Domain decomposition for parallel processing of spatial problems

Spatial models often are not used to their fullest potential because they have massive computational requirements. Existing workstations and microcomputers often must solve these models in batch mode and, consequently, decision makers are unable to explore and resolve complex spatial problems in an interactive and graphical environment similar to that provided by general purpose business software. Parallel processing can solve spatial models at high speed, however, greatly decreasing turnaround times and enabling decision makers quickly to see the results of revising parameters and criteria. To reap these benefits in a parallel processing environment, researchers must recast modelling procedures from their existing sequentially-oriented form to one in which parallelism can be exploited. This process, referred to as domain decomposition, is a fundamental enterprise in parallel spatial modelling. Domain decomposition for spatial problems can be structured by a set of general principles which are described and illustrated using an example from location-allocation modelling.     

Domain decomposition and multigrid solvers for flow simulation in porous media on distributed memory parallel processors

Realistic simulations of fluid flow in oil reservoirs have been proven to be computationally intensive. In this work, techniques for solving large sparse systems of linear equations that arise in simulating these processes are developed for parallel computers such as INTEL hypercubes iPSC/2 and iPSC/860. This solver is based on a combined multigrid and domain decomposition approach. The Algorithm uses line corrections solved using a multigrid method, line Jacobi and block incomplete domain decomposition as an overall preconditioner for a conjugate gradient-like acceleration method, ORTHOMIN (k). This is shown to be a factor of ten times faster on a 32-processor hypercube compared to widely used sequential solvers. Three test problems are used to validate the results which include implicit wells and faults: The first is based on highly heterogeneous two-phase flow, the second on the SPE Third Comparative Solution and the third on real production compositional data.     

An efficient cluster and decomposition algorithm for mining association rules

Conventional algorithms for mining association rules operate in a combination of smaller large itemsets. This paper presents a new efficient which combines both the cluster concept and decomposition of larger candidate itemsets, while proceeds from mining the maximal large itemsets down to large 1-itemsets, named cluster-decomposition association rule (CDAR). First, the CDAR method creates some clusters by reading the database only once, and then clustering the transaction records to the kth cluster, where the length of a record is k. Then, the large k-itemsets are generated by contrasts with the kth cluster only, unlike the combination concept that contrasts with the entire database. Experiments with real-life databases show that CDAR outperforms Apriori, a well-known and widely used association rule.     

A noniterative solution approach for parallel pseudospectral domain decomposition

A simple superposition technique is proposed for the solution of spectral collocation equations in multi-nonoverlapping subdomains. It is based on a property of linear differential equations that allows the interface conditions to be fully decoupled; thus yielding a strategy with a very high level of concurrency suitable for parallel computations. Numerical experiments indicate, for a fixed total number of collocation points, a significant degradation of spectral accuracy as the number of subdomains increases. While the technique generally yields reasonably good solution, a compromise between accuracy and geometric flexibility must be realized.     

Efficient parallel solution of structural eigenvalue problems

An efficient parallelisation of an existing sequential method for obtaining the eigenvalues of a structure by an exact analytical procedure is presented. Results are given which illustrate finding the undamped natural frequencies of a rigidly jointed plane frame, but the method is also applicable to buckling problems and to other types of structure. The parallel method is suited to both distributed and shared-memory parallel machines. It seeks to equate the workload of each processor (node) by initially sharing out the work and by subsequently passing work from working nodes to idle nodes. Experimental runs on an nCUBE2 computer show that reasonably high levels of efficiency are possible.     

Helmholtz方程的楔形基区域分解法

基于楔形基函数和无网格配点法,提出了一种求解Helmholtz型方程区域分解法。该方法克服了在求解大规模问题时用一般的全域配点法所带来的配置矩阵为非对称满阵,且高度病态的问题。通过数值结果表明,该算法在求解Helmholtz型方程降低系数矩阵条件数的同时,也能够降低误差,并达到满意的收敛效果。     

An extension of the Bene filter and some identification problems solved by nonlinear filtering methods

An extension of Benne ' filtering theorem to time dependent, parameter dependent and unknown initial conditions filtering problems is proved. A new approach for the identification of linear diffusions with unknown drift coefficients follows. New finite dimensional filters, which may also be regarded as identification filters, are derived and related to W.S. Wong's earlier work which used algebraic methods.     

Domain decomposition algorithms for spectral methods

In this paper we review some multiple-domain decomposition algorithrns for spectral methods. The alternating Schwarz algorithm and a flux preserving domain decomposition algorithm are considered. For both of them, some theoretical convergence results are given. These domain decomposition algorithms allow the use of iterative procedures for block matrices that can be efficiently implemented on parallel processors. Several numerical experiences based on the Chebyshev collocation method are carried out. A comparison of the performances of the two algorithms on several test problems is presented.     

Particle-field decomposition and domain decomposition in parallel particle-in-cell beam dynamics simulation

Particle-in-cell (PIC) simulation is widely used in many branches of physics and engineering. In this paper, we give an analysis of the particle-field decomposition method and the domain decomposition method in parallel particle-in-cell beam dynamics simulation. The parallel performance of the two decomposition methods was studied on the Cray XT4 and the IBM Blue Gene/P Computers. The domain decomposition method shows better scalability but is slower than the particle-field decomposition in most cases (up to a few thousand processors) for macroparticle dominant applications. The particle-field decomposition method also shows less memory usage than the domain decomposition method due to its use of perfect static load balance. For applications with a smaller ratio of macroparticles to grid points, the domain decomposition method exhibits better scalability and faster speed. Application of the particle-field decomposition scheme to high-resolution macroparticle-dominant parallel beam dynamics simulation for a future light source linear accelerator is presented as an example.     

区域分解算法在CoLM模式并行计算中的应用

结合通用陆面模式（CoLM）的特点,针对模式原始数据区域较大、计算精度要求较低的情况,提出基于加权平均的数据区域分解算法。算法根据网格的地表覆盖类型对网格进行分解,对各参数采用时间一维有限差分法进行离散,并对每个区域所含有的地块数进行加权累加,得到每个区域的最终输出结果。通过CoLM模式中的地表感热通量及蒸散模拟结果对区域分解算法进行验证,并进行并行算法性能分析。     

Parallel efficiency of domain decomposition methods

Data communication among processors is a key problem to increase performance of parallel algorithms for multiprocessors. In this paper, we investigate different types of domain decomposition methods on the distributed memory multiprocessor and give the estimates of their speedup and efficiency based on the computational complexity and the communication overheads.     

Analysis of parallel kinematic machine with kinetostatic modelling method

A new method, called Kinetostatic Modelling Method is proposed for analysis of parallel kinematic machines (PKMs) in the paper. First, system modelling includes mobility study, kinematic model and inverse kinematic is conducted. Then, kinetostatic modelling is presented. It includes a complete compliant model developed for the analysis of the PKMs with fixed-length legs, with the model, two global compliance indices are introduced to provide a new mean of measuring the PKM's compliance over the workspace. These two global indices are the mean value and the standard deviation of the trace of the generalized compliance matrix. The mean value represents the average compliance of the PKM over the workspace, while the standard deviation indicates the compliance fluctuation relative to the mean value. The effect of three types of compliance is investigated in order to identify a dominant factor. The proposed method is implemented to analyze the compliance of the Tripod-based PKM prototype built at the Integrated Manufacturing Technologies Institute of the National Research Council of Canada. It is shown that the proposed method is an effective means for evaluating the kinetostatic behaviour of PKMs globally.     

Domain decomposition for wave propagation problems

The problem posed by domain decomposition methods is to find the correct modeling of physical phenomena across the interfaces separating the subdomains. The technique described here for wave propagation problems is based on physical grounds since it relies on the fact that the wave equation can be decomposed into incoming and outgoing wave modes at the boundaries of the subdomains. The inward propagating waves depend on the solution exterior to the subdomains and therefore are computed from the appropriate boundary conditions, while the behavior of the outward propagating waves is determined by the solution inside the subdomains. The technique is applied to the anisotropic-viscoelastic wave equation, which practically includes all the possible rheologies of one-phase media.     

Applying formal methods to standard development: The open distributed processing experience

Since their introduction, formal methods have been applied in various ways to different standards. This paper gives an account of these applications, focusing on one application in particular: the development of a framework for creating standards for Open Distributed Processing (ODP). Following an introduction to ODP, the paper gives an insight into the current work on formalising the architecture of the Reference Model of ODP (RM-ODP), highlighting the advantages to be gained. The different approaches currently being taken are shown, together with their associated advantages and disadvantages. The paper concludes that there is no one all-purpose approach which can be used in preference to all others, but that a combination of approaches is desirable to best fulfil the potential of formal methods in developing an architectural semantics for ODP.     

Domain decomposition preconditioners for the spectral collocation method

We propose and analyze several block iteration preconditioners for the solution of elliptic problems by spectral collocation methods in a region partitioned into several rectangles. It is shown that convergence is achieved with a rate that does not depend on the polynomial degree of the spectral solution. The iterative methods here presented can be effectively implemented on multiprocessor systems due to their high degree of parallelism.     

Domain decomposition methods for the neutron diffusion problem

The neutronic simulation of a nuclear reactor core is performed using the neutron transport equation, and leads to an eigenvalue problem in the steady-state case. Among the deterministic resolution methods, simplified transport (S_PN$S{P}_N$) or diffusion approximations are often used. The MINOS solver developed at CEA Saclay uses a mixed dual finite element method for the resolution of these problems, and has shown his efficiency. In order to take into account the heterogeneities of the geometry, a very fine mesh is generally required, and leads to expensive calculations for industrial applications. In order to take advantage of parallel computers, and to reduce the computing time and the local memory requirement, we propose here two domain decomposition methods based on the MINOS solver. The first approach is a component mode synthesis method on overlapping subdomains: several eigenmodes solutions of a local problem on each subdomain are taken as basis functions used for the resolution of the global problem on the whole domain. The second approach is an iterative method based on a non-overlapping domain decomposition with Robin interface conditions. At each iteration, we solve the problem on each subdomain with the interface conditions given by the solutions on the adjacent subdomains estimated at the previous iteration. Numerical results on parallel computers are presented for the diffusion model on realistic 2D and 3D cores.     

Domain decomposition and the numerical solution of partial differential equations defined on irregular domains using segmented forms of the tau lines method

The Tau Lines Method, a numerical technique based on the combination of the Tau Method and the Method of Lines is used, in connection with the domain decomposition technique, to solve problems in partial differential equations defined on irregular domains. Two nontrivial problems have been considered. The first is a curved crack defined on a square domain and the second is defined on a kite-shaped domain. The domain of interest is subdivided into appropriate Semidiscretized elements so to efficiently deal with any appearance of boundary and/or interior singularities. Numerical application is carried out on the Poisson's equation. The approximate solutions are sought along segmented lines as finite expansions in terms of a given orthogonal polynomial basis. Two types of orthogonal expansions have been used alongside second and fourth order accurate Finite Difference Approximations. In both cases good and rapid convergence has been achieved.     

Explicit/implicit and Crank–Nicolson domain decomposition methods for parabolic partial differential equations

Two parallel non-overlapping domain decomposition algorithms for solving parabolic partial differential equations are proposed. The algorithms combine Crank–Nicolson scheme with implicit Galerkin finite element methods in sub-domains and explicit flux approximation along inner boundaries at each time step. Thus, parallelism can be easily achieved. $L^2$-norm error estimates for these explicit/implicit procedures are presented, in which time step constraints are proved to be less severe than that of fully explicit schemes. Numerical experiments are also performed to verify the theoretical analysis.     

A new parallel domain decomposition method for the adaptive finite element solution of elliptic partial differential equations

We present a new domain decomposition algorithm for the parallel finite element solution of elliptic partial differential equations. As with most parallel domain decomposition methods each processor is assigned one or more subdomains and an iteration is devised which allows the processors to solve their own subproblem(s) concurrently. The novel feature of this algorithm however is that each of these subproblems is defined over the entire domain—although the vast majority of the degrees of freedom for each subproblem are associated with a single subdomain (owned by the corresponding processor). This ensures that a global mechanism is contained within each of the subproblems tackled and so no separate coarse grid solve is required in order to achieve rapid convergence of the overall iteration. Furthermore, by following the paradigm introduced in 15 , it is demonstrated that this domain decomposition solver may be coupled easily with a conventional mesh refinement code, thus allowing the accuracy, reliability and efficiency of mesh adaptivity to be utilized in a well load-balanced manner. Finally, numerical evidence is presented which suggests that this technique has significant potential, both in terms of the rapid convergence properties and the efficiency of the parallel implementation. Copyright © 2001 John Wiley & Sons, Ltd.