claim. We turn now to the task of constructing LLI bases for the spaces V r;r j for j = 1; Gamma r. We make use of the cofactor approach used in [21 23]. Without loss of generality, we may assume that v = 0; 0) and the cell is rotated so that all of the coordinates (x i ; y i ) of the points v i are nonzero. Let y ff i x = 0 be the equation of the i th edge e i attached to v, where ff i = Gammay i =x i . Then every spline g 2 S r (4 v ) ....

....) ff 2 i Gamma1 ; and D e i denotes the derivative in the normal direction to e i , i.e. D e i : 1 ff 2 i ) Gamma1=2 (D y ff i D x ) 6 Then for any spline g 2 S r (4 v ) a P jk = P jk g and a [i] jk = i] jk g for i = 1; n. By Theorem 2. 2 in [23], n j : dimV r;r j = nj (r j 1 Gamma je) j = 1; Gamma r; where e is the number of edges attached to v with different slopes. We distinguish three cases. Case 1: Suppose r j 1 je. Then n j = r j 1 (n Gamma e)j. If e = n, i.e. all edges attached to v have ....

[Article contains additional citation context not shown here]

Schumaker, L. L., Dual bases for spline spaces on cells, Comput. Aided Geom. Design 5 (1988), 277--284.

....x1. Introduction Given a regular triangulation 4, let S r d (4) fs 2 C r( Omega Gamma : sj T 2 P d for all triangles T 2 4g; where P d is the space of polynomials of degree d, and Omega is the union of the triangles in 4. Such spline spaces have been heavily studied, cf. e.g. [1 10] and references therein. Of particular interest for applications are spline spaces that possess a basis where every spline is supported only on the star of a vertex. The star of a vertex is the set of triangles sharing that vertex. Using such bases in applications leads to sparse linear systems. ....

.... in rings R 0 (v) R 2 (v) with the B net of a spline in S 1 2 (4H ) The subset of points which are marked with a box in the figure form a minimal determining set for S 1 2 (4H ) This follows from the general theory of minimal determining sets for spline spaces on vertex stars given in [10], but can also easily be verified directly. For a spline s 2 V 1 4 (4H ) not all of these coefficients can be set independently, since the smoothness conditions coupled with the boundary conditions imply that certain coefficients in the second ring must be automatically zero. In particular, the ....

[Article contains additional citation context not shown here]

Schumaker, L. L., Dual bases for spline spaces on cells, Comput. Aided Geom. Design 5 (1988), 277--284.

.... surface are discussed in [5] Even though we are working in IR 3 , because of the nature of homogeneous polynomials which are essentially bivariate functions the entire development is closely modelled after the analysis of the bivariate spaces of splines S r d ( Delta) carried out in [7, 15, 16]. 2. Homogeneous Spline Spaces We begin by introducing some notation, closely following [4] Definition 2.1. Let fv 1 ; v 2 ; v 3 g be a set of linearly independent unit vectors in IR 3 . We call T = fv 2 IR 3 : v = b 1 v 1 b 2 v 2 b 3 v 3 with b i 0g (2:1) the trihedron generated ....

....In view of these properties, we say that such splines have local support. The duality property (2. 9) assures that the splines s for 2 are linearly independent, and since there are precisely #(G) of them, they form a basis for H 0 d (T ) To obtain analogous results for H r d (T ) we follow [7, 15, 16]. To get an upper bound on dimension, we construct a determining set Gamma ae such that if s 2 H r d (T ) fls = 0 for all fl 2 Gamma implies s j 0: 2:10) Then as shown in [7] dim H r d (T ) is bounded above by the cardinality of Gamma. We can get a lower bound for the dimension (and ....

[Article contains additional citation context not shown here]

Schumaker, L. L., Dual bases for spline spaces on cells, Comput. Aided Geom. Design 5 (1988), 277--284.

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Schumaker, L. L. (1988), Dual bases for spline spaces on cells, Comp. Aided Geom. Design, 5, pp. 277--284.

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Schumaker, L. L. (1988), Dual bases for spline spaces on cells, Comp. Aided Geom. Design, 5, pp. 277--284.

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