As for homogeneous lenses, for axial gradients the analysis of the Seidel and chromatic aberration coefficients can be very useful in lens design. However, at present few commercial optical design programs list the Seidel aberrations of GRIN lenses and none of them lists the chromatic aberrations. In order to facilitate the computer implementation of the chromatic aberrations of axial GRIN lens...

In this paper, we present an analysis of a multigrid method for nonsym-metric and/or indeenite elliptic problems. In this multigrid method various types of smoothers may be used. One type of smoother which we consider is deened in terms of an associated symmetric problem and includes point and line, Jacobi and Gauss-Seidel iterations. We also study smoothers based entirely on the original opera...

In this paper we study a multigrid (MG) method for the solution of a linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods, and we give a detailed analysis of the convergence for different block-relaxation strategies. We find that pointwise block-partitioning gives much better results than the classical cellwise partitioning. Both for the Baumann–Oden method ...

We say that a simple graph G is Seidel integral if its spectrum consists entirely of integers. In this work we establish characterization graphs which belong to the class ?Ka ? ?Kb,b, where mG denotes m-fold union G.

Multigrid solves a set of simultaneous linear equations optimally. That is the time and space used by the multigrid computation are proportional to N , the number of unknowns. In reality the performance of any computation is not just dictated by how many operations performed. The order of the operations and the layout of data in memory determine how well levels of cache in the target architectu...

In the present paper we concentrate on an important issue in constructing a good multigrid solver: the choice of an efficient smoother. We will introduce all-at-once multigrid solvers for optimal control problems which show robust convergence in the grid size and in the regularization parameter. We will refer to recent publications that guarantee such a convergence behavior. These publications ...

Methods of interval arithmetic can be used to reliably find with certainty all solutions to nonlinear systems of equations. In such methods, the system is transformed into a linear interval system and a preconditioned interval Gauss-Seidel method may then be used to compute such solution bounds. In this work, a new heuristic for solving polynomial systems is presented, called reordering techniq...

The original TPABLO algorithms are a collection of algorithms which compute a symmetric permutation of a linear system such that the permuted system has a relatively full block diagonal with relatively large nonzero entries. This block diagonal can then be used as a preconditioner. We propose and analyze three extensions of this approach: we incorporate a nonsymmetric permutation to obtain a la...

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...