Cubical multiprocessor architecture for solving partial differential equations in three dimensions

A cubical architecture is proposed for multi-microprocessor systems for solving partial differential equations in three dimensions, for possible use in weather forecasting and other similar applications. The architecture makes use of hundreds of processing elements to increase throughput. The system can achieve high performance and does not suffer greatly from interproccssor propagation delay.     

On solving elliptic stochastic partial differential equations

A model elliptic boundary value problem of second order, with stochastic coefficients described by the Karhunen–Loève expansion is addressed. This problem is transformed into an equivalent deterministic one. The perturbation method and the method of successive approximations is analyzed. Rigorous error estimates in the framework of Sobolev spaces are given.     

Finite-element neural networks for solving differential equations

The solution of partial differential equations (PDE) arises in a wide variety of engineering problems. Solutions to most practical problems use numerical analysis techniques such as finite-element or finite-difference methods. The drawbacks of these approaches include computational costs associated with the modeling of complex geometries. This paper proposes a finite-element neural network (FENN) obtained by embedding a finite-element model in a neural network architecture that enables fast and accurate solution of the forward problem. Results of applying the FENN to several simple electromagnetic forward and inverse problems are presented. Initial results indicate that the FENN performance as a forward model is comparable to that of the conventional finite-element method (FEM). The FENN can also be used in an iterative approach to solve inverse problems associated with the PDE. Results showing the ability of the FENN to solve the inverse problem given the measured signal are also presented. The parallel nature of the FENN also makes it an attractive solution for parallel implementation in hardware and software.     

Moving mesh methods for solving parabolic partial differential equations

A new adaptive method is described for solving nonlinear parabolic partial differential equations with moving boundaries, using a moving mesh with continuous finite elements. The evolution of the mesh within the interior of the spatial domain is based upon conserving the distribution of a chosen monitor function across the domain throughout time, where the initial distribution is selected based upon the given initial data. The mesh movement at the boundary is governed by a second monitor function, which may or may not be the same as that used to drive the interior mesh movement. The method is described in detail and a selection of computational examples are presented using different monitor functions applied to the porous medium equation (PME) in one and two space dimensions.     

A multiprocessor architecture for solving nonlinear partial differential equations

This paper presents the architecture of a special-purpose multiprocessor system, which we call the Broadcast Cube System (BCS), for solving non-linear Partial Differential Equations (PDEs). The BCS has the following important features: (a) Being based on the conceptually simple bus interconnection scheme it is easily understood. The use of homogeneous Processing Elements (PEs) which can be realized as standard VLSI chips makes the hardware less costly. (b) The interconnection network is simple and regular. The network can easily be extended to vast number of PEs by adding buses with new PEs on them and by slightly increasing the number of PEs on existing buses. The interconnection pattern is highly redundant to support fault tolerance in the event of PE failures. (c) In terms of the switching delays, the delay a message undergoes between a pair of PEs connected to a common bus is zero. The maximum delay between any pair of PEs is one unit and thus a strong localization of communicating tasks is not needed to avoid long message delays even in networks of thousands of PEs. The effectiveness of the BCS has been demonstrated by both analytical and simulation methods using heat transfer and fluid flow simulation, which requires solution of non-linear PDEs, as a benchmark program.     

A comparison of the method of lines to finite difference techniques in solving time-dependent partial differential equations

Steady state solutions to two time dependent partial differential systems have been obtained by the Method of Lines (MOL) and compared to those obtained by efficient standard finite difference methods: (i) Burger's equation over a finite space domain by a forward time—central space explicit method, and (ii) the stream function—vorticity form of viscous incompressible fluid flow in a square cavity by an alternating direction implicit (ADI) method. The standard techniques were far more computationally efficient when applicable. In the second example, converged solutions at very high Reynolds numbers were obtained by MOL, whereas solution by ADI was either unattainable or impractical. With regard to “set up” time, solution by MOL is an attractive alternative to techniques with complicated algorithms, as much of the programming difficulty is eliminated.     

The neural network collocation method for solving partial differential equations

Neural Computing and Applications - This paper presents a meshfree collocation method that uses deep learning to determine the basis functions as well as their corresponding weights. This method is...     

Explicit group over-relaxation methods for solving elliptic partial differential equations

Previous block (or line) iterative methods have been implicit in nature where a group of equations (or points on the grid mesh) are treated implicitly [2] and solved directly by a specialised algorithm, this has become the standard technique for solving the sparse linear systems derived from the discretisation of self-adjoint elliptic partial differential equations by finite difference/element techniques.The aim of this paper is to show that if a small group of points (i.e. 2, 4, 9, 16 or 25 point group) is chosen then each group can easily be initially inverted leading to a new class of Group Explicit iterative methods. A comparison with the usual 1-line and 2-line block S.O.R. schemes for the model problem confirm the new techniques to be computationally superior.     

Neural network approach to intricate problems solving for ordinary differential equations

We consider the problems arising in the construction of the solutions of singularly perturbed differential equations. Usually, the decision of such problems by standard methods encounters significant difficulties of various kinds. The use of a common neural network approach is demonstrated in three model problems for ordinary differential equations. The conducted computational experiments confirm the effectiveness of this approach.     

The role of the multiquadric shape parameters in solving elliptic partial differential equations

This study examines the generalized multiquadrics (MQ), φ_j(x) = [(x−x_j)²+c_j²]^β in the numerical solutions of elliptic two-dimensional partial differential equations (PDEs) with Dirichlet boundary conditions. The exponent β as well as c_j² can be classified as shape parameters since these affect the shape of the MQ basis function. We examined variations of β as well as c_j² where c_j² can be different over the interior and on the boundary. The results show that increasing ,β has the most important effect on convergence, followed next by distinct sets of (c_j²)Ω∂Ω ≪ (c_j²)∂Ω. Additional convergence accelerations were obtained by permitting both (c_j²)Ω∂Ω and (c_j²)∂Ω to oscillate about its mean value with amplitude of approximately 1/2 for odd and even values of the indices. Our results show high orders of accuracy as the number of data centers increases with some simple heuristics.     

Mixed discontinuous Legendre wavelet Galerkin method for solving elliptic partial differential equations

By incorporating the Legendre multiwavelet into the mixed discontinuous Galerkin method, in this paper, we present a novel method for solving second-order elliptic partial differential equations (PDEs), which is known as the mixed discontinuous Legendre multiwavelet Galerkin method, derive an adaptive algorithm for the method and estimate the approximating error of its numerical fluxes. One striking advantage of our method is that the differential operator, boundary conditions and numerical fluxes involved in the elementwise computation can be done with lower time cost. Numerical experiments demonstrate the validity of this method. The proposed method is also applicable to some other kinds of PDEs.     

An environment to develop parallel code for solving partial differential equations based-problems

Since the beginning of computing, the use of computers to simulate physical phenomena has been a driving force to advance the field of computing. Computational scientists demand more computational power and more facility to implement applications. The conventional library interface does not hide the complexity of the underlying parallel architectures and their programming paradigms from users. Our investigation group is currently implementing an environment for solving partial differential equations (PDEs). Using this high-level tool the user can easily develop parallel applications for solving PDEs based problems using multigrid techniques without knowledge on the underlying computer hardware or software. This paper provides the first steps towards the creation of the environment.     

Towards developing robust algorithms for solving partial differential equations on MIMD machines

Methods are proposed for efficient computation of numerical algorithms on a wide variety of MIMD machines. These techniques reorganize the data dependency patterns so that the processor utilization is imporved.The model problem examined finds the time-accurate solution to a parabolic partial differential equation discretized in space and implicitly marched forward in time. The algorithms investigated are extensions of Jacobi and SOR. The extensions consist of iterating over a window of several timesteps, allowing efficient overlap of computation with communication.The methods suggested here increase the degree to which work can be performed while data are communicated between processors. The effect of the window size and of domain partitioning on the system performance is examined both analytically and experimentally by implementing the algorithm on a simulated multiprocessor system.     

Optimization free neural network approach for solving ordinary and partial differential equations

Current work introduces a fast converging neural network-based approach for solution of ordinary and partial differential equations. Proposed technique eliminates the need of time-consuming optimization procedure for training of neural network. Rather, it uses the extreme learning machine algorithm for calculating the neural network parameters so as to make it satisfy the differential equation and associated boundary conditions. Various ordinary and partial differential equations are treated using this technique, and accuracy and convergence aspects of the procedure are discussed.

     

A new method for solving boundary value problems for partial differential equations

This paper proposes a symmetry–iteration hybrid algorithm for solving boundary value problems for partial differential equations. First, the multi-parameter symmetry is used to reduce the problem studied to a simpler initial value problem for ordinary differential equations. Then the variational iteration method is employed to obtain its solution. The results reveal that the proposed method is very effective and can be applied for other nonlinear problems.     

A new homotopy perturbation method for solving systems of partial differential equations

In this paper, a new homotopy perturbation method (NHPM) is introduced for obtaining solutions of systems of non-linear partial differential equations. Theoretical considerations are discussed. To illustrate the capability and reliability of the method three examples are provided. Comparison of the results of applying NHPM with those of applying HPM reveal the effectiveness and convenience of the new technique.