Compact Local Integrated Radial Basis Function (CLIRBF) methods based on Cartesian grids can be effective numerical methods for solving Elliptic Partial Differential Equations (EPDEs) for fluid flow problems. The combination of the domain decomposition technique and function approximation using CLIRBF methods yields an effective coarse-grained parallel processing approach. This feature has enabled not only each sub-domain in the original analysis domain to be discretised by a separate CLIRBF Network but also Compact Local stencils to be independently treated. The present algorithm, namely parallel CLIRBF, achieves higher throughput in solving large scale problems by, firstly, parallel processing of sub-regions which comprise the original domain and, secondly, accelerating the convergence rate within each sub-region using groups of CLIRBF stencils in which function approximations are carried out by parallel processes. The procedure is illustrated with several numerical examples of EPDEs using Message Passing Interface (MPI) supported by MATLAB

Hoang-Trieu, T.-T.

Mai-Duy, N.

Pham-Sy, N.

Tran, C.-D.

Tran-Cong, T.

English

University of Southern Queensland Repository

  The 4th International Conference on Computational Methods (ICCM2012), Gold Coast, Australia www.ICCM-2012.org  November 25-28, 2012, Gold Coast, Australia www.ICCM-2012.org  Development of Parallel algorithm for Boundary Value Problems using Compact Local Integrated RBFN and Domain Decomposition  N. Pham-Sy   , T.-T. Hoang-Trieu  , C-D. Tran  , N. Mai-Duy   and T. Tran-Cong       Computational Engineering and Science Research Centre,  Faculty of Engineering and Surveying,  University of Southern Queensland, Toowoomba, QLD 4350, Australia    Corresponding author: nam.pham-sy@usq.edu.au   Abstract  Compact Local Integrated Radial Basis Function (CLIRBF) methods based on Cartesian grids can be effective numerical methods for solving Elliptic Partial Differential Equations (EPDEs) for fluid flow problems. The combination of the domain decomposition technique and function approximation using CLIRBF methods yields an effective coarse-grained parallel processing approach. This feature has enabled not only each sub-domain in the original analysis domain to be discretised by a separate CLIRBF Network but also Compact Local stencils to be independently treated. The present algorithm, namely parallel CLIRBF, achieves higher throughput in solving large scale problems by, firstly, parallel processing of sub-regions which comprise the original domain and, secondly, accelerating the convergence rate within each sub-region using groups of CLIRBF stencils in which function approximations are carried out by parallel processes. The procedure is illustrated with several numerical examples of EPDEs using Message Passing Interface (MPI) supported by MATLAB.  Keywords: Integrated RBFs, Compact local stencils, Domain decomposition, Parallel algorithm.  1. Introduction Radial Basis Functions (RBFs) have traditionally been used to provide a continuous interpolation of scattered data sets (Franke, 1982; Kansa, 1990). The Differential RBF (DRBF) based methods have been successfully used to solve a wide variety of differential equations. For this approach, once the field variables are known, its derivatives can be calculated through differentiation. Another approach namely the Integrated RBF (IRBF) method, which was proposed by Mai-Duy and Tran-Cong (2001, 2003), is based on the approximation of the highest-order derivatives of the ODE/PDE using RBF at the first step, and subsequently its lower-order derivatives and the dependent variable itself are obtained by integration. The IRBF based methods can outperform other approximation methods based on the DRBF technique owing to its ability to produce very accurate solutions using relatively small number of data nodes.  Although full-domain IRBF methods are highly flexible and exhibit high order convergence rates in their basic implementation, the associated fully-populated matrix systems can lead to poor numerical conditioning as the scale of a problem increases (Mai-Duy and Tran-Cong, 2008). The problem becomes critical with increasingly large data sets. Many techniques have also been developed to reduce the effect of the problem, including domain decompositions (Ingber et al., 2004; Tran et al., 2009), adaptive selection of data-centres (Ling et al., 2006), RBF preconditioners (Brown, 2005) and RBF based compact local stencil methods (Mai-Duy and Tran-Cong, 2011; Hoang-Trieu et al., 2012; Thai-Quang et al., 2012). While a reliable method of controlling numerical ill-conditioning and particularly computational cost, as problem scale increases, can be based on domain decomposition (DD) (Ingber et al., 2004; Tran et al., 2009), the use of compact local approximations facilitates the solution of differential equations without having to deal with large systems of global equations. In this work, a parallel algorithm based on Compact Local Integrated RBF (CLIRBF) and DD techniques is developed for the solution of Boundary Value Problems (BVP). A large problem is firstly decomposed into many smaller manageable problems each of which is analysed in parallel and secondly the acceleration of the convergence rate within each sub-region using groups of CLIRBF stencils is carried out by parallel processes.  This paper is organized as follows. In section 2, a brief review of a CLIRBF method and the domain decomposition technique is described. Section 3 presents a parallel algorithm with domain decomposition and local stencils. Numerical examples are then discussed in section 4 with a conclusion in section 5.  2. Review of domain decomposition technique and compact local IRBF methods For the ease of presentation, consider a second-order ordinary differential equation (ODE) with Dirichlet boundary condition as follows.                         (1)            (2)  where   and      are continuous and prescribed functions, respectively;   a second order differential operator;   an known quantity;   and    the domain under consideration and its boundary.  Domain decomposition Domain decomposition is a general method for solution of systems of equations. In a BVP governed by differential equations, DD aims to split the whole domain into smaller subdomains of analysis while guaranteeing the continuity at the splitting boundary. DD algorithms can be grouped into two classes: (i) non-overlapping methods and (ii) overlapping methods (Smith et al., 1996). In this paper, the additive overlapping method is implemented on the domain under consideration.    Figure 1. 1D Domain Decomposition For illustrative purposes, the domain   is divided into two subdomains    and   . Let     and     be boundaries of    and    respectively and               the artificial boundaries between    and    (Figure 1). Here the boundary condition imposed on the interface is of Dirichlet type. The DD algorithm for n-   step can be written as                                 (3)                   and                                (4)                   where         (             ) is the value of   in    obtained from the solution in    at a step     (see Figure 1). The right most and left most sides of the overlapping zone are the artificial boundaries   ,    of   ,    respectively. While the middle part of it is for the purpose of continuity check. The BVPs (3) and (4) are solved separately using the CLIRBF method which is presented in the next section. Compact local IRBF methods The idea of using local RBF collocation approaches to solve DEs can be seen as a generalization of the compact finite difference scheme in terms of scattered nodes (Wright and Fornberg, 2006). In this work, the compact local  –point stencil technique which described in (Hoang-Trieu et al., 2012) is employed to discretise BVPs.    Consider three consecutive points                associated with grid point    (       ) in a typical global 1D Cartesian grid line.            is used to represent a stencil in local 1D Cartesian grid line (node      ). The second-order derivative of the function   in Eq. (1) is decomposed into a set of RBFs as                                  (5)  where          are the set of RBF weights, and                the set of RBFs. The superscript (.) is used to indicate the associated derivative order. In this work, the multiquadric (MQ) RBF is used and given by                           (6)  where    and    are the MQ center and width of RBFs, respectively. Integrating expression (5) over the stencil yields the expression for lower order derivatives and the function itself as                             (7)                            (8)  where    and    are constants of integration. Expression (5), (7) and (8) can be rewritten into the matrix form as follows.                                                      (9)                                                   (10)                                                (11)  where                    . The conversion matrix   is formed as                     (12)  where              ,         is a set of three equations representing nodal values of   over the stencil,      is employed to represent values of (1) at    and   .  As a result, values of function   and its derivatives at an arbitrary point   on the stencil are calculated in the physical space as                          (13)                       (14)                    (15) 3. Implementation The parallelization is carried out using the   node CLIRBF scheme and based on the domain decomposition technique as detailed in section 2 above. In this work, the overlapping DD scheme is employed. It is noted that each stencil does not needthe data from other stencils to start its calculation. This data independency allows stencils calculation to run on parallel CPU cores. The flowchart of the algorithm is shown in Figure 2. It can be seen that the algorithm is divided into three main parts. The first one is initialization, which allocates the memory and calculates initial values. The second part is the parallel part of algorithm where stencils are assigned equally to CPU cores. As all cores share the same memory, they can store the results of calculation in a predefined matrix. The final part of the algorithm is to solve the final system of equations to obtain the approximated result. Parallel implementation of the algorithm is based on the single program, multiple data (SPMD) paradigm with message passing interface (MPI) for parallel communication. In this paper, we use the MPI parallel programming within the MATLAB environment.  4. Numerical examples Consider the following second-order ODE.                                                                (16)  with an analytic solution                      . The problem is solved using the present method for two different types of boundary condition. The domain is partitioned into   sub-domains and a wide range of grids                 is considered.  Dirichlet boundary condition  The Dirichlet conditions are        and                       . Figure 3 depicts a comparison of the results obtained by the present method, the CLIRBF and the analytic ones. The results show that the present method achieves almost the same accuracy level as the CLIRBF method. In fact, convergence rate       of the present method is     , and the CLIRBF is     .       Figure 2. Parallel algorithm: Flow-chart of CLIRBF with parallel stencils (left figure); flowchart of domain decomposition with   sub-domains (right figure)   Figure 3. Second order problem with Dirichlet boundary condition: Solutions obtained by the     nodes CLIRBF method and the present method (left figure); Relative    errors of the solution   against the grid size by the     nodes CLIRBF method and the present method (right figure).   Dirichlet and Neumann Boundary conditions   The Dirichlet condition is imposed on the left end        and the Neumann condition on the right end                                               .   While the result, described in the figure 4 by the present method, is in very good agreement with the analytic solution, the convergence rate displayed in the right figure shows that the present method yields a higher accuracy in comparison with the   node CLIRBF method.   Figure 4. Second order problem with Dirichlet-Neuman boundary conditions: Solutions obtained by the     nodes CLIRBF method and the present method (left figure); Relative    errors of the solution   against the grid size by the     nodes CLIRBF method and the present method (right figure).  5. Conclusion In this paper, we have developed a parallel algorithm based on the combination of domain decomposition technique and function approximation using 3 point compact local IRBF method. The proposed algorithm allows not only a large scale problem to be discretised by parallel CLIRBF networks but also compact local stencils to be independently treated in multiple-core CPUs. Advantages of the new approach include (i) to alleviate the ill-conditioned problem associated with global IRBF methods; (ii) to avoid the reduction in convergence rate caused by differentiation and (iii) to achieve higher throughput in solving large scale problems. The method is verified with several numerical examples using MPI supported Matlab.   Acknowledgement N. Pham-Sy would like to thank CESRC, FoES and the University of Southern Queensland for a PhD scholarship. This research was supported by the Australian Research Council.     References Brown, D. (2005). On approximate cardinal preconditioning methods for solving pdes with radial basis functions. Engineering Analysis with Boundary Elements, 29(4):343–353. Franke, R. (1982). Scattered data interpolation: Tests of some methods. Mathematics of Computation, 38 (157). Hoang-Trieu, T.-T., Mai-Duy, N., and Tran-Cong, T. (2012). Several compact local stencils based on integrated RBFs for fourth-order ODEs and pdes. Computer Modeling in Engineering & Sciences, 84(2):171–203. Ingber, M., Chen, C., and Tanski, J. (2004). A mesh free approach using radial basis functions and parallel domain decomposition for solving three-dimensional diffusion equations. International Journal for Numerical Methods in Engineering, 60:2183–2201. Kansa, E. (1990). Multiquadrics - A scattered data approximation scheme with applications to computational fluid-dynamics - I Surface approximations and partial derivative estimates. Computers & Mathematics with Applications, 19(8-9):127–145. Ling, L., Opfer, R., and Schaback, R. (2006). Results on meshless collocation techniques. Engineering Analysis with Boundary Elements, 30(4):247–253. Mai-Duy, N. and Tran-Cong, T. (2001). Numerical solution of differential equations using multiquadric radial basis function networks. Neural Networks, 14:185–199. Mai-Duy, N. and Tran-Cong, T. (2003). Approximation of function and its derivatives using radial basis function networks. Applied mathematical modeling, 27:197–220. Mai-Duy, N. and Tran-Cong, T. (2008). A multidomain integrated-radial-basis-function collocation method for elliptic problems. Numerical Methods for Partial Differential Equations, 24(5):1301–1320. Mai-Duy, N. and Tran-Cong, T. (2011). Compact local integrated-RBF approximations for second-order elliptic differential problems. Journal of Computational Physics, 230:4772–4794. Smith, B. F., Bjorstad, P. E., and Gropp, W. D. (1996). Domain Decomposition Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press. Thai-Quang, N., Le-Cao, K., Mai-Duy, N., and Tran-Cong, T. (2012). A high-order compact local integrated-rbf scheme for steady-state incompressible viscous flows in the primitive variables. Computer Modeling in Engineering & Sciences, 84(6):528–557. Tran, C. D., Phillips, D. G., and Tran-Cong., T. (2009). Computation of dilute polymer solution flows using bcf-rbfn based method and domain decomposition technique. Korea-Australia Rheology Journal, 21:1–12. Wright, G. and Fornberg, B. (2006). Scattered node compact finite difference-type formulas generated from radial basis functions. Journal of Computational Physics, 212(1):99–123.  

4th International Conference on Computational Methods (ICCM 2012)

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4th International Conference on Computational Methods (ICCM 2012)

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