A linearly semi-implicit compact scheme for the Burgers–Huxley equation

In this paper, a linearly semi-implicit compact scheme is developed for the Burgers–Huxley equation. The equation is decomposed into two subproblems, i.e. a Burgers equation and a nonlinear ODE, by the operator splitting technique. The Burgers equation is solved by a linearly self-starting compact scheme which is fourth-order accurate in space and second-order accurate in time. The nonlinear ODE is discretized by a third-order semi-implicit Runge–Kutta method, which possesses good numerical stability with low computational cost. The numerical experiments show that the scheme provides the expected convergence order. Finally, several experiments are conducted to simulate the solutions of the Burgers–Huxley equation to validate our numerical method.     

Fast Parallel Algorithms for a Broad Class of Nonlinear Variational Diffusion Approaches

Variational segmentation and nonlinear diffusion approaches have been very active research areas in the fields of image processing and computer vision during recent years. In the present paper, we review recent advances in the development of efficient numerical algorithms for these approaches. The performance of parallel implementations of these algorithms on general-purpose hardware is assessed. A mathematically clear connection between variational models and nonlinear diffusion filters is presented that allows to interpret one approach as an approximation of the other, and vice versa. Extending this continuous connection to the fully discrete setting enables us to derive many structural similarities between efficient numerical algorithms for both frameworks. These results provide a perspective for uniform implementations of nonlinear variational models and diffusion filters on parallel architectures.     

Ensuring fault-tolerant computations in distributed control systems

This paper demonstrates that the fault tolerance of distributed control systems (DCSs) can be improved by scheduling of processes representing functional segments with guaranteed operation of the mechanisms of process reexecution and parallel execution based on checkpoints. Moreover, we suggest the methodological approach to assessing the fault tolerance level of DCSs, which proceeds from the probabilistic modeling of systems having the time triggered architecture (TTA). Finally, we derive numerical formulas for qualitative and quantitative estimation of the fault tolerance level for different modifications of DCSs at the design stage.     

A Para-Functional Programming Interface for a Parallel Computer Algebra Package

We describe the application of pD, a small para-functional language that we developed as a high-level programming interface for the parallel computer algebra package PACLIB. pD provides several facilities to express parallel algorithms in a flexible way on different levels of abstraction. The compiler translates a pD module into statically typed parallel C code with explicit task creation and synchronization constructs. This target code can be linked with the PACLIB kernel, the multi-processor runtime system of the computer algebra library SACLIB. The parallelization of several computer algebra algorithms on a shared memory multi-processor demonstrates the elegance and efficiency of this approach.     

Using the BSP cost model to optimise parallel neural network training

We derive cost formulae for three different parallelisation techniques for training both supervised and unsupervised networks. These formulae are parameterised by properties of the target computer architecture. It is therefore possible to decide both which technique is best for a given parallel computer, and which parallel computer best suits a given technique. One technique, exemplar parallelism, is far superior to almost all parallel computer architectures. Formulae also take into account optimal batch learning as the overall training approach. Cost predictions are made for several of today's popular parallel computers.     

Certifying spatially uniform behavior in reaction–diffusion PDE and compartmental ODE systems

We present a condition that guarantees spatial uniformity for the asymptotic behavior of the solutions of a reaction–diffusion PDE with Neumann boundary conditions. This condition makes use of the Jacobian matrix of the reaction terms and the second Neumann eigenvalue of the Laplacian operator on the given spatial domain, and eliminates the global Lipschitz assumptions commonly used in mathematical biology literature. We then derive numerical procedures that employ linear matrix inequalities to certify this condition, and illustrate these procedures on models of several biochemical reaction networks. Finally, we present an analog of this PDE result for the synchronization of a network of identical ODE models coupled by diffusion terms. From a systems biology perspective, the main contribution of the paper is to blend analytical and numerical tools from nonlinear systems and control theory to derive a relaxed and verifiable condition for spatial uniformity of biological processes.     

OLYMPUS for a BESM-6 — the development of medium-scale programs within the OLYMPUS framework: system OLYDES

The OLYMPUS methodology and system have been adopted as the precondition and framework for a software-development system OLYDES, which is being constructed to provide a unified methodological and technological approach to the numerical solution of small-to-medium-scale evolutionary problems. By a strong emphasis of the structural point of view, with a consequent separation of the physics and the numerics as the central designing principle, a considerable acceleration of ‘daily’ scientific computing is intended. Starting with some finite-difference methods for diffusion problems, this approach proved to allow concentration of the work on either the physical model or the numerical scheme, without any inconvenient and dangerous mixing of both aspects. The OLYDES project is introduced as a methodological adaptation of the ‘Olympian’ philosophy. The BESM-6 version of the OLYMPUS system is briefly reported, especially the new features resulting from replacement of the non-available NAMELIST facility by another, similarly powerful tool.     

Construction of a parallel computational model by composition of computational facilities

We consider a general algorithm for constructing a parallel numerical model by composition of several computational objects. The only requirement for the applicability of the proposed algorithm is for the disturbance propagation speed to be finite. We present one possible approach to map a numerical model onto an object-oriented program model. Some results of numerical experiments on solving the heat conduction equation in linear and nonlinear cases are discussed.     

Performance modeling of epidemic routing

In this paper, we develop a rigorous, unified framework based on ordinary differential equations (ODEs) to study epidemic routing and its variations. These ODEs can be derived as limits of Markovian models under a natural scaling as the number of nodes increases. While an analytical study of Markovian models is quite complex and numerical solution impractical for large networks, the corresponding ODE models yield closed-form expressions for several performance metrics of interest, and a numerical solution complexity that does not increase with the number of nodes. Using this ODE approach, we investigate how resources such as buffer space and the number of copies made for a packet can be traded for faster delivery, illustrating the differences among various forwarding and recovery schemes considered. We perform model validations through simulation studies. Finally we consider the effect of buffer management by complementing the forwarding models with Markovian and fluid buffer models.     

A Function-Composition Approach to Synthesize Fortran 90 Array Operations

An increasing number of programming languages, such as Fortran 90 and APL, are providing a rich set of intrinsic array functions and array expressions. These constructs which constitute an important part of data parallel languages provide excellent opportunities for compiler optimizations. In this paper, we present a new approach to combine consecutive array operations or array expressions into a composite access function of the source arrays. Our scheme is based on the composition of access functions, which is analogous to a composition of mathematic functions. Our new scheme can handle not only data movements of arrays with different numbers of dimensions and with multiple-clause array operations but also masked array expressions and multiple-source array operations. As a result, our proposed scheme is the first synthesis scheme which can collectively synthesize Fortran 90 RESHAPE, EOSHIFT, MERGE, array reduction operations, and WHERE constructs. In addition, we also discuss the case that the synthesis scheme may result in a performance anomaly in the presence of common subexpressions and one-to-many array operations. A solution is proposed to avoid such a performance anomaly. Experimental results show speedups from 1.21 to 2.95 over the base code for code fragments from real applications on a Sequent multiprocessor machine and also show comparable performance improvements on an 8-node SGI Power Challenge by incorporating our proposed optimizations     

Computation of travelling wave solutions of scalar conservation laws with a stiff source term

In this paper we propose a nonoscillatory numerical technique to compute the travelling wave solution of scalar conservation laws with a stiff source term. This procedure is based on the dynamical behavior described by the associated stationary ODE and it reduces/avoids numerical errors usually encountered with these problems, i.e., spurious oscillations and incorrect wave propagation speed. We combine this treatment with either the first order Lax-Friedrichs scheme or the second order Nessyahu-Tadmor scheme. We have tested several model problems by LeVeque and Yee for which the stiffness coefficient can be increased. We have also tested a problem with a nonlinear flux and a discontinuous source term.     

Solidification and control of a liquid metal column

In this paper, two different 1D mechanistic models for the solidification of a pure substance are presented. The first model is based on the two-domain approach, resulting in 2 partial differential equations (PDEs) and one ordinary differential equation (ODE) with 2 boundary conditions, 2 interface conditions, and one initial condition: the Stefan problem.In the second model, the metal column is considered as one-domain, and one PDE is valid for the whole domain. The result is one PDE with two boundary conditions.The models are implemented in MATLAB, and the ODE solver ode23s is used for solving the systems of equations. The models are developed in order to simulate and control the dynamic response of the solidification rate. The control scheme is based on a linear PI controller.     

A computationally efficient technique for transient analysis of repairable Markovian systems

A technique to design efficient methods using a combination of explicit (non-stiff) and implicit (stiff) ODE methods for numerical transient analysis of repairable Markovian systems is proposed. Repairable systems give rise to stiff Markov chains due to extreme disparity between failure rates and repair rates. Our approach is based on the observation that stiff Markov chains are non-stiff for an initial phase of the solution interval. A non-stiff ODE method is used to solve the model for this phase and a stiff ODE method is used to solve the model for the rest of the duration until the end of solution interval. A formal criterion to determine the length of the non-stiff phase is described. A significant outcome of this approach is that the accuracy requirement automatically becomes a part of model stiffness. Two specific methods based on this approach have been implemented. Both the methods use the Runge-Kutta-Fehlberg method as the non-stiff method. One uses the TR-BDF2 method as the stiff method while the other uses an implicit Runge-Kutta method as the stiff method. Numerical results obtained from solving dependability models of a multiprocessor system and an interconnection network are presented. These results show that the methods obtained using this approach are much more efficient than the corresponding stiff methods which have been proposed to solve stiff Markov models.     

HPC optimal parallel communication algorithm for the simulation of fractional-order systems

A parallel numerical simulation algorithm is presented for fractional-order systems involving Caputo-type derivatives, based on the Adams–Bashforth–Moulton predictor–corrector scheme. The parallel algorithm is implemented using several different approaches: a pure MPI version, a combination of MPI with OpenMP optimization and a memory saving speedup approach. All tests run on a BlueGene/P cluster, and comparative improvement results for the running time are provided. As an applied experiment, the solutions of a fractional-order version of a system describing a forced series LCR circuit are numerically computed, depicting cascades of period-doubling bifurcations which lead to the onset of chaotic behavior.

     

Kernel-Splitting Technique for Enclosing the Solution of Fredholm Equations of the First Kind

We derive a numerical method for solving linear Fredholm integral equations of the first kind. Based on series expansion techniques, the kernel of the corresponding integral equation is splitted into a finite rank degenerate part and an infinite dimensional, normwise small remainder. By enclosing the remainder term, the original problem, is transformed into a degenerate set-valued problem. For this problem, we derive a numerical method that provides a rigorous control of approximation and roundoff errors. We show that this approach provides a regularization scheme.     

JACK: an asynchronous communication kernel library for iterative algorithms

This article presents a new communication library developed to ease the implementation of both asynchronous and synchronous iterative methods. A mathematical and algorithmic framework about fixed-point methods is described to introduce this class of parallel iterative algorithms, although this library can be used for a larger class of parallel algorithms. After an overview of the main features, we describe detailed implementation aspects arising from the asynchronous context. While the library is mainly based on top of Message Passing Interface library, it has been designed to be easily extended to other types of communication middleware. Finally, some numerical experiments validate this new library, used for implementing both a classical parallel scheme and a sub-structuring approach of the Jacobi iterative method.     

Numerical solution of coupled nonlinear Schrödinger equation by Galerkin method

The coupled nonlinear Schrödinger equation models several interesting physical phenomena presents a model equation for optical fiber with linear birefringence. In this paper we derive a finite element scheme to solve this equation, we test this method for stability and accuracy, many numerical tests have been conducted. The scheme is quite accurate and describe the interaction picture clearly.     

Stability properties of a higher order scheme for a GKdV-4 equation modelling surface water waves

This work is devoted to the study of a higher order numerical scheme for the critical generalized Korteweg-de Vries equation (GKdV with p=4) in a bounded domain. The KdV equation and some of its generalizations as the GKdV type equations appear in Physics, for example in the study of waves on shallow water. Based on the analysis of stability of the first order scheme introduced by Pazoto et al. (Numer. Math. 116:317–356, 2010), we add a vanishing numerical viscosity term to a semi-discrete scheme so as to preserve similar properties of stability, and thus able to prove the convergence in L ⁴-strong. The semi-discretization of the spatial structure via central finite difference method yields a stiff system of ODE. Hence, for the temporal discretization, we resort to the two-stage implicit Runge-Kutta scheme of the Gauss-Legendre type. The resulting system is unconditionally stable and possesses favorable nonlinear properties. On the other hand, despite the formation of blow up for the critical case of GKdV, it is known that a localized damping term added to the GKdV-4 equation leads to the exponential decay of the energy for small enough initial conditions, which is interesting from the standpoint of the Control Theory. Then, combining the result of convergence in L ⁴-strong with discrete multipliers and a contradiction argument, we show that the presence of the vanishing numerical viscosity term allows the uniform (with respect to the mesh size) exponential decay of the total energy associated to the semi-discrete scheme of higher-order in space with the localized damping term. Numerical experiments are provided to illustrate the performance of the method and to confirm the theoretical results.     

Parallel Multigrid for Anisotropic Elliptic Equations

In this paper two well-known robust multigrid solvers for anisotropic operators on structured grids are compared: alternating-plane smoothers combined with full coarsening and plane smoothers combined with semi-coarsening. The study has taken into account not only numerical properties but also architectural ones, focusing on cache memory exploitation and parallel characteristics. Experimental results for the sequential algorithms have been obtained on two different systems based on the MIPS R10000 processor, but with different L2 cache sizes and memory bandwidths (an SGI O2 workstation and an SGI Origin 2000 system). Although the alternating-plane approach is the best choice for sequential implementations, experimental estimations show poor parallel efficiencies. For the semicoarsening alternative two different parallel implementations have been considered. The first one has optimal parallel characteristics but due to deterioration of the convergence properties its realistic efficiency is not satisfactory. In the second one, some processors remain idle during a short period of time on every multigrid cycle. However, the second parallel algorithm is more efficient since it preserves the numerical properties of the sequential version. Parallel experiments have also been taken on a Cray T3E system.     

PVaniM: a tool for visualization in network computing environments

Network computing has evolved into a popular and effective mode of high performance computing. Network computing environments have fundamental differences from hardware multiprocessors, involving a different approach to measuring and characterizing performance, monitoring an application's progress and understanding program behavior. In this paper, we present the design and implementation of PVaniM, an experimental visualization environment we have developed for the PVM network computing system. PVaniM supports a two-phase approach whereby on-line visualization focuses on large-grained events that are influenced by and relate to the dynamic network computing environment, and postmortem visualization provides for detailed program analysis and tuning. PVaniM's capabilities are illustrated via its use on several applications and a comparison with single-phase visualization environments developed for network computing. Our experiences indicate that, for several classes of applications, the two-phase visualization scheme can provide valuable insight into the behavior, efficiency and operation of distributed and parallel programs in network computing environments. © 1998 John Wiley & Sons, Ltd.