Analysis of radial basis collocation method and quasi-newton iteration for nonlinear elliptic problems

Transcription

1 Analysis of radial basis collocation method and quasi-newton iteration for nonlinear elliptic problems H. Y. Hu and J. S. Chen December, 5 Abstract This work presents a global radial basis collocation combining with the quasi- Newton iteration method for solving semilinear elliptic partial differential equations. A convergence analysis for such a meshfree discretization has been established. The main result is that there exists an exponential convergence rate with respect to the number and the shape of the radial basis functions. In addition, a superlinear convergence rate with respect to quasi-newton iteration is obtained. Some results are given to illustrate the efficiency and accuracy of the proposed method. Keywords: radial basis functions, collocation method, quasi-newton iteration, superlinear convergence, meshfree, semilinear equations. Introduction Nonlinear elliptic partial differential equations represent many scientific and engineering problems. Some literatures concerning the numerical approaches for nonlinear equations have been given, for example, finite difference method [33, 35, 6], finite element method [5, 6, 7], boundary element method [], and the related literatures in their references. The mesh-based methods are most widely used in solving partial differential equations. However, the construction of an appropriate mesh in arbitrary geometry is a hard and time-consume task. Consequently, meshfree methods have been introduced in last decade based on approximation methods that do not require a mesh topology, for example, kernel estimate and modified kernel estimate [5, 5, 9], moving least-squares [3, 4, 7,, 3], partition of unity [9,, 3, 3, ], natural neighbor interpolation [38], radial basis functions [3, 4], and non-uniform rational B-splines []. The discrete Corresponding author, address: huhy@math.cts.nthu.edu.tw, Department of Mathematics, National Tsing Hua University, Hsinchu 343, Taiwan address: jschen@seas.ucla.edu, Department of Civil and Environmental Engineering, University of Califormia, Los Angeles, CA , U.S.A.

2 equations of the meshfree methods can be constructed based on smoothed collocation [5], Galerkin method [4, 5, 3, 9, 3, 38], least-squares method [34], collocation of strong form [4], stabilized collocation of Galerkin method []. In this work we propose a discretization using radial basis collocation with the quasi- Newton iteration scheme to solve a semilinear elliptic partial differential equation and perform convergence analysis of this approach. For the collocation method using radial basis functions as admissible functions, we simply name such a scheme the radial basis collocation method. In the past two decades, there have been many applications for the radial basis functions, such as surface fitting, turbulence analysis, neural network, meteorology and partial differential equations, in which the lastest one is of interest to us. For the application in solving partial differential equations, a large amount of the literatures concentrated on the implementation scheme, and the detailed analysis on convergence or stability is still lacking. Very limited results were presented in application to nonlinear problems using the radial basis collocation method, for example, in computational biophysics or computational biochemistry. No error analysis and convergence study for such a discretized scheme was established to date. The objective of this study is to investigate how such a meshfree method can be used to solve the elliptic partial differential equation involving nonlinearity and whether the sequence solutions converge. We follow the ideas of previous works [9, ] for the semilinear elliptic equations. The main result is that there exists an exponential convergence rate with respect to the number and the shape factor of radial basis functions, and the accuracy of solution is controlled by such two parameters. Further, there exists a superlinear reduction in iteration errors using quasi-newton iteration scheme. A relevant work concerning the Newton iteration was given in []. The collocation method can be interpreted as an extension of the least squares methods. We view it as the least squares method with quadrature rules []. The numerical solutions by this method exhibit high accuracy, nevertheless it accompanies instability due to the large condition number. Several approaches have been proposed to address this issue, for example, the preconditioning techniques can be used to improve the stability. However, the real instability of collocation methods may not be so severe as the large condition number displays. A new effective condition number was proposed to provide a better upper bound of condition number, which may be small, and to display a good stability []. Various admissible functions, such as Fourier series, Chebyshev polynomials, and Legendre polynomials have been used as the admissible functions in solving partial differential equations with success. The radial basis functions have also been employed in the construction of approximations for solving linear elliptic partial differential equation under the framework of collocation method. The convergence rates obtained by this method are typically beyond those obtained by the conventional finite element and finite difference methods. Analogous to other meshfree methods, in radial basis collocation method the source points of radial basis functions and collocation nodes can be scattered or clustered in a rather arbitrary fashion.

3 The remaining part of this paper is organized as follows. An introduction to radial basis functions is given in Section. In Section 3, the convergence properties of radial basis collocation method and quasi-newton iteration scheme are established. Section 4 presents the implementation scheme and numerical experiments to demonstrate the effectiveness of the methods proposed. Some conclusions are summarized in Section 5. The radial basis functions (RBF) There have been many developments for RBF in the past two decades. Such basis functions can be used as the interpolatory tools for smooth functions. The convergence of interpolants for a given continuous function has been discussed, for example, Madych [8]. Some applications to partial differential equations (PDEs) have been given. Kansa [3, 4] presented a series of applications in computational fluid dynamics. Franke and Schaback [4] provided some theoretical foundation of RBF method for solving PDE. Hon and Mao [8] presented an application to Burgers equation. Wendland [39] derived error estimates for the solution of smooth problems. Hon [7] developed a quasi-radial basis functions method for solving the Black-Scholes equation. Hu et al. [9] presented a radial basis collocation method including the combined and alternative schemes for singularity problems.. Rationale of the RBF Several forms of RBF have been introduced recently. A few examples are: multiqudrics (MQ), Gaussian, thin plate splines and logarithmic RBFs as shown below g i (x) = (r i + c ) n 3, n =,,3,..., (.) g i (x) = exp( r i ), c (.) g i (x) = ri n, n =,,3,..., (.3) g i (x) = ri n lnr i, n =,,3,..., (.4) where x = (x,y) denotes the Cartesian coordinate in R, and the radius is given by r i = {(x x i ) + (y y i ) }, where (x i,y i ) is called the source point of the RBF, denoted by x i. The constant c involved in forms (.)-(.) is called the shape parameter of the RBF. These two functions become much flat as the parameter c increases. In other literatures, the radius and basis functions are denoted as r i = r(x,x i ) and g i (x) = g(x,x i ), where x i is the location of the ith source point. For the first form in (.), we may rewrite it as g i (x) = ri + ν c, (.5) 3

4 where ν = n 3, and the shape functions are called reciprocal MQ RBF if n =, linear MQ RBF if n =, and cubic MQ RBF if n = 3, and so forth. Among them, the linear MQ RBF is most widely used. For a given smooth function, we can represent it by a linear combination of RBF functions. An exponential convergence rate was presented by Madych [8], where there are two ways to increase the accuracy, either by adding more basis functions or by increasing the shape parameter. We assume that R is a closed region with the boundary. Let X s be a set of m source points, X s = {x,x,...,x m }. For a smooth function u(x), we may approximate it by m v(x) = a i g i (x), x i X s, (.6) i= where v(x) is the approximation of u(x), m > and a i are expansion coefficients. The convergence rate is given by [8] u(x) v(x) O(η c/δ ), (.7) where < η < is a real number, c denotes a shape parameter of the RBF, and δ denotes a radial distance defined as δ := δ(, X s ) = sup min {(x x i ) + (y y i ) }. (.8) x i X s x Note that η = exp( θ) with θ >. Equation (.7) indicates that the parameters c and δ affect the rate of convergence.. Fitting and other applications To approximate a given smooth function, say u(x), we represent it by RBF as in (.6). The coefficients a i in (.6) can be determined by solving a set of linear equations arise from the collocation method (CM). The basic idea of CM is to force the residuals to be zero at some nodes, so-called the collocation points, of a given domain. We choose the collocation points located both in the region and on boundary so that a discrete scheme can be constructed. Let X c be a set of N collocation points, X c = {x,x,...,x N }. The sets X s and X c may or may not have the common points. We express such an approximation as follow (u v)(x j ) =, for all x j X c. (.9) 4

5 The set of collocation points X c can be arbitrary scattered. Equation (.9) can be written in a general form as m a i g i (x j ) = u(x j ), for all x j X c. (.) i= For the number of collocation points, we may choose it to be larger than the number of the source points. Equation (.) leads to an algebraic system in the form Ga = b, (.) where a is an unknown vector consisting of coefficients a i in (.6). It is clear that G is a N by m matrix with entries g i (x j ), and b is a vector associated with function u(x j ) with dimension N. The resultant linear system (.) is an overdetermined system, and methods such as QR decomposition, singular value decomposition or least squares method can be considered. Remark. We may consider the RBF with variant shape parameter c i in the first two forms of RBF in (.) and (.). In the work of Buhmann and Micchelli [8], it has been shown that the convergence rate is accelerated for monotonically ordered c i. In present work we use the constant shape parameter for simplicity. While the radial basis functions offer good approximation of global functions, applications such as in computational biophysics and computational biochemistry often exhibit local properties, and the employment of small shape parameter c in basis functions for the solution of these problems yields very acute results. The Poisson-Boltzmann equation [37, 36] is typical of this kind, where dielectric discontinuity, nonlinearity and singularity exist. One possible approach for solving Poisson-Boltzmann equation is to introduce radial basis functions with collocation method. This meshfree scheme is more adequate for such application as the complex affine map in meshing and the need of fine mesh in finite element [3, 6, 37] or finite difference [36] yield high computational costs. In this work an equation involving a nonlinear term is first taken into consideration. We concentrate on the use of the radial basis collocation method and its corresponding convergence properties in this work. Other equations involving the singularity or the dielectric discontinuity will be discussed elsewhere. 3 RBF collocation method with iteration scheme Consider the semilinear elliptic partial differential equation of the form u = f(x,u) in, (3.) u = on, 5

6 where R is a bounded domain with boundary. A straightforward numerical treatment for such a semilinear problem is as follows: Given an initial guess u (x) and solve, { un+ (x) = f(x,u n (x)), in, u n+ (x) =, on, for n =,,,... At each iteration u n+ (x) is solved numerically. For problems with strong nonlinearities, an alternative numerical treatment is introduced as follows: Given an initial guess u (x) and solve, { un+ (x) λ u n+ (x) = f(x,u n (x)) λ u n (x), in, (3.) u n+ (x) =, on, for n =,,,... Such an iterative scheme falls into three types: () the monotone iteration if λ is chosen to be a constant; () the quasi-newton iteration if λ is chosen to be a quotient, f(x,un) f(x,u n ) u n u n ; (3) the Newton iteration if λ is chosen to be a derivative, f u (x,u n ) = f(x,un) u. In this work, we consider the use of the radial basis collocation method [9] to solve this semilinear problem. 3. Description of radial basis collocation method Denote by H k () the Sobolev space equipped with the norms k,,k =,,. Under some suitable assumptions, it can be shown that there exists a generalized solution of (3.). We summarize the conditions necessary for our discussion in the following assumption. Assumption 3. Let u H () be a generalized solution of the problem (3.), where the function f(x,u) is nonlinear in u. There exists constant γ and λ such that, for sufficiently small ε >, where =, and where B is defined as λ < γ f u (x,u) λ, (x,u) B ε, B ε = {v H () : v u B ε + u u B }, B =, +, λ > is the smallest eigenvalue of operator in subject to the homogeneous Dirichlet boundary condition satisfying γλ < λ < γλ +, and u H is a given function. 6

7 where The generalized solution u H () satisfies the following variational formulation B(u,v) = F(u,v) = The solution in can be expanded as B(u,v) = F(u,v), v H (), (3.3) ( u v + λ uv) dx + uv dl, (λ uv f(x,u) v) dx. u = a i g i (x), i= where g i (x) are RBF described as before, and a i are the expansion coefficients. Such an expansion converges exponentially to the true solution u. Its properties of convergence have been addressed in previous section. The admissible functions can be chosen as L v = ã i g i (x) in, (3.4) i= where ã i are unknown coefficients to be sought. Denote by V L the finite dimensional collection of such admissible functions, i.e., which is a subspace of H (). V L = span{g (x),g (x),...,g L (x)}, Choosing an initial u, a sequence of radial basis function iterates can be generated. The formulation is given as follow: To seek u n+ V L such that B(u n+,v) = F(u n,v), v V L, (3.5) for u n V L, n =,,,..., where B(, ) and F(, ) have the same definitions as in formulation (3.3). The u n in (3.5) is specified as u n := u n (x) = u (n) L L (x) = a (n) i g i (x), where a (n) i denote the coefficients of the nth iterate u n. The existence and uniqueness of the solution of (3.5) follows immediately from the Lax-Milgram lemma since the bilinear form B(, ) is coercive and bounded and the F(, ) is continuous. The formulation (3.5) can be described equivalently: To choose u, a sequence of solution u n+ V L can be generated by i= E(u n+ ) = min v V L E(v), (3.6) 7

8 where E(v) = ( v + λv + f(x,u n ) λu n ) dx + v dl, (3.7) for u n V L,n =,,,... The collocation method can be interpreted as a least squares method involving integration quadrature [9, ]. The integrals in problem (3.7) are numerically evaluated using quadrature rules, for example, the Gauss quadratures rules or the Newton-Cotes quadratures rules. The number of quadrature points in domain and on boundary are denoted by N a and N b. For the numbers N a,n b, N a P(x) dx = w k P(ξ k ), for all ξ k, P(x) dl = k= N b l= w l P( ξ l ), for all ξl, where P(x) is a given function, w k and w l denote positive weights, and ξ k and ξ l denote the quadrature points located in domain and on boundary, respectively. The problem (3.6) involving quadrature approximation leads to a discrete problem as follow: To seek ū n+ V L such that where Ẽ(v) = Ẽ(ū n+ ) = min v V L Ẽ(v), (3.8) ( v + λv + f(x,ū n ) λū n ) dx + for ū n V L,n =,,,..., in which the ū n is specified as ū n := ū n (x) = L i= ā (n) i g i (x), where ā (n) i denote the coefficients of the nth iterate ū n. v dl, (3.9) Let N c the total number of collocation points located in solution domain and on its boundary, N c = N a + N b. The discrete functional Ẽ(v) can further be manipulated as Ẽ(v) = N a w k {( v + λv + f(x,ū n ) λū n )(ξ k )} + N b w l {v( ξ l )}, k= l= = at M T WMa a T M T Wb + d, where the vector a consists of unknown coefficients with dimension L, and [ ] [ ] M [ ] M =, M T = M M T W MT, W =, b = W [ b b ], 8

9 where M is an N a by L matrix associated with v +λv and M is an N b by L matrix associated with v, and consequently M is an N c by L matrix. However, W is an N a by N a diagonal weight matrix consisting of w k and W is an N b by N b diagonal weight matrix consisting of w l. The b is the known vector associated with f(x,ū n ) λū n, b is a zero vector, and d is the known value associated with (f(x,ū n ) λū n ). Therefore, an implementation scheme of the radial basis collocation method can be derived from such a discrete minimization problem. Taking variation with respect to the undetermined coefficient vector a, we obtain a linear system M T WM a = M T W b, (3.) where M T WM is symmetric and positive definite. The system (3.) is called the normal equation. The solution by the following linear system W M a = W b is exactly equal to the solution of (3.). In real computation, we solve Ma = b, (3.) directly by QR decomposition or singular value decomposition. The solution by (3.), in fact, approximates the solution of (3.). Remark 3. We assume that a e and a o are the solutions of (3.) and (3.), respectively. Then there exists a relative error a o a e a e E Cond(M T WM) M T W M T W b, where stands for the Euclidean norm, and the value of E is an energy norm defined as E = Ma o b, and the Cond( ) means the condition number of a given matrix. The solution a o will approximate the solution a e as the energy error E is very small. The overdetermind system (3.) can be rewritten as following form { ū n+ + λū n+ + f(x,ū n ) λū n }(ξ k ) =, for all ξ k, {ū n+ } ( ξ l ) =, for all ξl. Such a form can also be constructed in a weighted residual formulation for problem (3.) using Dirac delta function as the test function. In general, the collocation points, i.e., the quadrature points, can be scattered in a rather arbitrary fashion. 9

10 3. Convergence analysis for general scheme In this section, we discuss the convergence properties of the sequence ū n+ of the problem (3.8) to the solution u. The sequence with any arbitrary initial guess in B ε will converge to u in general. The minimization problems are equivalent to the variational formulations. The analysis made in this section based on variational formulations consists of two steps: The first one is for the original formulation without quadrature approximation; the second is for the discrete formulation where the quadrature rules are involved. For the purpose of error analysis, we assume that there exists a projection u L V L satisfying B(u L,v) = F(u,v), v V L. (3.) Combining with formulation (3.5) and using mathematical induction, we obtain an estimate as follows []. Lemma 3. Let Assumption 3. hold. If u,u,...,u n all lie in B ε, u H () H k (),k, and u L satisfies (3.), then there exists where < α = λ γλ <. Moreover, u L u n+ B α u u n B, (3.3) u u n+ B αn+ α u u L B + α n+ u u B. (3.4) Next, a variational formulation involving the Newton-Cotes quadrature rules is taken into account. The discrete problem (3.8) in previous section can be described equivalently: To seek ū n+ V L such that B(ū n+,v) = F(ū n,v), v V L, (3.5) where B(, ) and F(, ) stand for the B(, ) and F(, ) evaluated by quadratures rules, B(u,v) = F(u,v) = ( u v + λ uv) dx + (λ uv f(x,u) v) dx. Suppose that there exists positive constant C such that uv dl, v k, CL k v,, v V L, (3.6) v k, CL k v, v V L, (3.7)

11 see [9], where L denotes the number of admissible functions in (3.4). It can be derived that there exist the following bounds, ( ( ( )( v), C h r+ L r+5 v,, (3.8) ) v, C h r+ L r+ v,, (3.9) ) v, C h r+ L r+ v,, (3.) where h denotes the maximal spacing between integration nodes, i.e., collocation points, and r denotes the order of the Newton-Cotes quadrature rule. The projection u L satisfies B(u L,v) = F(u,v), whereas B(u L,v) F(u,v). By using the inequalities (3.8) and (3.9), we obtain a relation where B(u L,v) F(u,v) + T( h,l), T( h,l) = C h r+ L r+5 u L, v, + C 3 λ h r+ L r+ ( u L, + u, ) v, +C 4 h r+ L r+3 f, v, + C 5 h r+ L r+ u L, v,, where C i are generic constants. Moreover, we obtain where B(u L,v) F(u,v) + C h r+ L r+5 v B, C = max{c u L,,C 3 λ( u L, + u, ),C 4 f,,c 5 u L, }. Assume that there exists another projection, denoted by ū L, which satisfies discrete variational formulation B(ū L,v) = F(u,v). (3.) Note that the projection u L is a limit of the sequence {u n }, and ū L is a limit of the sequence {ū n }. For the latter projcetion, a crucial condition h r+ L r+5 = o(), (3.) should be fulfilled. Following the idea for linear elliptic problems [9, ], we may derive form (3.3) and (3.) that there exists a lemma as follows. Lemma 3. Let condition (3.) hold. There exists positive constants C k,k =, independent of c, δ and L such that ( ) ( ) u c/δ ū L k C k η k C k, where η k are real numbers, < η k <. η c L k

12 Remark 3. We conclude from Lemma 3. that the errors in Sobolev norms for linear elliptic partial differential equations using radial basis collocation method are bounded by the truncation error of the RBF multiplied by a constant. Furthermore, we may derive form (3.5) and (3.) that an estimate can be established. Theorem 3. Let Lemma 3. and condition (3.) hold. If ū = u,ū,ū,...,ū n all lie in B ε. There exist where α is defined in Lemma 3.. u ū n+ B αn+ α u ū L B + α n+ u u B, (3.3) Remark 3.3 Theorem 3. implies that the ū n+ remains in the ball B ε provided the collocation nodes are sufficiently dense, so that the first term on the right hand side of equation (3.3) is less than or equal to ε. Corollary 3. Let Lemma 3. and Theorem 3. hold. Then, for sufficiently large L, ( ) u ū n+ B C η c L + α n+ u u B, where C = C α n+ α is a constant independent of c and L, C is given in Lemma 3.. Remark 3.4 Corollary 3. implies that the sequence solution ū n+ for semilinear elliptic partial differential equations using radial basis collocation iteration scheme converges exponentially to u with an exponential convergence rate with respect to c and L. 3.3 Superlinaer convergence for quasi-newton s iteration In the beginning of this section we provide three iteration schemes for solving a semilinear partial differential equation. They converge to the exact solution with different convergence rate. For the first iteration scheme, the iterates converge monotonically to the solution. The rate of convergence is slow and cannot be determined in general. For better performance, the quasi-newton and Newton iteration schemes can be adopted to accelerate the convergence. Their iterates converge superlinearlly to the solution. The latter two schemes are closely related, and their conditions for convergence are almost the same. Therefore, the error estimates for these two schemes are very similar. Newton s iteration converges more rapidly than quasi-newton s iteration in general. On the other hand, Newton s iteration is generally more expensive per iteration. Besides, Newton s iteration is not guaranteed to converge due to its sensitivity to the initial guess.

13 In present work the quasi-newton iteration scheme is considered. Its rate of convergence is superlinear, of order.6. Here we merely provide an estimate in Sobolev zero norm,, abbreviated as. For further discussion, one more assumption is needed. Assumption 3. Let Assumption 3. hold. There exist constants γ and γ such that γ f uu (x,u) γ, (x,u) B ε, where f uu denotes the second partial derivative of f with respect to u and B ε is defined in Assumption 3.. Letting G(u) = u f(x,u), we assume that the projection u L satisfies the formulation G(u L )v dx =, v V L. Its divide difference form is given by { G(u n ) + G[u n,u n ](u L u n ) + } G uu(ξ n )(u L u n )(u L u n ) v dx =, (3.4) where G uu ( ) denotes the second derivative of G( ) with respect to u, ξ n is a function belong to B ε, and the divide difference is G [a,b] := G(b) G(a). b a The corresponding quasi-newton s iteration scheme is {G(u n ) + G[u n,u n ](u n+ u n )} v dx =. (3.5) Subtracting (3.5) from (3.4) and combining definition of G[, ], we obtain G[u n,u n ](u L u n+ )v dx = G uu (ξ n )(u L u n )(u L u n )v dx and G(u n ) G(u n ) (u L u n+ )v dx u n u n = u ((u L u n )(u L u n )) v dx + f uu (ξ n )(u L u n )(u L u n )v dx. Multiplying above equation by, we obtain (u n u n ) f(x,u n ) f(x,u n ) (u L u n+ )v dx + (u L u n+ )v dx u n u n u n u n = ((u L u n )(u L u n )) v dx f uu (ξ n )(u L u n )(u L u n )v dx. u 3

14 It follows that λ (u n u n ) (u L u n+ ) f(x,u n ) f(x,u n ) v dx + (u L u n+ )v dx u n u n u n u n ((u L u n )(u L u n )) v dx f uu (ξ n )(u L u n )(u L u n )v dx. u By using Assumptions 3. and 3., and λ being the smallest eigenvalue of, we have (λ + γ) u L u n+ v {C u L u n u L u n v + γ u L u n u L u n v } } {CL 3 u L u n u L u n v + γ u L u n u L u n v, in which an inverse estimate is used, v k CL k v, k, for all v V L, (3.6) where C is a generic constant. According to the above, we have the estimate u L u n+ CL3 + γ (λ + γ) u L u n u L u n. We write down above estimate as a lemma below. Lemma 3.3 Let Assumptions 3. and 3. hold and u L be a projection for the semilinear problem (3.). If u and u is sufficiently near u L, then the solution u n+ of formulation (3.5) converges to u L. There exists an error bound where τ = CL3 +γ (γ+λ ) u L u n+ τ u L u n u L u n, (3.7) is a constant independent of n. An estimate for formulation involving quadrature rules can be derived similarly. We have one more lemma as follow. Lemma 3.4 Let Lemma 3.3 and the condition (3.) hold. Let ū L be a discrete projection for the semilinear problem (3.). If ū and ū are sufficiently near ū L, then the solution ū n+ of formulation (3.5) converges to ū L. There exists an error bound where τ is defined as in Lemma 3.3. ū L ū n+ τ ū L ū n ū L ū n, (3.8) 4

15 Denote by ε k the error ū L ū k for all k. Equation (3.8) leads to ε n+ τ ε n ε n. (3.9) To determine the order of convergence, we assume that there exists an asymptotic relationship as follow Therefore, it is clear that ε n+ ω ε n q. ε n ω ε n q and ε n ω q ε n q. Substituting the equations into equation (3.9) leads to and ω ε n q τ ω q ε n q ε n ω + q ε n q τ ε n + q. We conclude that q = + q, or q = + 5.6, and τ = ω q. Hence, the quasi-newton s rate of convergence is superlinear. Equation (3.9) can be written as ε n+ τ q ε n.6. Above results can be summarize as an important theorem below. Theorem 3. Let the conditations in Lemma 3.4 hold. There exists a constant ω such that ū L ū n+ ω ū L ū n q, (3.3) where q.6 and ω = τ q = τ q is a constant independent of n. Remark 3.5 In computation, such a q-rate for numerical results can be calculated via the formula q = ln ε n+ s ln ε n s ln ε n s ln ε n s, (3.3) where s stands for one kind of the Sobolev norms. Numerical experiments concerning the superlinear convergence will be given in the next section. 5

16 4 Numerical experiments In this section we provide an algorithm for the radial basis collocation method combining with the quasi-newton iteration and present an example in a two-dimensional setting. Finally, a comparison about the convergence between this scheme and the other Newton scheme is given. 4. Implementation scheme The quasi-newton iteration scheme for the semilinear problem (3.) is given by ū n+ (x k ) f(x,ū n(x k )) f(x,ū n (x k )) ū n (x k ) ū n (x k ) ū n+ (x k ) (4.) = f(x k,ū n (x k ) f(x,ū n(x k )) f(x,ū n (x k )) ū n (x k ), x k, ū n (x k ) ū n (x k ) ū n+ (x l ) =, x l, for n =,,..., with initial guesses ū (x) and ū (x), in which ū n+ (x) is specified as follow where ū n+ (x) = L i= Φ(x) = [g (x),g (x),...,g L (x)] T, and g i (x) are radial basis functions mentioned in Section. ā (n+) i g i (x) = Φ(x) T a (n+), (4.) a (n+) = [ā (n+),ā (n+),...,ā (n+) L ] T, Let {x k } Na k= and {x l} Na+N b l=n a+ be the sets of collocation points located in the solution domain and on its boundary, respectively. Denote by N c = N a + N b the total number of collocation points. The analysis in previous section states that the sequence ū n+ will converge to solution u as long as the condition (3.) is fulfilled. This condition requires that the maximal spacing of collocation points satisfy the relation h = o(l (+ 4 r+ ) ). To ensure that an optimal solution can be obtained, the number of collocation points should be chosen much larger than the number of basis functions, i.e., N c L. where The algebraic linear system (4.) can further be written as M (n) a (n+) = b (n), (4.3) M (n) = M Q(a (n ),a (n) ) and b (n) = b(a (n) ) Q(a (n ),a (n) )a (n), 6

17 with two initial guesses a () and a (), where M and Q(, ) are N c by L matrices, and the vector b( ) is with dimension N c. The definition of each row for the matrix M is M j = and for the matrix Q(, ) is Q(a (n ),a (n) ) j = The components of the vector b( ) are given by { Φxx (x j ) T + Φ yy (x j ) T, j N a Φ(x j ) T, N a + j N c, f(x j,φ(x j ) T a (n) ) f(x j,φ(x j ) T a (n ) ) Φ(x j ) T a (n) Φ(x j ) T a (n ) Φ(x j ) T, j N a,, N a + j N c. b(a (n) ) j = { f(xj,φ(x j ) T a (n) ), j N a, N a + j N c. We simply name the iteration scheme (4.3) as radial basis collocation quasi-newton (RBC-QN) iteration scheme. The stopping criterion for the iteration is chosen as ū n+ (x) ū n (x), 6. (4.4) Numerical results for solving a semilinear elliptic equation will be given in the following section. 4. Some results on a test problem Consider a two-dimensional model problem [33] of the form u = f(x,y,u) in, u x = on x =, y, u = on y =, < x, u = on x =, y, u = on y =, x, where = {(x,y) < x <, < y < }, the function f(x,y,u) is given by f(x,y,u) = π u( u) + q(x,y), 4 in which the function q(x,y) = sin( πy ) + π 4 ( x ) sin ( πy ). The analytical solution is given by u(x,y) = ( x )sin( πy ). 7

18 We utilize the RBC-QN iteration scheme (4.3) with two initial guesses a () = [,,, ] T and a () = [,,,] T and use the following basis functions g i (x) = r i + c, which are known as the reciprocal MQ RBF. Numerical results have been obtained by taking L = N to be 6 to, N c to be 5 to 9, and c to be. to be 3.. Equally spaced collocation and source points are adopted so that the radial distance δ has an approximate relation like δ O( N ) O( L ). The number of collocation points, N c, should be greater than the number of basis functions, L. An overdetermined algebraic system is constructed, the least squares method is used to solve the algebraic system at each step, and the iteration is performed until criterion (4.4) is satisfied. L = N 6 8 u ū n, 4.48(-3).53(-4).3(-5).74(-6) u ū n,.7(-3) 5.8(-5) 6.87(-6) 8.93(-7) u ū n, 9.47(-3) 4.94(-4) 3.(-5).94(-6) Cond. 4.4(5).(8) 8.55(8).63(9) No. of iterations Table : The error norms and condition numbers with c =.6 and N c = 3 5. The computed errors in the nth iterate and condition number of the matrix M (n) with various values of L are listed in Table. We see from this table that there exist the following asymptotic relations u ū n, = O((.7) /δ ) = O((.7) L ), (4.5) u ū n, = O((.3) /δ ) = O((.3) L ), u ū n, = O((.6) /δ ) = O((.6) L ). Cond. = O((4.8) /δ ) = O((4.8) L ). Above relations indicate that the errors decay exponentially as L increases, i.e., δ decreases, while the condition number grow exponentially as L increases. Next, we consider the changes of the number of collocation points N c and the shape parameter c of RBF. The computed results are listed in Tables and 3, respectively. We notice form Table that the number of collocation points N c has minor effect on 8

19 N c u ū n, 5.7(-4).3(-5).7(-5).63(-5) u ū n,.77(-4) 6.87(-6) 4.7(-6) 6.43(-6) u ū n, 8.8(-4) 3.(-5) 4.34(-5) 7.7(-5) Cond. 8.54(8) 8.55(8).4(9) 4.37(9) No. of iterations Table : The error norms and condition numbers with L = and c =.6. c u ū n, 9.99(-5).4(-5).4(-5) 8.54(-6) 6.4(-6) u ū n,.47(-5) 7.4(-6) 6.9(-6) 4.5(-6).7(-6) u ū n, 4.3(-4).3(-5) 3.39(-5).77(-5).9(-5) Cond. 8.4(8).7(9) 9.7(8).43(9).5(9) No. of iterations Table 3: The error norms and condition numbers with L = and N c = 35. the accuracy, as long as N c 3 5 = 35. From Table 3, we have obtained the asymptotic relations as follows u ū n, = O((.8) c ), (4.6) u ū n, = O((.5) c ), u ū n, = O((.5) c ), The errors decay exponentially as c increasing. Theoretical analysis states that the flatter RBF used for interpolation, the better accuracy in the approximation is obtained. It is seen form above computational results that the number of the RBF and its shape have major effects on accuracy. The solution convergence can be achieved either by adding the basis functions or by increasing the shape parameter. Equations (4.5) and (4.6) demonstrate the exponential convergence rate, which agrees with the analysis given in Theorem 3., also Remark 3.4. The results presented so far focused on the total errors. Next, we investigate the iteration errors. The th iterate is chosen to approximate the ū L and the iteration errors for L = are provided in Table 4. By virtue of the formula (3.3), the q-rates for the iteration errors are calculated. We can observe form Table 4 that there exists a superlinear reduction. The predicted q-rates agree well with the convergence analysis result given in Theorem 3.. The number of iterations required to converge increase as L increases. The total errors for the case of L = are listed in Table 5. The errors contain discretization errors and iteration errors. The profiles of approximate solutions and the 9

20 nth iteration n = n = n = 3 n = 4 n = 5 n = 6 ū L ū n,.93(-) 6.9(-) 5.(-3) 9.56(-5).45(-7) 6.49(-) q-rate / ū L ū n,.3(-) 4.43(-) 3.6(-3) 6.46(-5) 9.69(-8) 3.5(-) q-rate / ū L ū n, 5.47(-).8(-) 8.8(-3).58(-4).37(-7).84(-) q-rate / / Table 4: The iteration errors and q-rate for L =, c =.6 and N c = 35. nth iteration n = n = n = 3 n = 4 n = 5 n = 6 n = 7 u ū n,.39(-) 6.9(-) 5.(-3) 8.3(-5).3(-5).3(-5).3(-5) u ū n,.3(-) 4.43(-) 3.6(-3) 5.8(-5) 6.78(-6) 6.87(-6) 6.87 (-6) u ū n, 5.47(-).8(-) 8.8(-3).43(-4) 3.9(-5) 3.(-5) 3.(-5) Table 5: The total errors for L =, c =.6 and N c = 35. errors for the case of L = are plotted in Figures and, respectively. We can see from Figure that there seems to have larger errors occurred on the boundary. This can be improved by considering higher weight on the boundary in functionals E(v) and Ê(v) defined in (3.7) and (3.9), for example, E(v) = ( ) dx + P c v dl, where P c denotes a large and positive number. The profiles of the errors by such an approach with P c = 6 are illustrated in Figure 3. Indeed, we can observe that the errors near boundary have been improved. More details will appear elsewhere. 4.3 Discussion and comparison The Newton and the quasi-newton iteration schemes are closely related. The radial basis collocation Newton (RBC-N) iteration scheme is given by Ma (n+) J (a (n) )a (n+) = b(a (n) ) J (a (n) )a (n), (4.7) with an initial guess a (). The J ( ) is exactly the Jacobian matrix of the vector b( ), which is defined as { ( J (a (n) fu x ) j = j,φ(x j ) T a (n)) Φ(x j ) T, j N a,, N a + j N c. If the approximation f u (x,ū n ). = f(x,ū n) f(x,ū n ) ū n ū n

21 Approximation V Approximation V x Approximation V y.5.5 Figure : The profiles of solution and solution derivatives by RBC-QN iteration scheme with L =, N c = 35 and c =.6. is used in the Newton iteration scheme (4.7), we then obtain a quasi-newton iteration scheme (4.3). We note that the Newton iteration requires N a function evaluations per iteration for f(x,ū n ) and f u (x,ū n ), whereas the quasi-newton requires only N a function evaluations per iteration for f(x,ū n ). Therefore, the Newton scheme is generally more expensive per iteration. Other disadvantage of Newton iteration is related to the sensitivity to the initial guess. For example, for the case a () = [,,..., ] T, the Newton scheme (4.7) fails to converge. Let us make a brief conclusion about the superlinear convergence by using these two iteration schemes. Suppose that the initial guesses are quite close to the desired solution ū L. Let {ū N n } and {ū Q n } be generated by scheme (4.7) and scheme (4.3) tending to a limit ū L, respectively. There exists the following estimates ū L ū N n τ ū L ū N n, ū L ū Q n ω ū L ū Q n q, q.6, where τ and ω are defined as in Lemma 3.3 and Theorem 3.. It follows that ū L ū N n τ ++ +n { ū L u } n = τ n { ū L u } n = τ {τ ū L u } n,

22 x 6 Error for U.5 Error for U x Error for U y x 5 x Figure : The errors of solution and solution derivatives by RBC-QN iteration scheme with L =, N c = 35 and c =.6. ū L ū Q n ω +q+ +qn { ū L u } qn = ω qn q { ū L u } qn = τ {τ ū L u } qn, in which a crucial relation ω = τ q is used. To satisfy a desired tolerance ǫ, we must have n κ, for the RBC-N iteration scheme, ln n κ, for the RBC-QN iteration scheme, ln q where κ is given by { κ = ln ln τ + lnǫ ln τ + ln ū L u }. We denote the total errors for these two iteration schemes E N := u ū N n u ū L + ū L ū N n =: E + E N, E Q := u ū Q n u ū L + ū L ū Q n =: E + E Q, where E stands for the discretization error, and the E N and EQ stand for the iteration errors for Newton and the quasi-newton, respectively. Their relationship as illustrated

23 x 6 Error for U.5 Error for U x Error for U y x 5 x Figure 3: The errors of solution and solution derivatives by penalty RBC-QN iteration scheme with penalty number P c = 6. in Figure 4 indicates that the Newton scheme converges more rapidly when we fix the L. Consequently, it requires fewer iterations to attain the desired accuracy. Taking the initial guess a () = [,,...,] T, the iteration errors and the total errors for the case of L = are given in Tables 6 and 7, respectively. Indeed, we can observe from Table 6 that there exists a quadratic error reduction and it needs fewer iterations to achieve the desired accuracy. nth iteration n = 7 n = 8 n = 9 n = n = ū L ū n,.3 3.3(-).5(-).8(-4) 9.39(-9) q-rate / / ū L ū n,.3.5(-).73(-).(-4) 6.3(-9) q-rate / /.7.. ū L ū n, (-) 4.(-).98(-4).5(-8) q-rate / / Table 6: The iteration errors and the quadratic rate for L =, c =.6 and N c = 35 by using the Newton iteration scheme. 3

24 Error E N E Q E n Figure 4: The relationship between the iteration number and the Newton and the quasi- Newton iteration schemes when L is fixed. nth iteration n = 7 n = 8 n = 9 n = n = n = u ū n,.3 3.3(-).5(-).69(-4).3(-5).3(-5) u ū n,.3.5(-).73(-).6(-4) 6.87(-6) 6.87(-6) u ū n, (-) 4.(-).83(-4) 3.(-5) 3.(-5) Table 7: The total errors for L =, c =.6 and N c = 35 by using the Newton iteration scheme. 5 Conclusion remarks The present work is aimed at providing a convergence analysis and implementation schemes of the radial basis collocation method in conjunction with the quasi-newton iteration for solving semilinear elliptic problems. The main results of this work indicate that there exists an exponential convergence rate with respect to the number and the shape factor of the radial basis functions, and a superlinear convergence rate with respect to the number of iteration. The numerical results by using this iteration scheme (4.3) agree well with the analytical results provided in Section 3. We summarize the merits for such numerical results as follows.. The source points and collocation points may be scattered in a rather arbitrary fashion. However, the number of collocation points should be chosen to be much larger than the number of source points. Such a scheme is not confined by problems with regular domain. This is a meshfree technique with considerable flexibilities, and it can be easily applied to higher dimensions.. The proposed method provides high accuracy. It can be seen from numerical results that the errors exhibit exponential convergence rates. Typical mesh-based approaches, for example, finite difference or finite element methods, offer algebraic convergence rates. 3. The simplicity of the proposed method allows easy computer coding. It is known 4

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