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A note on the meshless method using radial basis functions
Computers and Mathematics with Applications 55 (2008) 66–75 www.elsevier.com/locate/camwa
A note on the meshless method using radial basis functions Yong Duan School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu 610054, PR China Received 11 July 2006; received in revised form 9 March 2007; accepted 16 March 2007
Abstract In this paper, we prove some inverse inequalities for the finite dimensional subspace interpolated by radial basis functions. The properties of these inequalities are worse than those in finite element spaces. These bad properties result in a huge condition number for both the collocation method and Galerkin method using radial basis functions. As applications, theoretical estimates of the condition numbers of a meshless method using radial basis functions are given. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Radial basis function; Inverse inequalities; Collocation method; Galerkin method; Meshless
1. Introduction In the last decade, meshless methods have received much attention because of their flexibility in the construction of finite dimensional subspaces. When the dominant domain is extremely complex, traditional numerical method for partial differential equations (PDEs), such as the finite element method, finite difference method etc, will be difficult to carry out. Another virtue is that the meshless method does not need the generation of a mesh. It is a complicated work in FEM. Those complications had been described recently by Griebel and Schweitzer [1] as “a very time-consuming portion of overall computation is the mesh generation from CAD input data. Typically, more than 70 percent of overall computation is spent by mesh generators”. Meshless methods mainly include GFEM (see Babuˇska [2]), EFGM (see Belytschko [3]) and meshless methods using radial basis functions (RBFs) (see Kansa [4]). For details, we refer readers to Liu [5]. In the present paper, we restrict ourselves to RBFs. Radial basis functions are powerful tools in multi-variable approximation. The corresponding theory had been extensively studied by Narcowich [6], Schaback [7,8], Wendland [9,10], Buhmann [11] etc. Consequently, numerical methods for PDEs using RBFs are concentrated on by mathematicians and engineers because of their property of being meshless. Indeed, it is: 1. A truly meshless method. We only need a set of distinct centers to construct finite dimensional spaces. 2. Dimensional independence. In other words, it can be easily extended to higher dimensional space as onedimensional ones.
E-mail address: [email protected]. c 2007 Elsevier Ltd. All rights reserved. 0898-1221/$ - see front matter doi:10.1016/j.camwa.2007.03.011
Y. Duan / Computers and Mathematics with Applications 55 (2008) 66–75
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Both collocation and the Galerkin method using RBFs are discussed in a series of papers, for example [12–17]. In contrast to the Galerkin method, the Collocation method is more successful in numerical applications, with a lack of theoretical analysis. For the unsymmetric collocation method, we cannot ascertain that the linear system that arises is solvable. Actually, there are counterexamples which indicate that the matrix of the linear system is singular [18]. In [17], the author gives an analysis of a meshless Galerkin method for a Helmholtz problem with natural boundary conditions using radial basis functions. FEM-like convergent estimates are obtained. Some techniques, for example, Lagrange multipliers and the penalty method, are advised for Dirichlet problems. The Lagrange multipliers method combined with RBFs needs inverse estimates in finite dimensional subspaces interpolated by the RBFs. The penalty method is similar, which in FEM, including Babuˇska’s method [19] and Nistche’s method [20]. It is well known that inverse inequality plays an important role in FEM analysis. It is used in basic theory for FEM analysis in estimating the condition number of the stiffness matrix and domain decomposition method etc. In this paper, we establish some inverse inequalities for finite dimensional spaces constructed by RBFs. Then, we give estimates of condition numbers for both the collocation method and Galerkin method using RBFs. Lagrange multipliers method coupled with RBFs and the Penalty method for Dirichlet problems are also discussed. Finally, we introduce an application to the domain decomposition method using our meshless Galerkin method. The rest of this paper is organized as follows. In Section 2, we introduce some basic aspects of interpolation by RBFs. In Section 3, we prove some inequalities for spaces spanned by RBFs. In Section 4, we introduce some applications for numerical methods for PDEs using RBFs, including the estimation of condition numbers of linear systems, convergence of the Lagrange method, and convergence of the domain decomposition method. At last, we draw a conclusion from this paper and point out some future work. 2. Interpolation by RBFs For a given function space V on Ω ⊂ Rd , we define the finite dimensional space Vh ⊂ V by Vh := span{φ(x − x1 ), φ(x − x2 ), . . . , φ(x − x N )} + Pdm
(1)
where φ : Rd R is at least a C 1 -function, Pdm denotes the space of polynomials of degree less than m, and X = {x1 , x2 , . . . , x N } ⊆ Ω¯ is a set of distinct centers. h denotes the density of X , defined by h = sup min kx − x j k
(2)
Ω xi ∈X
where norm k·k is the Euclidean norm. When m = 0, we will call φ strictly positive definiteness, otherwise conditional positive definiteness of order m. For a given function f ∈ V , assuming that f (x) =
N X
αi φ(x − xi ) +
i=1
M X
β j p j (x)
(3)
j=1
for a basis p j (x) of Pdm , where M is the number of the basis p j (x). The corresponding interpolated problem reads: solving the system N X
αi φ(xk − xi ) +
i=1 N X
M X
β j p j (xk ) = f (xk ),
1≤k≤N
(4)
j=1
αi p j (xi ) = 0,
1 ≤ j ≤ M.
(5)
i=1
In self-evident matrix formulation, the system above reads as A P T αE fE = . P 0 βE 0
(6)
The solvability of this system is usually guaranteed by the requirements rank(P) = M ≤ N , and λkyk2 ≤ y T Ay,
∀y ∈ Rd ,
P y = 0.
(7)
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Y. Duan / Computers and Mathematics with Applications 55 (2008) 66–75
The condition rank(P) ≤ N will be called the Pdm -non-degeneracy of X. Definition 2.1 ([7]). A set of centers X = {x1 , x2 , . . . , x N } is called quasi-uniform for a compact set Ω ⊂ Rd , if there exist δ > 0 and h > 0 such that 1. h(x) ≤ h ∀x ∈ Ω 2. q = 12 mini6= j kxi − x j k ≥ hδ 3. each set X \ {x j }, 1 ≤ j ≤ N is Pdm -non-degenerate where h(x) =
sup
min ky − x j k
ky−xk≤ρ xi ∈X
(8)
ρ > 0 is fixed. For a quasi-uniform set of centers X , the following property holds: G(h) ≤ λ ≤ F(h).
(9)
G(h), F(h) are given by Schaback in [7]. It is obvious that h −d ∼ N when the set of centers is uniformly distributed. N is the number of centers. In [17,21], Wendland gives the Sobolev type error estimate, which is important to numerical analysis for PDEs. Assuming that the RBFs satisfying φ ∈ C l with Fourier transform φˆ satisfies ˆ ∼ (1 + ktk)−2l . φ(t)
(10)
the following interpolation error holds. Lemma 2.1. Let Ω ⊆ R d be an open and bounded domain, having a Lipschitz boundary and satisfying the interior cone condition. Denote by Su the interpolant on X = {x1 , x2 , . . . , x N } ⊆ Ω to a function u ∈ H k (Ω ) with k > d/2. Then there exists a constant h 0 > 0 such that for all X with h < h 0 , where h is the density of X , the estimate ku − su k H j (Ω ) ≤ Ch k− j kuk H k (Ω ) is valid for 0 ≤ j ≤ k.
(11)
For the sake of completeness, we introduce some radial basis functions below: Gaussians exp(−ck · k2 ) c > 0 3 Multiquadrics (c2 + k · k2 )β− 2 β ∈ N , c 6= 0 Thin-plate splines k · kβ log k · k β ∈ 2N . The RBFs introduced above are globally supported RBFs. One of the most popular classes of CS-RBFs is introduced by Wendland in [9]. Another is given by Wu in [22]. These functions are strictly positive definite in Rd for all d less than or equal to some fixed value d0 , and can be constructed to have any desired amount of smoothness 2κ. For l = [ d2 ] + κ + 1 and κ = 0, 1, 2, 3, Wendland’s function reads: φl,0 (r ) = ˙ (1 − r )l+ , φl,1 (r ) = ˙ (1 − r )l+1 + [(l + 1)r + 1], 2 2 φl,2 (r ) = ˙ (1 − r )l+2 + [(l + 4l + 3)r + (3l + 6)r + 3], 3 2 3 2 2 φl,3 (r ) = ˙ (1 − r )l+3 + [(l + 9l + 32l + 15)r + (6l + 36l + 45)r + (15l + 45)r + 15].
Here and throughout, = ˙ denotes equality up to a constant factor. For example, the functions φ2,0 =(1 ˙ − r )2+ φ3,1 (r ) =(1 ˙ − r )4+ [4r + 1], φ4,2 (r ) =(1 ˙ − r )6+ [35r 2 + 18r + 3], φ5,3 (r ) =(1 ˙ − r )8+ [32r 3 + 25r 2 + 8r + 1] which are in C 0 , C 2 , C 4 , C 6 , respectively, and strictly positive definite in R3 .
Y. Duan / Computers and Mathematics with Applications 55 (2008) 66–75
69
Wu’s CS-RBF reads: φ(r ) = (1 − r )6+ (5r 5 + 30r 4 + 72r 3 + 82r 2 + 36r + 6) ∈ C 6 . 3. Some inverse inequalities in Vh In this paper, Sobolev spaces H s (Ω ), H s (∂Ω ) are defined in the standard way [23]. The norms are denoted by k · k H s (Ω ) , k · k H s (∂ Ω ) respectively. We recall that the well-known trace inequality and inverse trace inequality hold for suitable Sobolev spaces in Ω ⊂ Rd . The norm in H s (Ω ), s ≥ 0, s = k + t, 0 < t < 1, integer k ≥ 0 reads: X Z Z |∂ α u(x) − ∂ α u(y)|2 kuk2H s (Ω ) = kuk2H k (Ω ) + dxdy. (12) |x − y|d+2t |α|=k Ω Ω 1
1
We will also use the space H − 2 (∂Ω ), i.e, the dual space of H 2 (∂Ω ), with the norm kµk H
−1 2 (∂ Ω )
sup
=
1
z∈H 2 (∂ Ω )
hµ, zi kzk 1
(13)
H 2 (∂ Ω )
where the duality pairing is Z hµ, zi = µzds.
(14)
∂Ω
ˆ Theorem 3.1. Assuming a strictly positive definite RBF ψ with Fourier transform ψ(t) ∼ (1 + ktk)−l−d , then for ∀g ∈ Vh and two real numbers 0 ≤ s < k, the following inequality holds kgk H k (Ω ) ≤ Ch
−2l−3d 2
(15)
kgk H s (Ω )
where C is a constant independent of the density h. Proof. Without being P Nconfused, we will denote by C a generic constant independent of h throughout this paper. Assuming that g = i=1 ψ(kx − xi k)λi , by using the result in [24], we have E 2 kgk2H 0 (Ω ) ≥ Ch 2l+2d kλk Furthermore kgk2H k (Ω )
2
X
= ψ(kx − xi k)λi
i
H k (Ω )
!2 ≤
X
kψ(kx − xi k)λi k H k (Ω )
i
!2 ≤ max kψ(kx − xi k)k2H k (Ω )
X
i
|λi |
i
Pn Pn |ai |)2 ≤ n i=1 |ai |2 , we have By using ( i=1 E k2 kgk2H k (Ω ) ≤ N max kψ(kx − xi k)k2H k (Ω ) kλ ≤ Ch
i −d
E k2 . max kψ(kx − xi k)k2H k (Ω ) kλ i
This yields kgk2H 0 (Ω ) ≥ Ch 2l+3d kgk2H k (Ω ) . Then kgk H k (Ω ) ≤ Ch
−2l−3d 2
kgk H s (Ω ) .
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Y. Duan / Computers and Mathematics with Applications 55 (2008) 66–75
Remark 3.1. Unlike in FEM, the information about k, s is shown in the constant C. Actually, C depends on the embedding constant from H s (Ω ) to H 0 (Ω ) and the quantity maxi kψ(kx − xi k)k H k (Ω ) , which will be fixed whenever the RBF is chosen. The well-known inverse inequality in a suitable finite element space reads [25]: kgk H k (Ω ) ≤ Ch s−k kgk H s (Ω ) .
(16)
The inverse inequality is worse than that in FEM. It has been shown that this is true. An obvious result is that the condition number of a linear system arising in a meshless method using RBFs is worse. It has been verified in computational experiments [12]. As applications, we will estimate those condition numbers by using inverse inequalities in the next section. Theorem 3.2. Assuming that ψ a strictly positive definite RBF, then for ∀g ∈ Vh and two real numbers 0 ≤ s < k, the following inequality holds kgk H k (Ω ) ≤ Ch −d G −1 (h)kgk H s (Ω )
(17)
where C is a constant independent of the density h. PN Proof. Assuming that g = i=1 ψ(kx − xi k)λi , by using the Riemann sum, we have Z kgk2H 0 (Ω ) = |g|2 Ω
≥ Ch d
N X
|g(xi )|2
i=1
E k2 = Ch d kAλ where A is the interpolated matrix. As pointed out in Section 2, the eigenvalues of matrix A have lower bound G(h). This yields E k2 ≥ C G 2 (h)h d kλ E k2 . kAλ Following the same procedure as in the proof of Theorem 3.1, we have kgk H k (Ω ) ≤ Ch −d G −1 (h)kgk H s (Ω ) .
1
1
In the case of s < 0, the most popular inverse inequality is about H − 2 (∂Ω ) and H 2 (∂Ω ). In FEM, kgk Chkgk
H
−1 2 (∂ Ω )
1
H 2 (∂ Ω )
≤
, for a suitable finite element space. Denote Λh = span{ψ(x − xi )}, {x1 , . . . , x N } is a set of distinct
centers on ∂Ω with density h 2 and ψ is a strictly positive definite RBF, and we have ˆ Theorem 3.3. Assuming that ψ(t) ∼ (1 + ktk)−l−d ; then for ∀g ∈ Λh , the following inequality holds kgk
1
H 2 (∂ Ω )
≤ Ch −2l−3d+3 kgk 2
H
(18)
−1 2 (∂ Ω )
where C is a constant independent of the density h 2 . Proof. We will denote by C, a generic constant independent of h 2 . According to the definition of kgk have kgk H
−1 2 (∂ Ω )
sup
= µ∈H
1 2 (∂ Ω )\{0}
hg, gi ≥ kgk 1 =
H 2 (∂ Ω ) kgk2H 0 (∂ Ω )
kgk
1
H 2 (∂ Ω )
.
hg, µi kµk 1
H 2 (∂ Ω )
H
−1 2 (∂ Ω )
, we
Y. Duan / Computers and Mathematics with Applications 55 (2008) 66–75
Assuming that g = kgk2H 0 (∂ Ω )
P
≥
i
71
ψ(kx − xi k)λi , then by using the technique in [24], we have
E k2 Ch 2l+2d−2 kλ 2
Furthermore
2
X
kgk2H 1 (∂ Ω ) = ψ(kx − xi k)λi
i
H 1 (∂ Ω )
!2 ≤
X
kψ(kx − xi k)λi k H 1 (∂ Ω )
i
!2 ≤ max kψ(kx − i
xi k)k2H 1 (∂ Ω )
X
|λi |
i
Pn Pn by using ( i=1 |ai |)2 ≤ n i=1 |ai |2 , we have E 2 kgk2H 1 (∂ Ω ) ≤ N max kψ(kx − xi k)k2H 1 (∂ Ω ) kλk i
E k2 . ≤ Ch −d+1 max kψ(kx − xi k)k2H 1 (∂ Ω ) kλ 2 i
This yields kgk2H 0 (∂ Ω ) ≥ Ch 2l+3d−3 kgk2 2
1
H 2 (∂ Ω )
.
Then kgk
1
H 2 (∂ Ω )
≤ Ch −2l−3d+3 kgk 2
H
−1 2 (∂ Ω )
.
ˆ Without loss of generality, we always assume that RBFs satisfy ψ(t) ∼ (1 + ktk)−l−d in the rest of this paper. 4. Applications 4.1. Condition number of the meshless collocation method Consider the model problem: Lu = f in Ω Bu = g on ∂Ω
(19)
where Ω is a d-dimensional domain, with a smooth boundary ∂Ω , whose outer unit normal direction is denoted by n, f is a given function of L 2 (Ω ), X X ∂i (ai j (x, y)∂ j ) + bi ∂i + C0 (20) L= i, j
B = I,
i
or
∂ + Id ∂n L
is any linear elliptic operator. This means that: X ∃γ > 0, ai j ξi ξ j ≥ γ kξ k2 , ∀ξ ∈ Rd .
(21)
(22)
i, j
Further, assume P that the bilinear form induced by L, B is coercive and continuous. Let u = i λi φ(kx − xi k); the collocation method reads: Lu(xi ) = f (xi ) xi ∈ Ω Bu(x j ) = g(x j ) x j ∈ ∂Ω
(23)
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Y. Duan / Computers and Mathematics with Applications 55 (2008) 66–75
Denote by A = [Lφ(kxi − x j k), Bφ(kxi − xk k)]T = [A1 , A2 ]T
(24)
the linear system induced by collocation using RBF is E = [ fE, gE]T . Aλ
(25)
Theorem 4.1. Assume that matrix A is invertible. Then, the condition number Cond(A) has the following estimate: Cond(A) ≤ Ch −2l−3d+1
(26)
constant C depend on φ, Ω , L, B. Proof. For simplicity, we will denote by C a generic constant independent of h, which may depend on Ω and operators L, B. E ∈ RN ∀0 6= λ E T AT Aλ E = λ E T A1 T A1 λ E +λ E T A2 T A2 λ E λ 2 2 E k )k + kBu(x E j )k = kLu(x P where u = i λi φ(kx − xi k) E k )k = 0, kBu(x E k )k = 0, by using the invertibility of A, we have (1) if kLu(x Lu = 0;
Bu = 0
E=0 this implies that u = 0, then λ (2) E k )k2 + kBu(x E j )k2 E T AT Aλ E = kLu(x λ ≥ Ch −d kLuk2L 2 (Ω ) + h −d+1 kgk ˜ 2L 2 (∂ Ω ) where g(x ˜ k ) = Bu(xk ), xk ∈ ∂Ω ; in other words, g˜ is the interpolant of Bu on ∂Ω . By using the estimate for elliptic problems [26], we have E T AT Aλ E ≥ Ch −d kLuk2 2 + Ch −d+1 h 2l+3d−3 kgk λ ˜ 2H 1 (∂ Ω ) L (Ω ) ≥ Ch 2l+2d−2 (kLuk2L 2 (Ω ) + kBuk2
1
H 2 (∂ Ω )
)
≥ Ch 2l+2d−2 kuk2L 2 (Ω ) E k2 . ≥ Ch 4l+4d−2 kλ On the other hand E T AT Aλ E ≤ C1 h −2d kλ E k2 λ where C1 = supx,y∈Ω¯ {|Lφ(kx − yk)|, |Bφ(kx − yk)|}. Therefore cond(A) ≤ Ch −2l−3d+1 . C depends on Ω , φ, L, B and the constant induced by the embedding.
4.2. Condition number of the meshless Galerkin method Without loss of generality, consider the model problem: ( −∆u + u = f in Ω ∂u =g on ∂Ω . ∂n
(27)
Y. Duan / Computers and Mathematics with Applications 55 (2008) 66–75
By using Galerkin’s approximation, the corresponding linear system reads: Z E = fE, Aλ A= (∇φ(x − xi ) · ∇φ(x − x j ) + φ(x − xi )φ(x − x j ))dx
73
(28)
Ω
fE =
Z Ω
f (x)φ(x − xi )dx +
Z ∂Ω
g(x)φ(x − xi )ds
T
.
(29)
Theorem 4.2. The condition number Cond(A) has the following estimate: Cond(A) ≤ Ch −2l−3d
(30)
constant C depend on φ, Ω . E ∈ R N , assuming that w = Proof. For a given 0 6= λ
PN
i=1 λi φ(x
− xi ), we have
E 2. E T Aλ E = kwk2 1 ≥ h 2l+2d kλk λ H (Ω ) On the other hand E 2. kwk2H 1 (Ω ) ≤ Ch −d kλk This yields Cond(A) ≤ Ch −2l−3d . 4.3. Others applications It has been noted that the meshless Galerkin method cannot deal with Dirichlet problems directly. Lagrange multipliers and the penalty method are usual and legitimate. Combining inverse inequalities with Lemma 2.1, we can prove the following theorem in a similar way to traditional finite element analysis [27–29]. Theorem 4.3. Let f ∈ H k (Ω ), k ≥ 0, g ∈ H s (∂Ω ), s ≥ 12 , and u is the solution of the original problem. Further, let u h , λh be the approximate solution of the RBF method with Lagrange multipliers. Assuming that the centers of both Ω and ∂Ω are quasi-uniform with density h 1 , h 2 respectively, h 1 h 2 −2l−3d+3 is sufficiently small. Then
3 ∂u
ku − u h k H 1 (Ω ) + λh − ≤ C(h k+1 k f k H k (Ω ) + h s+ 2 kgk H s+1 (∂ Ω ) ) (31)
1 − ∂n H 2 (∂ Ω ) where h = max{h 1 , h 2 }, C = min{h 1 , h 2 }−2l−5d+6 .
Computational experiments [30] show that this method is successful to some extent. A key difficulty is that a smaller density cannot result in higher accuracy. Anyway, inverse inequalities provide another possible way to circumvent the Babuˇska–Brezzi condition to obtain stable scheme, like that in [31]. The Penalty method is an alternative way to deal with Dirichlet problems. It mainly consists of Lion’s method [32],Babuˇska’s method [19] and Nistche’s method [20]. According to Theorem 4.2, all of them will result in higher condition numbers than before, which may depend linearly on the chosen penalty parameters. Applications in the Domain decomposition method have been investigated in [24,33–36]. It has been proven that the convergence rate of traditional non-overlapping domain decompositions coupled with RBFs Galerkin methods will dramatically depend on the density h [34]. It is a direct conclusion of the inverse inequalities stated in Section 3. 5. Conclusions In this paper, we have proven several inverse inequalities in finite dimensional space spanned RBFs, at least strictly definite RBFs, including multi-quadrics, Gaussian, Sobolev splines and CS-RBFs. Another possible way follows the alternative definition of a Sobolev space using Fourier transforms, because we know more about the Fourier transforms of RBFs. Estimates of condition numbers indicate that the meshless method using RBFs may be unstable. Fortunately, the accuracy of meshless collocation is quite good. For example, Golberg and Chen [16] showed that the solution of
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Y. Duan / Computers and Mathematics with Applications 55 (2008) 66–75
a three dimensional Poisson equation could be obtined with only 60 randomly distributed knots to the same degree of accuracy as for an FEM solution with 71,000 linear elements. In [37–39], Kansa and Ling successfully circumvent the ill-conditioned of system by using improved multi-quadrics and adaptive techniques. In [40,41], Li combines domain decomposition with the meshless collocation method. In [24], Wu and Hon consider additive Schwarz approximations to get an FEM-like condition number estimate by a suitable choice of RBFs. These favorablw works motivate further in-depth research into meshless numerical methods using RBFs. Acknowledgements The author would like to thank the anonymous reviewers for their careful reading of the original manuscript and for providing very valuable comments and suggestions. This work was supported by the Young Scholar Foundation of UESTC under No. L08011001JX0646. References [1] M. 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[30] Y. Duan, Y.J. Tan, A meshless Galerkin method for Dirichlet problem using radial basis functions, J. Comput. Appl. Math. 196 (2006) 394–401. [31] H.J.C. Barbosa, T.J.R. Hughes, Boundary Lagrange multipliers in finite element methods: Error analysis in natural norms, Numer. Math. 62 (1992) 1–15. [32] J.L. Lions, Probl`emes aux limites non homog`enes a` don´ees irr´eguli`eres: Une m´ethode d’approximation, in: Numerical Analysis of Partial Differential Equations (C.I.M.E.2 Ciclo, Ispra, 1967), Edizioni Cremonese, Rome, 1968, pp. 283–292. [33] R.K. Beatson, W.A. Light, S. Billings, Fast solution of the radial basis function interpolation equations: Domain decomposition methods, SIAM J. Sci. Comput. 22 (2000) 1717–1740. [34] Y. Duan, Y.J. Tan, Meshless Galerkin method based on regions partitioned into subdomains using global RBF, in: Proceedings of ICNAAM 2004, Wiley-VCH, 2004, pp. 128–133. [35] L. Ling, E.J. Kansa, Preconditioning for radial basis functions with domain decomposition methods, Math. Comput. Modelling 40 (2004) 1413–1427. [36] L. Ling, Y.C. Hon, Improved numerical solver for Kansa’s method based on affine space decomposition, Eng. Anal. Bound. Elem. 29 (2005) 1077–1085. [37] E.J. Kansa, Y.C. Hon, Circumventing the Ill-conditioning problem with multiquadric radial basis functions: Applications for elliptic partial differential equations, Comput. Math. Appl. 39 (2000) 123–137. [38] L. Ling, R. Opfer, R. Schaback, Results on meshless collocation techniques, Eng. Anal. Bound. Elem. 30 (2006) 247–253. [39] L. Ling, E.J. Kansa, A least squares preconditioner for radial basis functions collocation methods, Adv. Comput. Math. 23 (2005) 31–54. [40] J. Li, Y.C. Hon, Domain decomposition for radial basis meshless methods, Numer. Methods Partial Differential Equations 20 (2004) 450–462. [41] X. Zhou, Y.C. Hon, Jichun Li, Overlapping domain decomposition method by radial basis functions, Appl. Numer. Math. 44 (2003) 241–255.
A note on the meshless method using radial basis functions Yong Duan School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu 610054, PR China Received 11 July 2006; received in revised form 9 March 2007; accepted 16 March 2007
Abstract In this paper, we prove some inverse inequalities for the finite dimensional subspace interpolated by radial basis functions. The properties of these inequalities are worse than those in finite element spaces. These bad properties result in a huge condition number for both the collocation method and Galerkin method using radial basis functions. As applications, theoretical estimates of the condition numbers of a meshless method using radial basis functions are given. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Radial basis function; Inverse inequalities; Collocation method; Galerkin method; Meshless
1. Introduction In the last decade, meshless methods have received much attention because of their flexibility in the construction of finite dimensional subspaces. When the dominant domain is extremely complex, traditional numerical method for partial differential equations (PDEs), such as the finite element method, finite difference method etc, will be difficult to carry out. Another virtue is that the meshless method does not need the generation of a mesh. It is a complicated work in FEM. Those complications had been described recently by Griebel and Schweitzer [1] as “a very time-consuming portion of overall computation is the mesh generation from CAD input data. Typically, more than 70 percent of overall computation is spent by mesh generators”. Meshless methods mainly include GFEM (see Babuˇska [2]), EFGM (see Belytschko [3]) and meshless methods using radial basis functions (RBFs) (see Kansa [4]). For details, we refer readers to Liu [5]. In the present paper, we restrict ourselves to RBFs. Radial basis functions are powerful tools in multi-variable approximation. The corresponding theory had been extensively studied by Narcowich [6], Schaback [7,8], Wendland [9,10], Buhmann [11] etc. Consequently, numerical methods for PDEs using RBFs are concentrated on by mathematicians and engineers because of their property of being meshless. Indeed, it is: 1. A truly meshless method. We only need a set of distinct centers to construct finite dimensional spaces. 2. Dimensional independence. In other words, it can be easily extended to higher dimensional space as onedimensional ones.
E-mail address: [email protected]. c 2007 Elsevier Ltd. All rights reserved. 0898-1221/$ - see front matter doi:10.1016/j.camwa.2007.03.011
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Both collocation and the Galerkin method using RBFs are discussed in a series of papers, for example [12–17]. In contrast to the Galerkin method, the Collocation method is more successful in numerical applications, with a lack of theoretical analysis. For the unsymmetric collocation method, we cannot ascertain that the linear system that arises is solvable. Actually, there are counterexamples which indicate that the matrix of the linear system is singular [18]. In [17], the author gives an analysis of a meshless Galerkin method for a Helmholtz problem with natural boundary conditions using radial basis functions. FEM-like convergent estimates are obtained. Some techniques, for example, Lagrange multipliers and the penalty method, are advised for Dirichlet problems. The Lagrange multipliers method combined with RBFs needs inverse estimates in finite dimensional subspaces interpolated by the RBFs. The penalty method is similar, which in FEM, including Babuˇska’s method [19] and Nistche’s method [20]. It is well known that inverse inequality plays an important role in FEM analysis. It is used in basic theory for FEM analysis in estimating the condition number of the stiffness matrix and domain decomposition method etc. In this paper, we establish some inverse inequalities for finite dimensional spaces constructed by RBFs. Then, we give estimates of condition numbers for both the collocation method and Galerkin method using RBFs. Lagrange multipliers method coupled with RBFs and the Penalty method for Dirichlet problems are also discussed. Finally, we introduce an application to the domain decomposition method using our meshless Galerkin method. The rest of this paper is organized as follows. In Section 2, we introduce some basic aspects of interpolation by RBFs. In Section 3, we prove some inequalities for spaces spanned by RBFs. In Section 4, we introduce some applications for numerical methods for PDEs using RBFs, including the estimation of condition numbers of linear systems, convergence of the Lagrange method, and convergence of the domain decomposition method. At last, we draw a conclusion from this paper and point out some future work. 2. Interpolation by RBFs For a given function space V on Ω ⊂ Rd , we define the finite dimensional space Vh ⊂ V by Vh := span{φ(x − x1 ), φ(x − x2 ), . . . , φ(x − x N )} + Pdm
(1)
where φ : Rd R is at least a C 1 -function, Pdm denotes the space of polynomials of degree less than m, and X = {x1 , x2 , . . . , x N } ⊆ Ω¯ is a set of distinct centers. h denotes the density of X , defined by h = sup min kx − x j k
(2)
Ω xi ∈X
where norm k·k is the Euclidean norm. When m = 0, we will call φ strictly positive definiteness, otherwise conditional positive definiteness of order m. For a given function f ∈ V , assuming that f (x) =
N X
αi φ(x − xi ) +
i=1
M X
β j p j (x)
(3)
j=1
for a basis p j (x) of Pdm , where M is the number of the basis p j (x). The corresponding interpolated problem reads: solving the system N X
αi φ(xk − xi ) +
i=1 N X
M X
β j p j (xk ) = f (xk ),
1≤k≤N
(4)
j=1
αi p j (xi ) = 0,
1 ≤ j ≤ M.
(5)
i=1
In self-evident matrix formulation, the system above reads as A P T αE fE = . P 0 βE 0
(6)
The solvability of this system is usually guaranteed by the requirements rank(P) = M ≤ N , and λkyk2 ≤ y T Ay,
∀y ∈ Rd ,
P y = 0.
(7)
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The condition rank(P) ≤ N will be called the Pdm -non-degeneracy of X. Definition 2.1 ([7]). A set of centers X = {x1 , x2 , . . . , x N } is called quasi-uniform for a compact set Ω ⊂ Rd , if there exist δ > 0 and h > 0 such that 1. h(x) ≤ h ∀x ∈ Ω 2. q = 12 mini6= j kxi − x j k ≥ hδ 3. each set X \ {x j }, 1 ≤ j ≤ N is Pdm -non-degenerate where h(x) =
sup
min ky − x j k
ky−xk≤ρ xi ∈X
(8)
ρ > 0 is fixed. For a quasi-uniform set of centers X , the following property holds: G(h) ≤ λ ≤ F(h).
(9)
G(h), F(h) are given by Schaback in [7]. It is obvious that h −d ∼ N when the set of centers is uniformly distributed. N is the number of centers. In [17,21], Wendland gives the Sobolev type error estimate, which is important to numerical analysis for PDEs. Assuming that the RBFs satisfying φ ∈ C l with Fourier transform φˆ satisfies ˆ ∼ (1 + ktk)−2l . φ(t)
(10)
the following interpolation error holds. Lemma 2.1. Let Ω ⊆ R d be an open and bounded domain, having a Lipschitz boundary and satisfying the interior cone condition. Denote by Su the interpolant on X = {x1 , x2 , . . . , x N } ⊆ Ω to a function u ∈ H k (Ω ) with k > d/2. Then there exists a constant h 0 > 0 such that for all X with h < h 0 , where h is the density of X , the estimate ku − su k H j (Ω ) ≤ Ch k− j kuk H k (Ω ) is valid for 0 ≤ j ≤ k.
(11)
For the sake of completeness, we introduce some radial basis functions below: Gaussians exp(−ck · k2 ) c > 0 3 Multiquadrics (c2 + k · k2 )β− 2 β ∈ N , c 6= 0 Thin-plate splines k · kβ log k · k β ∈ 2N . The RBFs introduced above are globally supported RBFs. One of the most popular classes of CS-RBFs is introduced by Wendland in [9]. Another is given by Wu in [22]. These functions are strictly positive definite in Rd for all d less than or equal to some fixed value d0 , and can be constructed to have any desired amount of smoothness 2κ. For l = [ d2 ] + κ + 1 and κ = 0, 1, 2, 3, Wendland’s function reads: φl,0 (r ) = ˙ (1 − r )l+ , φl,1 (r ) = ˙ (1 − r )l+1 + [(l + 1)r + 1], 2 2 φl,2 (r ) = ˙ (1 − r )l+2 + [(l + 4l + 3)r + (3l + 6)r + 3], 3 2 3 2 2 φl,3 (r ) = ˙ (1 − r )l+3 + [(l + 9l + 32l + 15)r + (6l + 36l + 45)r + (15l + 45)r + 15].
Here and throughout, = ˙ denotes equality up to a constant factor. For example, the functions φ2,0 =(1 ˙ − r )2+ φ3,1 (r ) =(1 ˙ − r )4+ [4r + 1], φ4,2 (r ) =(1 ˙ − r )6+ [35r 2 + 18r + 3], φ5,3 (r ) =(1 ˙ − r )8+ [32r 3 + 25r 2 + 8r + 1] which are in C 0 , C 2 , C 4 , C 6 , respectively, and strictly positive definite in R3 .
Y. Duan / Computers and Mathematics with Applications 55 (2008) 66–75
69
Wu’s CS-RBF reads: φ(r ) = (1 − r )6+ (5r 5 + 30r 4 + 72r 3 + 82r 2 + 36r + 6) ∈ C 6 . 3. Some inverse inequalities in Vh In this paper, Sobolev spaces H s (Ω ), H s (∂Ω ) are defined in the standard way [23]. The norms are denoted by k · k H s (Ω ) , k · k H s (∂ Ω ) respectively. We recall that the well-known trace inequality and inverse trace inequality hold for suitable Sobolev spaces in Ω ⊂ Rd . The norm in H s (Ω ), s ≥ 0, s = k + t, 0 < t < 1, integer k ≥ 0 reads: X Z Z |∂ α u(x) − ∂ α u(y)|2 kuk2H s (Ω ) = kuk2H k (Ω ) + dxdy. (12) |x − y|d+2t |α|=k Ω Ω 1
1
We will also use the space H − 2 (∂Ω ), i.e, the dual space of H 2 (∂Ω ), with the norm kµk H
−1 2 (∂ Ω )
sup
=
1
z∈H 2 (∂ Ω )
hµ, zi kzk 1
(13)
H 2 (∂ Ω )
where the duality pairing is Z hµ, zi = µzds.
(14)
∂Ω
ˆ Theorem 3.1. Assuming a strictly positive definite RBF ψ with Fourier transform ψ(t) ∼ (1 + ktk)−l−d , then for ∀g ∈ Vh and two real numbers 0 ≤ s < k, the following inequality holds kgk H k (Ω ) ≤ Ch
−2l−3d 2
(15)
kgk H s (Ω )
where C is a constant independent of the density h. Proof. Without being P Nconfused, we will denote by C a generic constant independent of h throughout this paper. Assuming that g = i=1 ψ(kx − xi k)λi , by using the result in [24], we have E 2 kgk2H 0 (Ω ) ≥ Ch 2l+2d kλk Furthermore kgk2H k (Ω )
2
X
= ψ(kx − xi k)λi
i
H k (Ω )
!2 ≤
X
kψ(kx − xi k)λi k H k (Ω )
i
!2 ≤ max kψ(kx − xi k)k2H k (Ω )
X
i
|λi |
i
Pn Pn |ai |)2 ≤ n i=1 |ai |2 , we have By using ( i=1 E k2 kgk2H k (Ω ) ≤ N max kψ(kx − xi k)k2H k (Ω ) kλ ≤ Ch
i −d
E k2 . max kψ(kx − xi k)k2H k (Ω ) kλ i
This yields kgk2H 0 (Ω ) ≥ Ch 2l+3d kgk2H k (Ω ) . Then kgk H k (Ω ) ≤ Ch
−2l−3d 2
kgk H s (Ω ) .
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Y. Duan / Computers and Mathematics with Applications 55 (2008) 66–75
Remark 3.1. Unlike in FEM, the information about k, s is shown in the constant C. Actually, C depends on the embedding constant from H s (Ω ) to H 0 (Ω ) and the quantity maxi kψ(kx − xi k)k H k (Ω ) , which will be fixed whenever the RBF is chosen. The well-known inverse inequality in a suitable finite element space reads [25]: kgk H k (Ω ) ≤ Ch s−k kgk H s (Ω ) .
(16)
The inverse inequality is worse than that in FEM. It has been shown that this is true. An obvious result is that the condition number of a linear system arising in a meshless method using RBFs is worse. It has been verified in computational experiments [12]. As applications, we will estimate those condition numbers by using inverse inequalities in the next section. Theorem 3.2. Assuming that ψ a strictly positive definite RBF, then for ∀g ∈ Vh and two real numbers 0 ≤ s < k, the following inequality holds kgk H k (Ω ) ≤ Ch −d G −1 (h)kgk H s (Ω )
(17)
where C is a constant independent of the density h. PN Proof. Assuming that g = i=1 ψ(kx − xi k)λi , by using the Riemann sum, we have Z kgk2H 0 (Ω ) = |g|2 Ω
≥ Ch d
N X
|g(xi )|2
i=1
E k2 = Ch d kAλ where A is the interpolated matrix. As pointed out in Section 2, the eigenvalues of matrix A have lower bound G(h). This yields E k2 ≥ C G 2 (h)h d kλ E k2 . kAλ Following the same procedure as in the proof of Theorem 3.1, we have kgk H k (Ω ) ≤ Ch −d G −1 (h)kgk H s (Ω ) .
1
1
In the case of s < 0, the most popular inverse inequality is about H − 2 (∂Ω ) and H 2 (∂Ω ). In FEM, kgk Chkgk
H
−1 2 (∂ Ω )
1
H 2 (∂ Ω )
≤
, for a suitable finite element space. Denote Λh = span{ψ(x − xi )}, {x1 , . . . , x N } is a set of distinct
centers on ∂Ω with density h 2 and ψ is a strictly positive definite RBF, and we have ˆ Theorem 3.3. Assuming that ψ(t) ∼ (1 + ktk)−l−d ; then for ∀g ∈ Λh , the following inequality holds kgk
1
H 2 (∂ Ω )
≤ Ch −2l−3d+3 kgk 2
H
(18)
−1 2 (∂ Ω )
where C is a constant independent of the density h 2 . Proof. We will denote by C, a generic constant independent of h 2 . According to the definition of kgk have kgk H
−1 2 (∂ Ω )
sup
= µ∈H
1 2 (∂ Ω )\{0}
hg, gi ≥ kgk 1 =
H 2 (∂ Ω ) kgk2H 0 (∂ Ω )
kgk
1
H 2 (∂ Ω )
.
hg, µi kµk 1
H 2 (∂ Ω )
H
−1 2 (∂ Ω )
, we
Y. Duan / Computers and Mathematics with Applications 55 (2008) 66–75
Assuming that g = kgk2H 0 (∂ Ω )
P
≥
i
71
ψ(kx − xi k)λi , then by using the technique in [24], we have
E k2 Ch 2l+2d−2 kλ 2
Furthermore
2
X
kgk2H 1 (∂ Ω ) = ψ(kx − xi k)λi
i
H 1 (∂ Ω )
!2 ≤
X
kψ(kx − xi k)λi k H 1 (∂ Ω )
i
!2 ≤ max kψ(kx − i
xi k)k2H 1 (∂ Ω )
X
|λi |
i
Pn Pn by using ( i=1 |ai |)2 ≤ n i=1 |ai |2 , we have E 2 kgk2H 1 (∂ Ω ) ≤ N max kψ(kx − xi k)k2H 1 (∂ Ω ) kλk i
E k2 . ≤ Ch −d+1 max kψ(kx − xi k)k2H 1 (∂ Ω ) kλ 2 i
This yields kgk2H 0 (∂ Ω ) ≥ Ch 2l+3d−3 kgk2 2
1
H 2 (∂ Ω )
.
Then kgk
1
H 2 (∂ Ω )
≤ Ch −2l−3d+3 kgk 2
H
−1 2 (∂ Ω )
.
ˆ Without loss of generality, we always assume that RBFs satisfy ψ(t) ∼ (1 + ktk)−l−d in the rest of this paper. 4. Applications 4.1. Condition number of the meshless collocation method Consider the model problem: Lu = f in Ω Bu = g on ∂Ω
(19)
where Ω is a d-dimensional domain, with a smooth boundary ∂Ω , whose outer unit normal direction is denoted by n, f is a given function of L 2 (Ω ), X X ∂i (ai j (x, y)∂ j ) + bi ∂i + C0 (20) L= i, j
B = I,
i
or
∂ + Id ∂n L
is any linear elliptic operator. This means that: X ∃γ > 0, ai j ξi ξ j ≥ γ kξ k2 , ∀ξ ∈ Rd .
(21)
(22)
i, j
Further, assume P that the bilinear form induced by L, B is coercive and continuous. Let u = i λi φ(kx − xi k); the collocation method reads: Lu(xi ) = f (xi ) xi ∈ Ω Bu(x j ) = g(x j ) x j ∈ ∂Ω
(23)
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Y. Duan / Computers and Mathematics with Applications 55 (2008) 66–75
Denote by A = [Lφ(kxi − x j k), Bφ(kxi − xk k)]T = [A1 , A2 ]T
(24)
the linear system induced by collocation using RBF is E = [ fE, gE]T . Aλ
(25)
Theorem 4.1. Assume that matrix A is invertible. Then, the condition number Cond(A) has the following estimate: Cond(A) ≤ Ch −2l−3d+1
(26)
constant C depend on φ, Ω , L, B. Proof. For simplicity, we will denote by C a generic constant independent of h, which may depend on Ω and operators L, B. E ∈ RN ∀0 6= λ E T AT Aλ E = λ E T A1 T A1 λ E +λ E T A2 T A2 λ E λ 2 2 E k )k + kBu(x E j )k = kLu(x P where u = i λi φ(kx − xi k) E k )k = 0, kBu(x E k )k = 0, by using the invertibility of A, we have (1) if kLu(x Lu = 0;
Bu = 0
E=0 this implies that u = 0, then λ (2) E k )k2 + kBu(x E j )k2 E T AT Aλ E = kLu(x λ ≥ Ch −d kLuk2L 2 (Ω ) + h −d+1 kgk ˜ 2L 2 (∂ Ω ) where g(x ˜ k ) = Bu(xk ), xk ∈ ∂Ω ; in other words, g˜ is the interpolant of Bu on ∂Ω . By using the estimate for elliptic problems [26], we have E T AT Aλ E ≥ Ch −d kLuk2 2 + Ch −d+1 h 2l+3d−3 kgk λ ˜ 2H 1 (∂ Ω ) L (Ω ) ≥ Ch 2l+2d−2 (kLuk2L 2 (Ω ) + kBuk2
1
H 2 (∂ Ω )
)
≥ Ch 2l+2d−2 kuk2L 2 (Ω ) E k2 . ≥ Ch 4l+4d−2 kλ On the other hand E T AT Aλ E ≤ C1 h −2d kλ E k2 λ where C1 = supx,y∈Ω¯ {|Lφ(kx − yk)|, |Bφ(kx − yk)|}. Therefore cond(A) ≤ Ch −2l−3d+1 . C depends on Ω , φ, L, B and the constant induced by the embedding.
4.2. Condition number of the meshless Galerkin method Without loss of generality, consider the model problem: ( −∆u + u = f in Ω ∂u =g on ∂Ω . ∂n
(27)
Y. Duan / Computers and Mathematics with Applications 55 (2008) 66–75
By using Galerkin’s approximation, the corresponding linear system reads: Z E = fE, Aλ A= (∇φ(x − xi ) · ∇φ(x − x j ) + φ(x − xi )φ(x − x j ))dx
73
(28)
Ω
fE =
Z Ω
f (x)φ(x − xi )dx +
Z ∂Ω
g(x)φ(x − xi )ds
T
.
(29)
Theorem 4.2. The condition number Cond(A) has the following estimate: Cond(A) ≤ Ch −2l−3d
(30)
constant C depend on φ, Ω . E ∈ R N , assuming that w = Proof. For a given 0 6= λ
PN
i=1 λi φ(x
− xi ), we have
E 2. E T Aλ E = kwk2 1 ≥ h 2l+2d kλk λ H (Ω ) On the other hand E 2. kwk2H 1 (Ω ) ≤ Ch −d kλk This yields Cond(A) ≤ Ch −2l−3d . 4.3. Others applications It has been noted that the meshless Galerkin method cannot deal with Dirichlet problems directly. Lagrange multipliers and the penalty method are usual and legitimate. Combining inverse inequalities with Lemma 2.1, we can prove the following theorem in a similar way to traditional finite element analysis [27–29]. Theorem 4.3. Let f ∈ H k (Ω ), k ≥ 0, g ∈ H s (∂Ω ), s ≥ 12 , and u is the solution of the original problem. Further, let u h , λh be the approximate solution of the RBF method with Lagrange multipliers. Assuming that the centers of both Ω and ∂Ω are quasi-uniform with density h 1 , h 2 respectively, h 1 h 2 −2l−3d+3 is sufficiently small. Then
3 ∂u
ku − u h k H 1 (Ω ) + λh − ≤ C(h k+1 k f k H k (Ω ) + h s+ 2 kgk H s+1 (∂ Ω ) ) (31)
1 − ∂n H 2 (∂ Ω ) where h = max{h 1 , h 2 }, C = min{h 1 , h 2 }−2l−5d+6 .
Computational experiments [30] show that this method is successful to some extent. A key difficulty is that a smaller density cannot result in higher accuracy. Anyway, inverse inequalities provide another possible way to circumvent the Babuˇska–Brezzi condition to obtain stable scheme, like that in [31]. The Penalty method is an alternative way to deal with Dirichlet problems. It mainly consists of Lion’s method [32],Babuˇska’s method [19] and Nistche’s method [20]. According to Theorem 4.2, all of them will result in higher condition numbers than before, which may depend linearly on the chosen penalty parameters. Applications in the Domain decomposition method have been investigated in [24,33–36]. It has been proven that the convergence rate of traditional non-overlapping domain decompositions coupled with RBFs Galerkin methods will dramatically depend on the density h [34]. It is a direct conclusion of the inverse inequalities stated in Section 3. 5. Conclusions In this paper, we have proven several inverse inequalities in finite dimensional space spanned RBFs, at least strictly definite RBFs, including multi-quadrics, Gaussian, Sobolev splines and CS-RBFs. Another possible way follows the alternative definition of a Sobolev space using Fourier transforms, because we know more about the Fourier transforms of RBFs. Estimates of condition numbers indicate that the meshless method using RBFs may be unstable. Fortunately, the accuracy of meshless collocation is quite good. For example, Golberg and Chen [16] showed that the solution of
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Y. Duan / Computers and Mathematics with Applications 55 (2008) 66–75
a three dimensional Poisson equation could be obtined with only 60 randomly distributed knots to the same degree of accuracy as for an FEM solution with 71,000 linear elements. In [37–39], Kansa and Ling successfully circumvent the ill-conditioned of system by using improved multi-quadrics and adaptive techniques. In [40,41], Li combines domain decomposition with the meshless collocation method. In [24], Wu and Hon consider additive Schwarz approximations to get an FEM-like condition number estimate by a suitable choice of RBFs. These favorablw works motivate further in-depth research into meshless numerical methods using RBFs. Acknowledgements The author would like to thank the anonymous reviewers for their careful reading of the original manuscript and for providing very valuable comments and suggestions. This work was supported by the Young Scholar Foundation of UESTC under No. L08011001JX0646. References [1] M. 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