ABSTRACT
In this work we analyze the approximation error in Sobolev norms for quasi-interpolation operators in spline spaces. We establish in a general way the hypotheses on a quasi-interpolant to achieve the optimal order of approximation. Finally, we propose simple but general constructions of such operators that satisfy the established hypotheses and illustrate their performance through some numerical tests.
Keywords:
spline approximation; quasi-interpolation; stability; B-splines
1 INTRODUCTION
Quasi-interpolation operators represent a practical and efficient method when calculating approximations using spline functions since they present a great simplicity and flexibility in their construction. To define them, B-splines bases are used and the coefficients are chosen locally. That is, each coefficient depends only on the data found in the support of the corresponding B-spline. This localization implies that changes in the data or space of splines have a low computational cost to recalculate the approximation, because the changes are only local.
Quasi-interpolation operators play a fundamental role from several points of view. On the one hand, they constitute a key tool in the theoretical analysis of the approximation power of spline spaces. On the other hand, they provide simple recipes for building good approximations of functions in spline spaces, that are needed in practical algorithms for solving partial differential equations; for example, for imposing boundary conditions and for keeping important information after coarsening in time dependent equations.
2 SPLINE SPACES AND B-SPLINE BASES
Let and let where ζ 1 = a, ζ N = b and , for . Let p be a polynomial degree. We associate a number m j , called multiplicity, to each breakpoint ζ j , such that and , for . Let 𝒮 be the space of piecewise polynomials of degree ≤ p on Z such that they have continuous derivatives at the breakpoint ζ j . It is known that 𝒮 is a vector space of finite dimension with .
Let be the associated (p + 1)-open knot vector, i.e.,
Let be the B-spline basis of degree p associated to the knot vector Ξ, see e.g., 22 C. de Boor. “A practical guide to splines”, volume 27. Springer-Verlag, New York, revised edition, 2001 (1978).), (55 L. Schumaker. “Spline functions: basic theory”. Cambridge University Press (2007).. We remark that B-splines are non-negative, locally supported, and form a convex partition of unity, namely,
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βi ≥ 0, for i = 1, 2, . . . , n.
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, for i = 1, 2, . . . , n.
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In Figure 1 we show some examples of cubic B-splines.
Examples of cubic B-splines of maximum smoothness (top) and with a triple internal knot (bottom).
Let ℐ be the mesh defined by . Notice that for each there exists a unique such that and . The union of the supports of the B-splines acting on I identifies the support extension, namely
3 QUASI-INTERPOLATION OPERATORS: STABILITY AND POWER OF APPROXIMATION
Let E ⊂ ℝ be a closed interval, r ∈ ℕ0 and . The Sobolev space W r,q (E) is defined by
Notice that .
A quasi-interpolation operator is defined by
for some linear functionals .
We consider the following usual assumption which is not too strong.
Assumption 1. There exists a constant such that
for all .
3.1 Local stability and approximation properties in L q -norms
The proofs in this section follow exactly the same lines in [44 T. Lyche , C. Manni & H. Speleers. Foundations of spline theory: B-splines, spline approximation, and hierarchical refinement. In “Splines and PDEs: From Approximation Theory to Numerical Linear Algebra”. Fond. CIME/CIME Found. Subser., Springer, Cham (2018), p. 1-76. Lecture Notes in Math, 2219.., Theorem 16], but we include them here for the sake of completeness.
In the following result we prove that Assumption 1 guarantees that the quasi-interpolation operator 𝒬 is locally L q -stable, .
Theorem 3.1 (LocalLq-stability).Let 𝒬 be a quasi-interpolation operator given by (3.1) that satisfies Assumption 1 with constant C 𝒬 , for some. Then, for,
Proof. Let and x ∈ I, then . Finally, taking the L q -norm on I we have that (3.2) holds. □
Let ℓ ∈ ℕ0 and 𝒫 ℓ be the space of polynomials of degree at most ℓ.
Theorem 3.2 (Local approximation inLq-norms).Let 𝒬 be a quasi-interpolation operator given by (3.1) that satisfies Assumption 1 with constant C 𝒬 , for some. Let. If 𝒬g = g, for g ∈ 𝒫 ℓ , then,
for all I ∈ ℐ.
Proof. Let g ∈ 𝒫 ℓ . Then, using that 𝒬g = g and the local L q -stability of 𝒬 (Theorem 3.1) we have that
If g is the Taylor polynomial of degree ℓ at ξ k−p to f in , the Taylor interpolation error [44 T. Lyche , C. Manni & H. Speleers. Foundations of spline theory: B-splines, spline approximation, and hierarchical refinement. In “Splines and PDEs: From Approximation Theory to Numerical Linear Algebra”. Fond. CIME/CIME Found. Subser., Springer, Cham (2018), p. 1-76. Lecture Notes in Math, 2219.., Theorem 15] implies that (3.3) holds. □
Theorem 3.3 (Global approximation inLq).Let 𝒬 be a quasi-interpolation operator given by (3.1) that satisfies Assumption 1 with constant C 𝒬 , for some. Let. If 𝒬g = g, for g ∈ 𝒫 ℓ , then,
where and .
Proof. The proof follows from Theorem 3.2 using that and the fact that each knot interval I belongs (at most) to 2p + 1 support extensions. □
3.2 Local stability and power of approximation in high order norms
Now we extend the results from the previous section by considering Sobolev seminorms in the right hand side of the inequalities.
Following the same lines in the proof of [44 T. Lyche , C. Manni & H. Speleers. Foundations of spline theory: B-splines, spline approximation, and hierarchical refinement. In “Splines and PDEs: From Approximation Theory to Numerical Linear Algebra”. Fond. CIME/CIME Found. Subser., Springer, Cham (2018), p. 1-76. Lecture Notes in Math, 2219.., Lemma 4] we obtain the next auxiliary result.
Lemma 3.1.Let 𝒬 be a quasi-interpolation operator given by (3.1) that satisfies Assumption 1 with constant C 𝒬 , for some. Then, for,
for all I ∈ ℐ where
Proof. Let and x ∈ I, then using the upper bound for the r-th derivative of the B-splines from [44 T. Lyche , C. Manni & H. Speleers. Foundations of spline theory: B-splines, spline approximation, and hierarchical refinement. In “Splines and PDEs: From Approximation Theory to Numerical Linear Algebra”. Fond. CIME/CIME Found. Subser., Springer, Cham (2018), p. 1-76. Lecture Notes in Math, 2219.., Proposition 2] we have that
Finally, using Assumption 1 we conclude the proof. □
Definition 3.1 (Locally quasi-uniform mesh).The mesh ℐ is locally quasi-uniform with parameter θ > 0 if
Recall that 𝒫 ℓ denotes the space of polynomials of degree at most ℓ.
Theorem 3.4 (Local stability in high order norms).Let 𝒬 be a quasi-interpolation operator given by (3.1) that satisfies Assumption 1 with constant C 𝒬 , for some. Let. If 𝒬g = g, for g ∈ 𝒫 ℓ , then, for,
for all I ∈ ℐ, where the constant C S > 0 depends on C 𝒬 , p, r and θ.1 1 Notice that (3.4) holds with C S = C 𝒬 when r = 0, due to Theorem 3.1.
Proof. Let . Let and let g be the Taylor polynomial of degree r − 1 at ξ k−p to f in . Since 𝒬g = g, using Lemma 3.1 and the Taylor interpolation error [44 T. Lyche , C. Manni & H. Speleers. Foundations of spline theory: B-splines, spline approximation, and hierarchical refinement. In “Splines and PDEs: From Approximation Theory to Numerical Linear Algebra”. Fond. CIME/CIME Found. Subser., Springer, Cham (2018), p. 1-76. Lecture Notes in Math, 2219.., Theorem 15] we have that
Considering that is bounded above by a constant that depends on p and θ, we conclude that (3.4) holds. □
The next result generalizes [44 T. Lyche , C. Manni & H. Speleers. Foundations of spline theory: B-splines, spline approximation, and hierarchical refinement. In “Splines and PDEs: From Approximation Theory to Numerical Linear Algebra”. Fond. CIME/CIME Found. Subser., Springer, Cham (2018), p. 1-76. Lecture Notes in Math, 2219.., Proposition 6] for general quasi-interpolation operators with essentially the same proof.
Theorem 3.5 (Local approximation in high order norms).Let 𝒬 be a quasi-interpolation operator given by (3.1) that satisfies Assumption 1 with constant C 𝒬 , for some. Let. If 𝒬g = g, for g ∈ 𝒫 ℓ , then, for,
for all I ∈ ℐ.
Proof. Let g ∈ 𝒫 ℓ . Then, using that 𝒬g = g and the local stability of 𝒬 given in Theorem 3.4 we have that
If g is the Taylor polynomial of degree ℓ at ξ k−p to f in , the Taylor interpolation error [44 T. Lyche , C. Manni & H. Speleers. Foundations of spline theory: B-splines, spline approximation, and hierarchical refinement. In “Splines and PDEs: From Approximation Theory to Numerical Linear Algebra”. Fond. CIME/CIME Found. Subser., Springer, Cham (2018), p. 1-76. Lecture Notes in Math, 2219.., Theorem 15] implies that (3.5) holds. □
4 CONSTRUCTION OF LOCALLY STABLE QUASI-INTERPOLATION OPERATORS
In this section we consider the construction of quasi-interpolation operators based on 11 A. Buffa, E.M. Garau, C. Giannelli & G. Sangalli. On quasi-interpolation operators in spline spaces. In G. Barrenechea, F. Brezzi, A. Cangiani & E. Georgoulis (editors), “Building bridges: connections and challenges in modern approaches to numerical partial differential equations”, volume 114. Springer, Cham (2016), p. 73-91.), (33 B.G. Lee, T. Lyche & K. Mørken. Some examples of quasi-interpolants constructed from local spline projectors. In Mathematical methods for curves and surfaces. Innov. Appl. Math., (2001), 243-252. and apply the results stated in the previous section to analize their local approximation properties in Sobolev norms.
In order to define a quasi-interpolation operator 𝒬 given by
we need to choose appropriate linear functionals λ β , for each β ∈ ℬ.
Local approximation method. For each knot interval I ∈ ℐ, let ΠI be the L 2-projection operator onto 𝒫 p (I), the set of polynomials of degree ≤ p on I. If denotes the set of B-splines restricted to I, we have that ℬ I is a basis for 𝒫 p (I) and
for some linear functionals . Notice that which is equivalent to say that the set is a dual basis for ℬ I in the sense that . In view of [11 A. Buffa, E.M. Garau, C. Giannelli & G. Sangalli. On quasi-interpolation operators in spline spaces. In G. Barrenechea, F. Brezzi, A. Cangiani & E. Georgoulis (editors), “Building bridges: connections and challenges in modern approaches to numerical partial differential equations”, volume 114. Springer, Cham (2016), p. 73-91., Theorem 1], there exists a constant independent of I ∈ ℐ such that
for q with .
Dual basis for B-splines and quasi-intepolation operator. As explained in [11 A. Buffa, E.M. Garau, C. Giannelli & G. Sangalli. On quasi-interpolation operators in spline spaces. In G. Barrenechea, F. Brezzi, A. Cangiani & E. Georgoulis (editors), “Building bridges: connections and challenges in modern approaches to numerical partial differential equations”, volume 114. Springer, Cham (2016), p. 73-91., Section 6], we can define each λ β as a convex combination of local projections onto some I ∈ ℐ such that . More specifically, for β ∈ ℬ, we define , and for each , where with is such that on I. Now, the functional λ β is given by
where , for all I ∈ ℐ β , and .
Then, the quasi-interpolation operator 𝒬 is given by (4.1). Since ΠI is a projection onto 𝒫 p (I), it is easy to check that for any choice of the coefficients c I,β we have that , which in turn implies that 𝒬 is a projection onto the spline space, i.e., 𝒬f = f, whenever f ∈ 𝒮. Moreover, (4.2) and (4.3) imply that Assumption 1 holds with a constant C 𝒬 > 0 that depends on p and θ . Finally, applying Theorems 3.4 and 3.5 we have that 𝒬 satisfies (3.4) and (3.5), which generalize [11 A. Buffa, E.M. Garau, C. Giannelli & G. Sangalli. On quasi-interpolation operators in spline spaces. In G. Barrenechea, F. Brezzi, A. Cangiani & E. Georgoulis (editors), “Building bridges: connections and challenges in modern approaches to numerical partial differential equations”, volume 114. Springer, Cham (2016), p. 73-91., Theorem 2] when considering high order norms in the left hand side.
5 NUMERICAL TESTS
In this section we propose two specific ways of defining the coefficients c I,β , for I ∈ ℐ β , in the framework of the previous section. Each of them, in turn, gives rise a particular quasi-interpolation operator.
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(i) Given β ∈ ℬ, we let the coefficient associated to the central knot interval in the support of β be equal 1, whenever the mumber of knot intervals within its support is odd; whereas we define as the coefficients associated to the two central knot intervals if such number is even (cf. Figure 2). More precisely, let us assume that ℐ β consists of k = k(β) consecutive knot intervals, namely, I 1 , I 2 , . . . ,I k . Then, we let if k is odd, and if k is even.
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(ii) We define , for I ∈ ℐ β . The operator defined with this specific choice of coefficients c I,β has been introduced in 66 D.C. Thomas, M.A. Scott, J.A. Evans, K. Tew & E.J. Evan. Bezier projection: a unified approach for local projection and quadrature-free refinement and coarsening of NURBS and T-splines with particular application to isogeometric design and analysis. Comput. Methods Appl. Mech. Engrg., 284 (2015), 55-105. where it was called Bézier projection.
The procedure from (i) for the choice of coefficients c I,β in (4.3) for some B-splines β (solid lines).
In Figure 3 we consider the L 2-error in approximations of the function f (x) = arctan(25x), for x ∈ [−1, 1], in spline spaces of degree 4 (left) and degree 5 (right) with maximum smoothness defined onto uniform partitions. We consider four kinds of approximations: the L 2-projection ΠL 2 onto the spline space, the quasi-interpolant 𝒬 0 defined in [44 T. Lyche , C. Manni & H. Speleers. Foundations of spline theory: B-splines, spline approximation, and hierarchical refinement. In “Splines and PDEs: From Approximation Theory to Numerical Linear Algebra”. Fond. CIME/CIME Found. Subser., Springer, Cham (2018), p. 1-76. Lecture Notes in Math, 2219.., Section 1.5.3.1], and the operators 𝒬 1 and 𝒬 2 obtained by picking the coefficients c I,β as described in (i) and (ii), respectively. In these cases, we obtain the optimal orders of convergence for all the considered methods. However, it is worth to notice that the behaviour of 𝒬 1 and especially 𝒬 2 are quite similar to the best approximation ΠL 2.
L2-error in different approximation methods for f in spline spaces of degrees 4 (left) and 5 (right) with maximum smoothness onto uniform meshes.
In Figure 4 we consider the L 2-error associated to the four mentioned operators in the approximation of the function , for x ∈ [0, 10], in spline spaces of degree 4 (left) and degree 5 (right) with maximum smoothness defined onto uniform partitions. Now, we obtain a suboptimal order of convergence in all cases due to the lack of regularity of the function g. But we notice that, as in the previous case, the errors for 𝒬 1 and 𝒬 2 are basically like the errorfor ΠL 2.
L2-error in different approximation methods for g in spline spaces of degrees 4 (left) and 5 (right) with maximum smoothness onto uniform meshes.
In Figure 5 we consider the number of degrees of freedom (DOFs) vs. . We compare the performance for splines of degree 4 (left) and degree 5 (right) with maximum smoothness with globally continuous splines of the same degrees defined both on the same type of mesh. First, we consider uniform meshes and we obtain suboptimal rates of convergence as before, but in this case the space of splines C 0 works better. For example, in order to get an error ≈ 10−4 using splines of degree 4 with maximum smoothness we requiere 2564 DOFs whereas using splines C 0 we only need 641 DOFs, i.e., four times less. Additionally, in Figure 5 we consider also adaptive meshes. Given the mesh ℐ, we compute the local errors , for I ∈ ℐ. Then, we refine dyadically the elements where the local error is greater than . We notice that using adaptive meshes we recover the optimal order of convergence in all cases. But now, using splines of maximum smoothness we can achieve a given accuracy with half of the DOFs used by C 0 splines.
Acknowledgments
This work was partially supported by Consejo Nacional de Investigaciones Científicas y Técnicas through grant PIP 2021-2023 (11220200101180CO), by Agencia Nacional de Promoción Científica y Tecnológica through grant PICT-2020-SERIE A-03820, by Universidad Nacional del Litoral through grant CAI+D-2020 50620190100136LI, and by Universidad Nacional de Tucumán through grant PIUNT 2020 E-685. This support is gratefully acknowledged.
REFERENCES
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1A. Buffa, E.M. Garau, C. Giannelli & G. Sangalli. On quasi-interpolation operators in spline spaces. In G. Barrenechea, F. Brezzi, A. Cangiani & E. Georgoulis (editors), “Building bridges: connections and challenges in modern approaches to numerical partial differential equations”, volume 114. Springer, Cham (2016), p. 73-91.
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2C. de Boor. “A practical guide to splines”, volume 27. Springer-Verlag, New York, revised edition, 2001 (1978).
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3B.G. Lee, T. Lyche & K. Mørken. Some examples of quasi-interpolants constructed from local spline projectors. In Mathematical methods for curves and surfaces. Innov. Appl. Math., (2001), 243-252.
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4T. Lyche , C. Manni & H. Speleers. Foundations of spline theory: B-splines, spline approximation, and hierarchical refinement. In “Splines and PDEs: From Approximation Theory to Numerical Linear Algebra”. Fond. CIME/CIME Found. Subser., Springer, Cham (2018), p. 1-76. Lecture Notes in Math, 2219..
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5L. Schumaker. “Spline functions: basic theory”. Cambridge University Press (2007).
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6D.C. Thomas, M.A. Scott, J.A. Evans, K. Tew & E.J. Evan. Bezier projection: a unified approach for local projection and quadrature-free refinement and coarsening of NURBS and T-splines with particular application to isogeometric design and analysis. Comput. Methods Appl. Mech. Engrg., 284 (2015), 55-105.
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1
Notice that (3.4) holds with C S = C 𝒬 when r = 0, due to Theorem 3.1.
Publication Dates
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Publication in this collection
27 Mar 2023 -
Date of issue
Jan-Mar 2023
History
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Received
30 Sept 2021 -
Accepted
30 June 2022