Numerical solution of non-linear elliptic boundary-value problems by isomorphic iterative methods

New implicit iterative methods are presented for the efficient numerical solution of non-linear elliptic boundary-value problems. Isomorphic iterative schemes in conjunction with preconditioning techniques are used for solving non-linear elliptic equations in two and three-space dimensions. The application of the derived methods on characteristic 2D and 3D non-linear boundary-value problems is discussed and numerical results are given.     

Solution of inverse boundary-value problems for multicomponent parabolic distributed systems

Computational algorithms implementing gradient methods based on solution of direct and adjoint problems in weak formulations are proposed for a number of complex-valued inverse problems of parameter renewal in multicomponent parabolic distributed systems. This approach makes it unnecessary to construct Lagrangian functionals explicitly and to use Green’s functions. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 49–73, July–August 2007.     

Normalized explicit finite element approximate inverse preconditioning

Normalized explicit approximate inverse matrix techniques for computing explicitly various families of normalized approximate inverses based on normalized approximate factorization procedures for solving sparse linear systems, which are derived from the finite difference and finite element discretization of partial differential equations are presented. Normalized explicit preconditioned conjugate gradient-type schemes in conjunction with normalized approximate inverse matrix techniques are presented for the efficient solution of linear and non-linear systems. Theoretical estimates on the rate of convergence and computational complexity of the normalized explicit preconditioned conjugate gradient method are also presented. Applications of the proposed methods on characteristic linear and non-linear problems are discussed and numerical results are given.     

Solving combined inverse problems for multicomponent parabolic distributed systems

The paper outlines a technique to construct computational algorithms for solving combined inverse problems for multicomponent parabolic systems with main and natural inhomogeneous interface conditions. Frechet derivatives are obtained in explicit form for quadratic residual functionals to construct gradient computational algorithms. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 48–71, September–October 2007.     

Solving inverse eigenvalue problems by a projected Newton method

The inverse eigenvalue problem for symmetric matrices (IEP) can be formulated as a system of two matrix equations. For solving the system a variation of Newton's method is used which has been proposed by Fusco and Zecca [Calcolo XXIII (1986), pp. 285–303] for the simultaneous computation of eigenvalues and eigenvectors of a given symmetric matrix. An iteration step of this method consists of a Newton step followed by an orthonormalization with the consequence that each iterate satisfies one of the given equations. The method is proved to convergence locally quadratically to regular solutions. The algorithm and some numerical examples are presented. In addition, it is shown that the so-called Method III proposed by Friedland, Nocedal, and Overton [SIAM J. Numer. Anal., 24 (1987), pp. 634–667] for solving IEP may be constructed similarly to the method presented here.     

The direct solution of periodic parabolic problems by boundary-value techniques

In this paper fast direct methods are presented for the solution of the parabolic boundary value problem. The algorithms used are block variants of the cyclic factorisation algorithm previously presented in Evans [1]     

Numerical solution of fractional differential equations using the generalized block pulse operational matrix

The Riemann-Liouville fractional integral for repeated fractional integration is expanded in block pulse functions to yield the block pulse operational matrices for the fractional order integration. Also, the generalized block pulse operational matrices of differentiation are derived. Based on the above results we propose a way to solve the fractional differential equations. The method is computationally attractive and applications are demonstrated through illustrative examples.     

Solution of integrated inverse problems for multicomponent hyperbolic distributed systems

The paper presents a technique for developing computational algorithms to solve inverse problems for multicomponent hyperbolic systems with principal and natural heterogeneous conjugation conditions. Explicit expressions for the Frechet derivatives of quadratic residual functionals are obtained to generate gradient computational algorithms. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 55–80, March–April 2008.     

A new solution to the generalized Sylvester matrix equation

This note deals with the problem of solving the generalized Sylvester matrix equation AV-EVF=BW, with F being an arbitrary matrix, and provides complete general parametric expressions for the matrices V and W satisfying this equation. The primary feature of this solution is that the matrix F does not need to be in any canonical form, and may be even unknown a priori. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in control systems theory.     

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.     

On the Solution of Boundary Value Problems by Using Fast Generalized Approximate Inverse Banded Matrix Techniques

A class of finite difference schemes in conjunction with approximate inverse banded matrix techniques based on the concept of LU-type factorization procedures is introduced for computing fast explicit approximate inverses. Explicit preconditioned iterative schemes in conjunction with approximate inverse matrix techniques are presented for the efficient solution of banded linear systems. A theorem on the rate of convergence and estimates of the computational complexity required to reduce the L-norm of the error is presented. Applications of the method on linear and non-linear systems are discussed and numerical results are given.     

A class of inverse problems for discontinuous systems

We analyze a class of inverse parametric problems for dynamic processes described by systems of ordinary differential equations whose form and piecewise-constant parameters depend on what subdomain in the state space the state of the process belongs to. The study was supported by the INTAS (Project No. 06-1000017-8909). Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 142–152, November–December 2008.     

Inversion of the generalized Fibonacci matrix by convolution

A convolution formula containing the generalized Fibonacci numbers and applications of this formula are investigated. Starting from the convolution formula, we derive combinatorial identities involving generalized and usual Fibonacci numbers, as well as the Lucas numbers. The inversion of a lower triangular matrix and the generalized inversion of strictly lower triangular Toeplitz matrix whose non-zero elements are generalized Fibonacci numbers are considered.     

On the solution of a nonlinear matrix equation for MIMO symmetric realizations

A result concerning a particular solution of a nonlinear solution matrix equation reported in a previous paper [2] and interesting for studying SISO systems, is extended to MIMO symmetric realizations. For such a class of systems other important SISO results can be generalized.     

Average consensus problems in networks of agents with delayed communications

The present paper is devoted to the study of average consensus problems for undirected networks of dynamic agents having communication delays. By focusing on agents with integrator dynamics, the accent is put here on the study of the time-delay influence: both constant and time-varying delays are considered, as well as uniform and non-uniform repartitions of the delays in the network. The main results provide sufficient conditions (also necessary in most cases) for the existence of average consensus under bounded communication delays. Simulations are provided that show adequation with these results.     

A class of explicit preconditioned conjugate gradient methods for solving large finite element systems

A class of Explicit Preconditioned Conjugate Gradient (EPCG) methods for solving large sparse linear systems of algebraic equations resulting from the Finite Element discretization of Elliptic and Parabolic PDE's is introduced. The EPCG methods are based on explicit Approximate Inverse Matrix techniques and are particularly suitable for solving numerically initial/boundary-value problems on multiprocessor systems. The application of the new methods on 2D-linear boundary-value problems is discussed and numerical results are given.     

Double ultraspherical solution of eigenvalue problems for partial differential equations

Particular spectral method based on an expansion in double ultraspherical polynomials is proposed to solve a number of eigenvalue problems defined by partial differential equations with constant and variable coefficients.We obtain comparable results with those computed by Liu and Oritz [14] by using Tau-Lines method, El-Hawary [8] by using El-Gendi-Lines method, El-Hawary [7] by using double Chebyshev approximation and Brandt et al.[1] by using multigrid method.     

FGMRES preconditioning by symmetric/skew-symmetric decomposition of generalized stokes problems

The generalized Stokes problem is solved for non-standard boundary conditions. This problem arises after time semi-discretization by ALE method of the Navier–Stokes system, which describes the flow of two immiscible fluids with similar densities but different viscosities in a horizontal pipe, when modeling heavy crude oil transportation. We discretized the generalized Stokes problem in space using the “Mini” finite element. The inf-sup condition is proved when the interface between the two fluids and its discretization match exactly. The linear system obtained after discretization is solved using different iterative Krylov methods with and without preconditioning. Numerical experiments with different meshes are presented as well as comparisons between the methods considered. The results suggest that FGMRES and a preconditioning technique based on symmetric/skew-symmetric decomposition is a promising candidate for solving large scale generalized Stokes problem.     

Numerical solution of a one-dimensional inverse problem by the discontinuous Galerkin method

In this paper, we combine a Tikhonov regularization with a discontinuous Galerkin method to solve an inverse problem in one-dimension. We show that the regularization is simpler than in the case of the inversion using continuous finite elements. We numerically demonstrate that there exist optimal step sizes and polynomial degrees for inversion using the DG method. Numerical results are compared with those obtained by applying the standard finite element method with B-splines as a basis.