Convergence proofs for the multi-grid iteration are known for the case of finite element equations and for the case of some difference schemes discretizing boundary value problems in a rectangular region. In the present paper we give criteria of convergence that apply to general difference schemes for boundary value problems in Lipschitzian regions. Furthermore, convergence is proved for the mu...

A batch training algorithm for feed-forward networks is proposed which uses Newton’s method to estimate a vector of optimal learning factors, one for each hidden unit. Backpropagation, using this learning factor vector, is used to modify the hidden unit’s input weights. Linear equations are then solved for the network’s output weights. Elements of the new method’s Gauss-Newton Hessian matrix ar...

The convergence rate of a multigrid method for the numerical solution of the Poisson equation on a uniform grid is estimated. The results are independent of the shape of the domain as long as it is convex and polygonal. On the other hand, pollution effects become apparent when the domain contains reentrant corners. To estimate the smoothing of the Gauss-Seidel relaxation, the smoothness is meas...

We present a systematic discussion of algorithms to multiply a vector by a matrix expressed as the Kronecker product of sparse matrices, extending previous work in a unified notational framework. Then, we use our results to define new algorithms for the solution of large structured Markov models. In addition to a comprehensive overview of existing approaches, we give new results with respect to...

Abstract: In this paper, we studied the numerical solution of a time-fractional Korteweg–de Vries (KdV) equation with new generalized fractional derivative proposed recently. The fractional derivative employed in this paper was defined in Caputo sense and contained a scale function and a weight function. A finite difference/collocation scheme based on Jacobi–Gauss–Lobatto (JGL) nodes was applie...

Based on numerical experiments with a cell-centred multigrid Poisson solver in one and two dimensions, we propose to use Alternating Coarse-Line Zebra ordering as a robust parallel smoother for the 2D case. The algorithm is presented based on a heuristic discussion, and the convergence rates obtained with diierent smoothers are compared. Local mode analysis is used to estimate the smoothing fac...

A fourth-order compact difference scheme with unequal mesh sizes in different coordinate directions is employed to discretize a two-dimensional Poisson equation in a rectangular domain. Multigrid methods using a partial semicoarsening strategy and line Gauss–Seidel relaxation are designed to solve the resulting sparse linear systems. Numerical experiments are conducted to test the accuracy of t...

We generalize existing Jacobi–Gauss–Lobatto collocation methods for variable-order fractional differential equations using a singular approximation basis in terms of weighted Jacobi polynomials of the form (1 ± x)μP a,b j (x), where μ > −1. In order to derive the differentiation matrices of the variable-order fractional derivatives, we develop a three-term recurrence relation for both integrals...

We analyze the structure of the Euler-Lagrange conditions of a lifted long-horizon optimal control problem. The analysis reveals that the conditions can be solved by using block Gauss-Seidel schemes and we prove that such schemes can be implemented by solving sequences of short-horizon problems. The analysis also reveals that a receding-horizon control scheme is equivalent to performing a singl...

The paper addresses quaternion-based energy and time optimal spacecraft reorientation in the presence of complex attitude constrained zones. Designing an optimal reorientation trajectory for a rigid body spacecraft is posed as a nonlinear optimal control problem. In this direction, attitude constrained zones are defined with respect to the onboard instrument under two categories, forbidden and ...