Performance considerations in the parallel execution of numerical algorithms on two processors

Summary Loosely coupled multiprocessor systems seem to offer an interesting alternative to the solution of large numerical problems. It is in the context of such an investigation that we treat here a special parallel-processing case concerning the solution of a numerical problem on two independent processors including simultaneous input-output operations. We present a short discussion of the underlying numerical algorithm, a modelling approach to the parallel computing system and a comparison of the theoretically obtained results with simulation and experimental results. The experimental setting is the XANTHOS multi-microprocessor system implemented at the Laboratoire de Recherche en Informatique, Université de Paris-Sud.This work was supported by a DGRST Research Scholarship at Université Paris-Sud     

On one type of finite element simulation in dynamic elasticity

The aim of the present paper is to generalize the finite element approximation for numerical simulation of special types of problems in the dynamic theory of elasticity, described by the two-dimensional mixed boundary value problems, and to demonstrate the property of their numerical solution. Physically the problems describe propagation of elastic waves, generated by a harmonic line source, in a nonhomogeneous anisotropic media. The existence and unicity of the weak solution as well as of the finite element approximation is proved. Convergence of the method is proved for any regular family of triangulations.     

Adams methods for the efficient solution of stochastic differential equations with additive noise

The application of Adams methods for the numerical solution of stochastic differential equations is considered. Especially we discuss the path-wise (strong) solutions of stochastic differential equations with additive noise and their numerical computation. The special structure of these problems suggests the application of Adams methods, which are used for deterministic differential equations very successfully. Applications to circuit simulation are presented.     

Algorithmus 31 Ein Algorithmus zur Bestimmung von Weganzahlen in Netzen

The current most economical solution to network problems is via Monte Carlo simulation. One of the main problems when using a computer is to answer the question of the storage space required. This depends roughly on the number of paths in the network. Here an algorithm will be presented to compute in a simple manner the number of paths in a network by constructing a sequence of special graphs.     

Mathematical models of elastic wave processes in seismology and seismic prospecting: forward and inverse problems

This paper presents a simulation system based on the solution of forward and inverse problems of elastic wave propagation. Forward and inverse modeling have become a useful tool for interpretation in exploration geophysics and seismology. By 3D modeling, field observations can be simulated numerically, and computed results can be compared to field data. It is known that 3D seismic modeling requires the up-to-date high-performance multiprocessor computer systems which are not readily available for many geophysical firms in Russia. This situation makes us focus on the creation of efficient numerical–analytical algorithms which allow the solution of 3D forward and inverse seismic problems without supercomputers. This paper presents some algorithms and numerical experiments for different models of media. A special emphasis is given to “non-ray” waves which play an important role in seismic interpretation theory.     

基于XML的数控仿真系统的NC代码解析

在直接通过NC代码来驱动仿真加工过程的数控仿真系统中,NC代码的解析处于系统的核心地位。然而各种NC代码之间的差异给系统的通用造成巨大的困难,系统各个模块的工作严格依赖于特定的NC代码格式,也使得系统难以扩展与变更。采用XML文档来作为用户提交的NC代码与系统之间的中间代码,很好地解决了以上问题,并获得了良好的应用效果,同时,对实现基于网络服务的数控仿真服务提供了可行性方案。     

Numerical verification of solutions for elasto-plastic torsion problems

In this paper, we consider a numerical technique which enables us to verify the existence of solutions for the elasto-plastic torsion problems governed by the variational inequality. Based upon the finite element approximations and the explicit a priori error estimates for a simple problem, we present an effective verification procedure that through numerical computation generates a set which includes the exact solution. This paper is an extension of the previous paper [1] in which we mainly dealt with the obstacle problems, but some special techniques are utilized to verify the solutions for nondifferentiable nonlinear equations concerned with the present problem. A numerical example is illustrated.     

Zur numerischen Behandlung gestörter Verzweigungsprobleme bei gewöhnlichen Differentialgleichungen

We present a method for the numerical solution of perturbed bifurcation problems of boundary-value problems for ordinary differential equations by standard techniques. This is achieved by reparametrisation by perturbation parameters and definition of appropriate boundary-value problems which have isolated solutions. The efficiency of the method is demonstrated by a numerical example —a special rod buckling problem.     

Normalization techniques for the SVE of the Green function of Helmholtz operator

Electromagnetic and acoustic scattering problems can be usually formulated by suitable integral equations, where the kernel is given in terms of the fundamental solution of the Helmholtz operator. We can consider a special analytic method for the singular value expansion (SVE) of this integral kernel. Note that this is an important tool for the numerical solution of scattering problems, in fact, from the knowledge of the SVE of the integral kernel, we can easily solve the corresponding integral equation. In this paper, we study the numerical approximation of the SVE of this integral kernel, where we have to consider the asymptotic behavior of the Bessel functions.     

大规模流体场景的真实感与实时模拟

目的 基于物理的流体动画模拟是计算机图形学领域中的研究热点,针对实际应用中仍难以实现大规模流体场景的真实感与实时模拟,提出了基于shallow water方程的物理模拟方法。方法 首先,给出shallow water方程的稳定欧拉数值求解方法,解决模拟过程中存在的毛刺、陡坡水滴斑点等数值求解的不稳定性问题;其次,提出刚体和粒子系统与流体高度场的稳定耦合模型,实现双向固流耦合和流体表面细节的真实感模拟;最后,设计高度场的多精度网格算法以及粒子的隔点采样方法,加速大规模流体的物理模拟计算。结果 实验结果表明,本文方法解决了传统欧拉方法求解shallow water方程的流体模拟过程中存在的不稳定和计算复杂等问题,在300×300网格分辨率和2.2×10⁴粒子数的规模下,达到了20帧/s的实时模拟速度。结论 本文算法具有良好的高效性和稳定性,适用于电子游戏和视景仿真等实时应用领域中的大规模流体场景的真实感模拟。     

Modeling physical systems using finite element Cell-DEVS

We propose a method to analyze complex physical systems using two-dimensional Cell-DEVS models. These problems are usually modeled with one or more Partial Differential Equations and solved using numerical methods. Our goal is to improve the definition of such problems by mapping them into the Cell-DEVS formalism, which permits easy integration with models defined with other formalisms, and its definition using advanced modeling and simulation tools. To show this, we used two methods for solving PDEs, and deduced the updating rules for their mapping to Cell-DEVS. As our simulation results show, the accuracy of the Cell-DEVS solution is the same of these previous methods, showing that we can use Cell-DEVS as a tool to obtain numerical solution for systems of PDEs. Simultaneously, this method provides us with a simpler mechanism for model definition, automated parallelism, and faster execution.     

Iterative solvers for Tikhonov regularization of dense inverse problems

According to the special demands arising from the development of science and technology, in the last decades appeared a special class of problems that are inverse to the classical direct ones. Such an inverse problem is concerned with the opposite way, usually followed by a direct one: finding the cause of a given effect or finding the law of evolution given the cause and effect. Very frequently, such inverse problems are modelled by Fredholm first-kind integral equations that give rise after discretization to (very) ill-conditioned linear systems, in classical or least squares formulation. Then, an efficient numerical solution can be obtained by using the Tikhonov regularization technique. In this respect, in the present paper, we propose three Kovarik-like algorithms for numerical solution of the regularized problem. We prove convergence for all three methods and present numerical experiments on a mathematical model of an inverse problem concerned with the determination of charge distribution generating a given electric field.     

Optimizing physical parameters in 1-D particle-in-cell simulations with Python

A particle-in-cell (PIC) simulation tool, OOPD1, is wrapped in the Python programming language, enabling automated algorithmic optimization of physical and numerical parameters. The Python-based environment exposes internal variables, enabling modification of simulation parameters, as well as run-time generation of new diagnostics based on calculations with internal data. For problems requiring an iterative optimization approach, this enables a programmable interactive feedback loop style simulation model, where the input to one simulation is a programmable function of the output of the previous one. This approach is applied to field-emission of electrons in a diode, in order to explore space charge effects in bipolar flow. We find an analytical solution for maximizing the space-charge limited current through a diode with an upstream ion current, and confirm the result with simulations, demonstrating the efficacy of the feedback scheme. We also demonstrate and analyze a modeling approach for scaling the ion mass, which can shorten simulation time without changing the ultimate result. The methods presented can be generalized to handle other applications where it is desirable to evolve simulation parameters based on algorithmic results from the simulation, including models in which physical or numerical parameter tuning is used to converge or optimize a system in one or more variables.     

An efficient class of direct search surrogate methods for solving expensive optimization problems with CPU-time-related functions

In this paper, we characterize a new class of computationally expensive optimization problems and introduce an approach for solving them. In this class of problems, objective function values may be directly related to the computational time required to obtain them, so that, as the optimal solution is approached, the computational time required to evaluate the objective is significantly less than at points farther away from the solution. This is motivated by an application in which each objective function evaluation requires both a numerical fluid dynamics simulation and an image registration process, and the goal is to find the parameter values of a predetermined reference image by comparing the flow dynamics from the numerical simulation and the reference image through the image comparison process. In designing an approach to numerically solve the more general class of problems in an efficient way, we make use of surrogates based on CPU times of previously evaluated points, rather than their function values, all within the search step framework of mesh adaptive direct search algorithms. Because of the expected positive correlation between function values and their CPU times, a time cutoff parameter is added to the objective function evaluation to allow its termination during the comparison process if the computational time exceeds a specified threshold. The approach was tested using the NOMADm and DACE MATLAB? software packages, and results are presented.     

Finite element approximation of nonlinear transient magnetic problems involving periodic potential drop excitations

This paper deals with the computation of nonlinear 2D transient magnetic fields when the data concerning the electric current sources involve potential drop excitations. In the first part, a mathematical model is stated, which is solved by an implicit time discretization scheme combined with a finite element method for space approximation. The second part focuses on developing a numerical method to compute periodic solutions by determining a suitable initial current which avoids large simulations to reach the steady state. This numerical method leads to solve a nonlinear system of equations which requires to approximate several nonlinear and linear magnetostatic problems. The proposed methods are first validated with an axisymmetric example and sinusoidal source having an analytical solution. Then, we show the saving in computational effort that this methodology offers to approximate practical problems specially with pulse-width modulation (PWM) voltage supply.     

On reachability and minimum cost optimal control

Questions of reachability for continuous and hybrid systems can be formulated as optimal control or game theory problems, whose solution can be characterized using variants of the Hamilton-Jacobi-Bellman or Isaacs partial differential equations. The formal link between the solution to the partial differential equation and the reachability problem is usually established in the framework of viscosity solutions. This paper establishes such a link between reachability, viability and invariance problems and viscosity solutions of a special form of the Hamilton-Jacobi equation. This equation is developed to address optimal control problems where the cost function is the minimum of a function of the state over a specified horizon. The main advantage of the proposed approach is that the properties of the value function (uniform continuity) and the form of the partial differential equation (standard Hamilton-Jacobi form, continuity of the Hamiltonian and simple boundary conditions) make the numerical solution of the problem much simpler than other approaches proposed in the literature. This fact is demonstrated by applying our approach to a reachability problem that arises in flight control and using numerical tools to compute the solution.     

A meshless multilayer model for a coastal system by radial basis functions

A multilayer computational model for simulating three-dimensional tidal flows in coastal waters is presented in this paper. A truly meshless numerical scheme based on radial basis functions (RBFs) is employed to obtain an accurate approximation to the solution of the model. For computational efficiency in solving large-scale problems, a noniterative domain decomposition method is combined with the use of the RBFs scheme. To smooth the numerical simulation of the advection across each subdomain, the commonly used upstream technique is also incorporated. Finally, for numerical verification, the proposed multilayer model is successfully used to obtain a stable and convergent simulation of the water flows in the Pearl River Estuary of the South China Sea. The numerical approximations exhibit good agreement with the observed tidal and current data.     

A multigrid optimization algorithm for the numerical solution of quasilinear variational inequalities involving the p-Laplacian

In this paper we propose a multigrid optimization algorithm (MG/OPT) for the numerical solution of a class of quasilinear variational inequalities of the second kind. This approach is enabled by the fact that the solution of the variational inequality is given by the minimizer of a nonsmooth energy functional, involving the $p$-Laplace operator. We propose a Huber regularization of the functional and a finite element discretization for the problem. Further, we analyze the regularity of the discretized energy functional, and we are able to prove that its Jacobian is slantly differentiable. This regularity property is useful to analyze the convergence of the MG/OPT algorithm. In fact, we demonstrate that the algorithm is globally convergent by using a mean value theorem for semismooth functions. Finally, we apply the MG/OPT algorithm to the numerical simulation of the viscoplastic flow of Bingham, Casson and Herschel–Bulkley fluids in a pipe. Several experiments are carried out to show the efficiency of the proposed algorithm when solving this kind of fluid mechanics problems.     

A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation

Many simulation algorithms (chemical reaction systems, differential systems arising from the modelling of transient behaviour in the process industries etc.) contain the numerical solution of systems of differential equations. For the efficient solution of the above mentioned problems, linear multistep methods or Runge-Kutta single-step methods are used. For the simulation of chemical procedures the radial Schrödinger equation is used frequently. In the present paper we will study a class of linear multistep methods. More specifically, the purpose of this paper is to develop an efficient algorithm for the approximate solution of the radial Schrödinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. Hence the main result of this paper is the development of an efficient multistep method for the numerical solution of systems of ordinary differential equations with oscillating or periodical solutions. The reason of their efficiency, as the analysis proved, is that the phase-lag and its derivatives are eliminated. Another reason of the efficiency of the new obtained methods is that they have high algebraic order     

Adaptive finite volume methods for displacement problems in porous media

In this paper we consider adaptive numerical simulation of miscible and immiscible displacement problems in porous media, which are modeled by single and two phase flow equations. Using the IMPES formulation of the two phase flow equation both problems can be treated in the same numerical framework. We discretise the equations by an operator splitting technique where the flow equation is approximated by Raviart-Thomas mixed finite elements and the saturation or concentration equation by vertex centered finite volume methods. Using a posteriori error estimates for both approximation schemes we deduce an adaptive solution algorithm for the system of equations and show the applicability in several examples.