An analytical method for solving linear Fredholm fuzzy integral equations of the second kind

In this paper, we use the parametric form of a fuzzy number and convert a linear fuzzy Fredholm integral equation to two linear systems of integral equations of the second kind in the crisp case. For fuzzy Fredholm integral equations with kernels, the sign of which is difficult to determine, a new parametric form of the fuzzy Fredholm integral equation is introduced. We use the homotopy analysis method to find the approximate solution of the system, and hence, obtain an approximation for fuzzy solutions of the linear fuzzy Fredholm integral equation of the second kind. The proposed method is illustrated by solving some examples. Using the HAM, it is possible to find the exact solution or the approximate solution of the problem in the form of a series.     

A class of iterative methods for finite element equations

A general method in the form of an accelerated preconditioned iterative refinement method (including some wellknown iterative methods and direct factorization methods) is presented for the solution of symmetric, sparse matrix problems. An analysis of one such approximate factorization, the SSOR method, is given, and some inherently advantageous properties of the conjugate gradient acceleration method are pointed out. A comparison is made of the computational complexity and storage in the SSOR preconditioned method with some direct methods applied to second order discretized boundary value problems. For plane problems of average size the direct methods are somewhat faster if enough right hand sides are present. For large enough problems (large number of nodes) the iterative method is faster. For three-dimensional problems no Cholesky factorization method can compete with the SSOR preconditioned method, not even for average sized problems.     

A Numerical Method to Solve Identification Problem for the Lower Coefficient and the Source in the Convection–Reaction Equation

We consider the problem of identifying simultaneously the kinetic reaction coefficient and source function depending only on a spatial variable in one-dimensional linear convection–reaction equation. As additional conditions, a non-local integral condition for the solution of the equation and condition of final overdetermination are given. This problem belongs to the class of combined inverse problems. By integrating the equation with the use of additional integral condition, the problem is transformed to a coefficient inverse problem with local conditions. The derivative with respect to the spatial variable is discretized and a special representation is proposed to solve the resultant semi-discrete problem. As a result, for each discrete value of the spatial variable, the semi-discrete problem splits into two parts: a Cauchy problem and a linear equation with respect to the approximate value of the unknown kinetic coefficient. To determine the source function, an explicit formula is also obtained. The numerical solution of the Cauchy problem uses the implicit Euler method. Numerical experiments are carried out on the basis of the proposed method.     

Coupling of optimal variation of parameters method with Adomian’s polynomials for nonlinear equations representing fluid flow in different geometries

In this study, we describe a modified analytical algorithm for the resolution of nonlinear differential equations by the variation of parameters method (VPM). Our approach, including auxiliary parameter and auxiliary linear differential operator, provides a computational advantage for the convergence of approximate solutions for nonlinear boundary value problems. We consume all of the boundary conditions to establish an integral equation before constructing an iterative algorithm to compute the solution components for an approximate solution. Thus, we establish a modified iterative algorithm for computing successive solution components that does not contain undetermined coefficients, whereas most previous iterative algorithm does incorporate undetermined coefficients. The present algorithm also avoid to compute the multiple roots of nonlinear algebraic equations for undetermined coefficients, whereas VPM required to complete calculation of solution by computing roots of undetermined coefficients. Furthermore, a simple way is considered for obtaining an optimal value of an auxiliary parameter via minimizing the residual error over the domain of problem. Graphical and numerical results reconfirm the accuracy and efficiency of developed algorithm.

     

A Matrix-free Two-grid Preconditioner for Solving Boundary Integral Equations in Electromagnetism

In this paper, we describe a matrix-free iterative algorithm based on the GMRES method for solving electromagnetic scattering problems expressed in an integral formulation. Integral methods are an interesting alternative to differential equation solvers for this problem class since they do not require absorbing boundary conditions and they mesh only the surface of the radiating object giving rise to dense and smaller linear systems of equations. However, in realistic applications the discretized systems can be very large and for some integral formulations, like the popular Electric Field Integral Equation, they become ill-conditioned when the frequency increases. This means that iterative Krylov solvers have to be combined with fast methods for the matrix-vector products and robust preconditioning to be affordable in terms of CPU time. In this work we describe a matrix-free two-grid preconditioner for the GMRES solver combined with the Fast Multipole Method. The preconditioner is an algebraic two-grid cycle built on top of a sparse approximate inverse that is used as smoother, while the grid transfer operators are defined using spectral information of the preconditioned matrix. Experiments on a set of linear systems arising from real radar cross section calculation in industry illustrate the potential of the proposed approach for solving large-scale problems in electromagnetism.     

The solution of the poisson-boltzmann equation between two spheres: Modified iterative method

We present a modification on the successive overrelaxation (SOR) method and the iteration of the Green's function integral representation for the solution of the (nonlinear) Poisson-Boltzmann equation between two spheres. In comparison with other attempts, which approximate the geometry or the nonlinearity, the computations here are done for the full problem and compared with those done by the finite element method as a typical method for such problems. For the parameters of general interest, while the SOR method does not work, and the iteration of the integral representation is limited in its convergence, our modification to these iterative schemes converge. The modified SOR surpasses both methods in simplicity and speed; it is about 100 times faster than the modified iteration of the integral representation, with the latter being still simpler and faster than the finite element method. These two examples further illustrate the advantage of our recent modification to iterative methods, which is based on an analytical fixed point argument.     

Legendre–Galerkin spectral approximation for a nonlocal elliptic Kirchhof-type problem

We propose a numerical scheme to obtain an approximate solution of a nonlocal elliptic Kirchhof-type problem. We first reduce the problem to a nonlinear finite dimensional system by a Legendre–Galerkin spectral method and then solve it by an iterative process. Convergence of the iterative process and an error estimation of the approximate solution is provided. Numerical experiments are conducted to illustrate the performance of the proposed method.     

Projections and preconditioning for inconsistent least-squares problems

In this paper we describe two iterative algorithms for the numerical solution of linear least-squares problems. They are based on a combination between an extension of the classical Kaczmarzs projections method (Popa [5]) and an approximate orthogonalization technique due to Kovarik. We prove that both new algorithms converge to any solution of an inconsistent and rank-defficient least-squares problem (with respect to the choice of the initial approxi-mation), the convergence being much faster than for the classical Kaczmarz - like methods. Some numerical experiments on a first kind integral equation are described.     

A boundary integral method for the simulation of two-dimensional particle coarsening

A boundary integral method for the solution of a time-dependent free-boundary problem in a two-dimensional, multiply-connected, exterior domain is described. The method is based on an iterative solution of the resulting integral equations at each time step, with the initial guesses provided by extrapolation from previous time steps. The method is related to a technique discussed by Baker for the study of water waves. The discretization is chosen so that the solvability conditions required for the exterior Dirichlet problem do not degrade the convergence rate of the iterative solution procedure. Consideration is given to the question of vectorizing the computation. The method is applied to the problem of the coarsening of two-dimensional particles by volume diffusion.     

A finite element method for thermoviscoelastic analysis of plane problems

A stress analysis for plane problems in linear thermoviscoelasticity using a finite element formulation is presented. The method employed is based on the assumptions that (1) the material is isotropic, homogeneous and linear, (2) the stress-strain laws are expressed in the hereditary integral form, and (3) the material is thermorheologically simple, which implies that the temperature-time equivalence hypothesis is valid. The associated computer program utilizes isoparametric plane elements.The element matrices that result in the equilibrium equations involve hereditary integrals, and these are approximated by a finite difference scheme for time marching. The solutions for two problems are compared with analytical results evaluated by the integral transform method.For approximate results which require less computer time an alternative form of equilibrium equations utilizing an iterative technique is presented and an example solution is included. Finally, the effect of incompressibility is considered for an axisymmetric numerical example.     

Difference approximation of optimal periodic control problems

The constrained optimal periodic control problem is approximated by a sequence of discretized problems in which the system of differential equations of the basic continuous problem is replaced by a system of one–step difference equations. Two kinds of approximate optimal controls are derived from the optimal solutions of discretized problems: the first in the form of a step function and the second in the form of a special trigonometric polynomial generated by a positive kernel. Sufficient conditions for approximate solutions to be weakly convergent to the optimal solution of the basic problem are given. Certain improvements in the difference approximation considered are discussed and potential applications given.     

Even- and odd-parity kinetic equations of particle transport. 1: Algebraic and centered forms of the scattering integral

We consider an equivalent formulation of the linear kinetic transport equation for neutral particles (neutrons, photons) as a system of two equations for even and odd parts of the distribution function. The particle scattering integral of even- and odd-parity transport equations is converted into a non-linear algebraic form and into a centered form. In the algebraic form of the integral we clearly identify the net result of two opposite processes, i.e., particle scattering from a beam and into the beam. In the centered form of the integral the principal terms of scattering processes are canceled out. An iterative method is proposed for the solution of the system of even- and odd-parity equations with these forms of the scattering integral. Convergence of iterations is studied for a one-dimensional plane problem.     

Numerical solution and spectrum of boundary-domain integral equation for the Neumann BVP with a variable coefficient

In this paper, a numerical implementation of a direct united boundary-domain integral equation (BDIE) related to the Neumann boundary value problem for a scalar elliptic partial differential equation with a variable coefficient is discussed. The BDIE is reduced to a uniquely solvable one by adding an appropriate perturbation operator. The mesh-based discretization of the BDIEs with quadrilateral domain elements leads to a system of linear algebraic equations (discretized BDIE). Then, the system is solved by LU decomposition and Neumann iterations. Convergence of the iterative method is discussed in relation to the distribution of eigenvalues of the corresponding discrete operators calculated numerically.     

Convergence Acceleration for Hyperbolic Systems Using Semicirculant Approximations

The iterative solution of systems of equations arising from systems of hyperbolic, time-independent partial differential equations (PDEs) is studied. The PDEs are discretized using a finite volume or finite difference approximation on a structured grid. A convergence acceleration technique where a semicirculant approximation of the spatial difference operator is employed as preconditioner is considered. The spectrum of the preconditioned coefficient matrix is analyzed for a model problem. It is shown that, asymptotically, the time step for the forward Euler method could be chosen as a constant, which is independent of the number of grid points and the artificial viscosity parameter. By linearizing the Euler equations around an approximate solution, a system of linear PDEs with variable coefficients is formed. When utilizing the semicirculant (SC) preconditioner for this problem, which has properties very similar to the full nonlinear equations, numerical experiments show that the favorable convergence properties hold also here. We compare the results for the SC method to those of a multigrid (MG) scheme. The number of iterations and the arithmetic complexities are considered, and it is clear that the SC method is more efficient for the problems studied. Also, the MG scheme is sensitive to the amount of artificial dissipation added, while the SC method is not.     

A new a posteriori error estimator in adaptive direct boundary element methods: the Dirichlet problem

In this paper we propose a new a posteriori error estimator for a boundary element solution related to a Dirichlet problem with a second order elliptic partial differential operator. The method is based on an approximate solution of a boundary integral equation of the second kind by a Neumann series to estimate the error of a previously computed boundary element solution. For this one may use an arbitrary boundary element method, for example, a Galerkin, collocation or qualocation scheme, to solve an appropriate boundary integral equation. Due to the approximate solution of the error equation the proposed estimator provides high accuracy. A numerical example supports the theoretical results. Received: June 1999 / Accepted: September 1999     

An efficient sensor quantization algorithm for decentralized estimation fusion

In this paper, we consider the design problem of optimal sensor quantization rules (quantizers) and an optimal linear estimation fusion rule in bandwidth-constrained decentralized random signal estimation fusion systems. First, we derive a fixed-point-type necessary condition for both optimal sensor quantization rules and an optimal linear estimation fusion rule: a fixed point of an integral operation. Then, we can motivate an iterative Gauss–Seidel algorithm to simultaneously search for both optimal sensor quantization rules and an optimal linear estimation fusion rule without Gaussian assumptions on the joint probability density function (pdf) of the estimated parameter and observations. Moreover, we prove that the algorithm converges to a person-by-person optimal solution in the discretized scheme after a finite number of iterations. It is worth noting that the new method can be applied to vector quantization without any modification. Finally, several numerical examples demonstrate the efficiency of our method, and provide some reasonable and meaningful observations how the estimation performance is influenced by the observation noise power and numbers of sensors or quantization levels.     

On the regularized Lagrange principle in iterative form and its application for solving unstable problems

For a convex programming problem in a Hilbert space with operator equality constraints, the Lagrange principle in sequential nondifferential form or, in other words, the regularized Lagrange principle in iterative form, that is resistant to input data errors is proved. The possibility of its applicability for direct solving unstable inverse problems is discussed. As an example of such problem, we consider the problem of finding the normal solution of the Fredholm integral equation of the first kind. The results of the numerical calculations are shown.     

迭代粒子群算法及其在间歇过程鲁棒优化中的应用

针对无状态独立约束和终端约束的间歇过程鲁棒优化问题，将迭代方法与粒子群优化算法相结合，提出了迭代粒子群算法．对于该算法，首先将控制变量离散化，用标准粒子群优化算法搜索离散控制变量的最优解．然后在随后的迭代过程中将基准移到刚解得的最优值处，同时收缩控制变量的搜索域，使优化性能指标和控制轨线在迭代过程中不断趋于最优解．算法简洁、可行、高效，避免了求解大规模微分方程组的问题．对一个间歇过程的仿真结果证明了迭代粒子群算法可以有效地解决无状态独立约束和终端约束的间歇过程鲁棒优化问题．     

Trigonometric approximation of optimal periodic control problems for distributed-parameter systems

The optimal periodic control problem for a system described by first order partial differential equations is approximated by a sequence of discretized optimization problems. Trigonometric polynomials in two variables are used in the latter problems to approximate the state trajectory, the control and functions appearing in differential equations and in the criterion of the basic problem. The state equations and the instantaneous constraints on the state and the control are taken into account by the mixed exterior-interior penalty function. Sufficient conditions are given for the convergence of solutions of discretized problems to the optimal solution of the basic problem. The possibility of applying the method to a class of optimal periodic control problems in chemical engineering is emphasized.     

压缩感知优化问题的等价表示及其目标罚函数方法

首先定义了压缩感知优化问题的一个等价表示问题,证明了这个等价表示问题的最优解也是压缩感知优化问题的最优解。然后定义了它的一个具有2阶以上的光滑性的目标罚函数,给出了一个迭代求解算法,证明了所提算法的收敛性定理。定理表明,可以通过求解目标罚函数来获得压缩感知优化问题的近似最优解,该方法为研究和解决实际的压缩感知问题提供了一个新的工具。