Essays

A matrix decomposition RBF algorithm: Approximation of functions and their derivatives

Description
A matrix decomposition RBF algorithm: Approximation of functions and their derivatives
Categories
Published
of 16
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
Share
Transcript
  Applied Numerical Mathematics 57 (2007) 304–319www.elsevier.com/locate/apnum A matrix decomposition RBF algorithm: Approximationof functions and their derivatives Andreas Karageorghis a , C.S. Chen b , Yiorgos-Sokratis Smyrlis a , ∗ a  Department of Mathematics and Statistics, University of Cyprus/  Π   ANE  Π   I  Σ  THMIO KY  Π  POY,PO Box 20537, 1678 Nicosia/  Λ  EYK  ΩΣ   IA, Cyprus/KY  Π  PO Σ  b  Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS 39406, USA Available online 18 May 2006 Abstract We propose an efficient algorithm for the approximation of functions and their derivatives using radial basis functions (RBFs).The interpolation points are placed on concentric circles and the resulting matrix has a block circulant structure. We exploit thiscirculant structure to develop an efficient algorithm for the solution of the resulting system using RBFs. As a result, extremely highaccuracy in approximating the given function and its derivatives can be achieved. The given algorithm is also capable of solvinglarge-scale problems with more than 100000 interpolation points in two dimensions. © 2006 IMACS. Published by Elsevier B.V. All rights reserved. Keywords:  Radial basis functions; Boundary meshless methods; Circulant matrices; Elliptic boundary value problems 1. Introduction During the last three decades, methods using radial basis functions (RBFs) have been gaining in popularity invarious areas of scientific computing such as neural networks, computer graphics, Computer Aided Design (CAD) andsurface reconstruction. RBFs are particularly effective for multivariate approximation [8,7,10,11,28,29]. The theoryof RBFs was not well understood until Franke [14] published a survey paper on the evaluation of 29 interpolatingmethods. This was followed by a paper by Micchelli [25] who established firm theoretical results on the invertibilityof the RBF interpolation matrix. Further details of these results can be found in [8].In contrast to the many attractive features of RBFs, one disadvantage is that most of the commonly used RBFs areglobally supported. This implies that the resulting interpolation matrix is dense. It appears that other than possiblysymmetry, the interpolation matrices have no obvious structure that could be exploited in the solution of the interpola-tion system. As a result, when using a direct symmetric solver the cost of solving the linear system is  N  3 / 6 + O (N  2 ) flops. Further, when the number of interpolation points used is large and they are densely distributed the matrix be-comes ill-conditioned. In practice, direct methods for solving the systems resulting from RBFs are inappropriate forproblems with more than 2000 interpolation points. To alleviate these difficulties, compactly supported radial basisfunctions (CS-RBFs) were developed in the last decade [6,29,33,34]. The use of CS-RBFs leads to sparse matrix * Corresponding author.  E-mail address:  smyrlis@ucy.ac.cy (Y.-S. Smyrlis).0168-9274/$30.00 © 2006 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.apnum.2006.03.028   A. Karageorghis et al. / Applied Numerical Mathematics 57 (2007) 304–319  305 systems which is desirable for large-scale problems, and these basis functions have been used in various applications.However, the convergence rate of these newly constructed CS-RBFs is slow and it is thus difficult to achieve highaccuracy. Also, the way that the scaling factor appearing in CS-RBFs needs to be chosen is not well understood.Most of the work to date on RBFs is related to interpolation of scattered data. To tackle large-scale interpolationproblemsusingRBFs,variousfastfittingalgorithmshavebeendeveloped[2–5].Theseapproachesareveryeffectiveinthe reconstruction and representation of three-dimensional objects. However, it is not clear how these fast algorithmsfor solving large-scale interpolation problems can be extended to solving science and engineering problems whichinvolve solving a series of partial differential equations (PDEs). During the past decade, there has been an increasinginterest in applying RBFs for solving such problems involving the solution of PDEs [15,20]. Methods using RBFs are meshless  and have therefore recently attracted much attention in the engineering community and RBFs have becomean important tool in scientific computing. Recently, pre-conditioning techniques [1,21] and domain decompositionmethods (DDM) [18,36] have been proposed to alleviate the conditioning and storage difficulties associated withsolving PDEs using RBFs when using a large number of degrees of freedom. One disadvantage of pre-conditioningand DDM techniques is the fact that their implementation is highly non-trivial.In general, when using RBFs interpolation schemes, randomly scattered points or uniform grid points are chosen.In the solution of PDEs using RBFs we have the freedom of selecting the collocation (interpolation) points. If thesecollocation points are chosen in a particular fashion, the resulting coefficient matrix has a certain structure which canbe exploited for the efficient solution of the corresponding system. For regularly spaced points, an RBF formulationcan lead to Toeplitz matrices, for which effective preconditioning techniques have been developed [1]. In this paper,by placing the interpolation points on concentric circles, we develop an efficient matrix decomposition algorithm byexploiting the properties of circulant matrices. Recently, efficient algorithms using the properties of circulant matriceshave been employed for solving homogeneous differential equations in the context of the Method of FundamentalSolutions (MFS) [12,13,31,32]. These algorithms make extensive use of Fast Fourier Transforms (FFTs). The basicidea of using FFTs for RBFs is not new as it has been used, albeit in a different context, by Jetter and Stöckler [19].Motivated by this simple and elegant circulant matrix approach, in this paper we extend these ideas for the reconstruc-tion of function surfaces and their derivatives. The technique developed here will be later extended to solving a largeclass of partial differential equations in a subsequent paper. In this paper, we focus on the approximation of functionsand their derivatives in the two-dimensional case. 2. Radial basis function interpolation We consider the interpolation of a multivariate function  f   : Ω  → R , where  Ω  ⊂ R 2 , from a set of sample values { f( x j  ) } N j  = 1  on a discrete set  X  ={ x j  } N j  = 1  ⊂ Ω . Such multivariate functions can be efficiently reconstructed if theyare approximated by linear combinations of univariate interpolation functions with Euclidean norm  · . This can beachieved by using translates  Φ( x  − x j  )  of a single continuous real valued function  Φ  defined on R , and by letting  Φ be radially symmetric; i.e., Φ( x ) := ϕ   x   , with a continuous function  ϕ  on  R + 0  .  In the mathematical literature,  ϕ  is often called a radial basis function withcenters  { x j  } N j  = 1  and  Φ  is the associated kernel.Interpolants  ˆ f  N   to  f   can be constructed as ˆ f  N  ( x ) = N   j  = 1 a j  ϕ   x  − x j     (2.1)with real coefficients { a j  } N j  = 1 . The coefficients on the right-hand side of (2.1) can be determined by interpolation; i.e., ˆ f  N  ( x i ) = f( x i ),  1  i  N, if the linear system N   j  = 1 a j  ϕ   x i  − x j    = f( x i ),  1  i  N,  (2.2)  306  A. Karageorghis et al. / Applied Numerical Mathematics 57 (2007) 304–319 is uniquely solvable. This is true if the symmetric  N   × N   matrix A ϕ  =  ϕ(  x 1 − x 1  ) ... ϕ(  x 1 − x n  ).........ϕ(  x n − x 1  ) ... ϕ(  x n − x n  )   (2.3)is nonsingular. In the RBF literature, it has been observed that, for certain choices of RBFs, (2.3) could be singular,for some configurations of interpolation points. As a result, a lot of interest arose in finding sufficient conditions toensure the existence of   A − 1 ϕ  [8]. Most of the globally defined RBFs are only conditionally positive definite [25], thusnot guaranteeing the invertibility of the matrix  A ϕ . In order to guarantee the unique solvability of the interpolationproblem, one needs to add a polynomial term to the interpolant (2.1) giving ˆ f  N  ( x ) = N   j  = 1 a j  ϕ   x  − x j    + t   k = 1 b k p k ( x ),  (2.4)along with the constraints N   j  = 1 a j  p k ( x j  ) = 0 ,  1  k  t,  (2.5)where  { p k } t k = 1  is a basis for P  m − 1 , the set of polynomials in two variables of degree  m − 1, and t   =  m + 12  is the dimension of  P  m − 1 . Let P  T =  p 1 ( x 1 ) ... p 1 ( x N  ).........p t  ( x 1 ) ... p t  ( x N  )  . The interpolation conditions f( x i ) = N   j  = 1 a j  ϕ   x i  − x j    + t   k = 1 b k p k ( x i ),  1  i  N, subject to (2.5) can be rewritten as the linear system  A ϕ  P P  T 0  ab  =  f  0  ,  (2.6)where  a  =[ a 1 ,...,a N  ] T , b =[ b 1 ,...,b t  ] T and  f   =[ f( x 1 ),...,f( x N  ) ] T .For convenience, we often replace the argument of   ϕ  in (2.1) by  r  := x  − x j   . In Table 1, we present a list of popular choices of basis functions  ϕ . In Table 1, polynomials up to degree  m − 1 are required to guarantee the re-construction of a given function. Despite the many attractive features of these RBFs, it is known that all the RBFsin Table 1 are globally supported. Thus, the direct solution of the system resulting from the interpolation equationsrequires O (N  3 )  operations and O (N  2 )  storage, and is thus impractical for large  N  . As stated earlier, in order to over-come these difficulties, CS-RBFs were developed [6,29,33,34]. The most popular CS-RBFs are the ones constructedby Wendland [33] as shown in Table 2 where the cut-off function  (r) +  is defined to be  r  if 0  r  1 and zero else-where. The application of CS-RBFs leads to a sparse interpolation matrices which are positive definite and which canbe reconstructed without additional polynomial terms. Despite the effectiveness of CS-RBFs, there are some issuesassociated with their slow rate of convergence, as mentioned earlier.Apart from the problems of storage and computer running time, RBF systems suffer from ill-conditioning whena large number of interpolation points is used. In general, the condition number is directly linked to the order of thebasis functions and density of the interpolation points. Schaback [30] has established the following   A. Karageorghis et al. / Applied Numerical Mathematics 57 (2007) 304–319  307Table 1Globally defined radial basis functionsGaussian  ϕ(r) = e − cr 2 for  c > 0Inverse multiquadrics  ϕ(r) = (r 2 + c 2 ) β/ 2 ,  c > 0 >β Sobolev splines  ϕ(r) = K ν (r)r ν , where  K ν  is the spherical Bessel function,  ν > 0Linear  ϕ(r) = r ,  m = 1Cubic  ϕ(r) = r 3 ,  m = 2Polyharmonic splines  ϕ(r) = r β log r ,  β  ∈ 2 Z ,  m>β/ 2Polyharmonic splines  ϕ(r) = r β ,  β  ∈ R > 0 \ 2 Z ,  m> [ β/ 2 ] Multiquadrics  ϕ(r) = (r 2 + c 2 ) β/ 2 ,  β  ∈ R > 0 \ 2 Z ,  c > 0 ,m> [ β/ 2 ] Table 2Wendland’s CS-PD-RBFs [33] d   = 1  ϕ  = ( 1 − r) +  C 0 ϕ  = ( 1 − r) 3 + ( 3 r  + 1 ) C 2 ϕ  = ( 1 − r) 5 + ( 8 r 2 + 5 r  + 1 ) C 4 d   = 3  ϕ  = ( 1 − r) 2 +  C 0 ϕ  = ( 1 − r) 4 + ( 4 r  + 1 ) C 2 ϕ  = ( 1 − r) 6 + ( 35 r 2 + 18 r  + 3 ) C 4 ϕ  = ( 1 − r) 8 + ( 32 r 3 + 25 r 2 + 8 r  + 1 ) C 6 Uncertainty Principle.  Either one goes for a small error and gets a bad sensitivity, or one wants a stable algorithmand has to take a comparably large error.The higher the order of the basis function, the worse the condition number, and the better the accuracy. If thenumber and positions of the interpolation points (data density) are fixed, the Uncertainty Principle suggests that theorder of the basis functions used should be chosen with great care. This order should be as low as the applicationpermits, and any excessive order will have negative effects on stability. Furthermore, for low density interpolationpoints, one can use high-order basis functions, and for high density interpolation points, one can use low-order basisfunctions to avoid numerical problems.In the next section, we propose an algorithm which will enable us to solve (2.3) for large  N  . 3. Matrix decomposition algorithm Our goal is to develop an efficient algorithm for the solution of the system arising from equations (2.2). Usually,the collocation points are uniformly distributed in a region containing  Ω . The resulting system (2.2) can be solvedusing a direct solver at a cost of  O (N  3 )  operations.For the new approach, we assume that the given function to be approximated can be extended outside its domain Ω .We then place the collocation points on the boundaries of concentric disks defined by Ω R i  =  x  ∈ R 2 :  | x | <R i  ,  1  i  m,  (3.1)where  R 1  <R 2  < ··· <R m  and  Ω  ⊂ Ω R m .On the circles  ∂Ω R i , 1  i  m , we define the  mn  collocation points  x i,j   ={ (x i,j  ,y i,j  ) } m,ni = 1 ,j  = 1  by x i,j   = R i  cos  2 (j   − 1 )πn +  2 α i πn  , y i,j   = R i  sin  2 (j   − 1 )πn +  2 α i πn  , j   = 1 ,...,n, where the position of the points on the circle of radius  R i  is shifted by an angle 2 α i π/n  with 0  α i  1. For  α i  = 0,we have a rotation for each circle. To ensure a uniform distribution of the collocation points, we choose the radii of the concentric circles as follows R i  =  imr max , i  = 1 ,...,m,  308  A. Karageorghis et al. / Applied Numerical Mathematics 57 (2007) 304–319 Fig. 1. Typical distribution of circulant collocation points. where  r max  is the radius of the largest circle. In Fig. 1 we present a typical distribution of collocation points with n = 20,  m = 15,  r max  = 1, and  α i  = 0 for  i  = 1 , 3 ,..., 15,  α i  = 0 . 5, for  i  = 2 , 4 ,..., 14.The collocation equations (2.2) yield a system of the form A ϕ a  = f  ,  (3.2)where the  mn × mn  matrix  A ϕ , has the structure A ϕ  =  A 11  A 12  ... A 1 ,m A 21  A 22  ... A 2 ,m ............A m 1  A m 2  ... A m,m  (3.3)where each of the  n × n  submatrices  A k,ℓ  is  circulant  , and A k,ℓ  = circ  ϕ  x k, 1 − x ℓ, 1  ,ϕ  x k, 1 − x ℓ, 2  ,...,ϕ  x k, 1 − x ℓ,n   , k,ℓ = 1 ,...,m. Also, we have f  (i − 1 )n + j   = f( x i,j  ), i  = 1 ,...,m, j   = 1 ,...,n. We shall exploit the property that these submatrices  A k,ℓ  are circulant by using the fact that circulant matricesare diagonalized in the following way. If   C  =  circ (c 1 ,...,c n ) , then  C  =  U  ∗ DU   where  D  =  diag (d  1 ,...,d  n ) , d  j   =  nk = 1 c k ω (k − 1 )(j  − 1 ) and the  n  ×  n  matrix  U   is the Fourier matrix which is the conjugate of the matrix (see[9,31]) U  ∗  =  1 n 1 / 2  1 1 1  ...  11  ω ω 2 ... ω n − 1 1  ω 2 ω 4 ... ω 2 (n − 1 ) ............ 1  ω n − 1 ω 2 (n − 1 ) ... ω (n − 1 )(n − 1 )  ,
Search
Similar documents
View more...
Related Search
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks