Piet Hemker

The Defect Correction Approach

This is an introductory survey of the defect correction approach which may serve as a unifying frame of reference for the subsequent papers on special subjects.

Multiple grid and Osher's scheme for the efficient solution of the steady Euler equations

Abstract An iterative method is developed for the solution of the steady Euler equations for inviscid flow. The system of hyperbolic conservation laws is discretized by a finite-volume Osher-discretization. The iterative method is a multiple grid (FAS) iteration with symmetric Gauss-Seidel (SGS) as a relaxation method. Initial estimates are obtained by full multigrid (FMG). In the pointwise relaxation the equations are kept in block-coupled form and local linearization of the equations and the boundary conditions is considered. The efficient formulation of Oshers discretization of the 2-D non-isentropic steady Euler equations and its linearization is presented. The efficiency of FAS-SGS iteration is shown for a transonic model problem. It appears that, for the problem considered, the rate of convergence is almost independent of the gridsize and that for all meshsizes the discrete system is solved up to truncation error accuracy in only a few (2 or 3) iteration cycles.

On the order of prolongations and restrictions in multigrid procedures

Abstract It is well known in the world of multigrid that the order of the prolongation and the order of the restriction in a multigrid method should satisfy certain conditions. A rule of thumb is that the sum of the orders of the prolongation and of the restriction should at least be equal to the order of the differential equation solved. In this note we show the correctness of this rule. We notice that we have to distinguish between low frequency and high frequency orders for the transfer operators. For the restriction, the low frequency order is related with its accuracy, whereas for the interpolation operator both orders are related with the accuracy of the result of the interpolation. If an interpolation rule leaves all polynomials of degree k − 1 invariant, then both the low and the high frequency order are equal to k. It is the high frequency order that plays a role in the above-mentioned rule of thumb.

Space Mapping and Defect Correction

Abstract In this paper we show that space-mapping optimization can be understood in the framework of defect correction. Then, space-mapping algorithms can be seen as special cases of defect correction iteration. In order to analyze the properties of space mapping and the space-mapping function, we introduce the new concept of flexibility of the underlying models. The best space-mapping results are obtained for so-called equally flexible models. By introducing an a±ne operator as a left preconditioner, two models can be made equally flexible, at least in the neighborhood of a solution. This motivates an improved space-mapping (or manifold-mapping) algorithm. The left preconditioner complements traditional space mapping where only a right preconditioner is used. In the last section a few simple examples illustrate some of the phenomena analyzed in this paper.

A singularly perturbed model problem for numerical computation

Abstract In this note we introduce a model problem for the numerical solution of a two-dimensional singular perturbation problem. To combine a number of typical difficulties in a relatively simple problem, we propose to solve the linear convection-diffusion problem in the domain exterior of a circle. We describe the analytical solution of the problem and we comment on its numerical evaluation. For small values of the parameter, asymptotic approximations of the solution are given based on expansions by Friedlander (1958) and Waechter (1968). This information gives insight into the behaviour of the solution and allows the computation of a reference solution for small values of the parameter.

The limits of simulation of the clotting system

Summary.  Objective: To investigate in how far successful simulation of a thrombin generation (TG) curve gives information about the underlying biochemical reaction mechanism. Results: The large majority of TG curves do not contain more information than can be expressed by four parameters. A limited kinetic mechanism of six reactions, comprising proteolytic activation of factor (F) X and FII, feedback activation of FV, a cofactor function of FVa and thrombin inactivation by antithrombin can simulate any TG curve in a number of different ways. The information content of a TG curve is thus much smaller than the information required to describe a physiologically realistic reaction scheme of TG. Consequently, much of the input information is irrelevant for the output. FVIII deficiency or activation of protein C can, for example, be simulated by a reaction mechanism in which these factors do not occur. Conclusion: A model that comprises not more than six reactions can simulate the same TG curve in a number of possible ways. The possibilities increase exponentially as the model grows more realistic. Successful simulation of experimental data therefore does not validate the underlying assumptions. A fortiori, simulation that is not checked against experimental data lacks any probative force. Simulation can be of use, however, to detect mistaken hypotheses and for parameter estimation in systems with fewer than five free parameters.

General Kinetics of Enzyme Cascades

The theory of the kinetics of enzyme cascades is developed. Two types of cascades are recognized, one in which the products are stable (open cascades) and another in which the products are broken down (damped cascades). It is shown that it is a characteristic of a cascade that the final product appears after a certain lag phase. After this lag phase, the velocity of product formation can be very rapid. It is shown that whereas open cascades will always show a complicated time–product relation, damped cascades can under certain circumstances resemble a simple enzymic reaction. Because the relation between the over-all reaction velocity in the extrinsic coagulation cascade and the concentration of any of the proenzymes in this cascade is a hyperbolic one, it is concluded that this cascade is of the damped type rather than the open type.

Multigrid methods: development of fast solvers

Numerical and programming aspects are discussed of multigrid algorithms for the solution of discretized linear elliptic equations. The aim is to obtain software that is perceived and can be used just like any standard subroutine for solving systems of linear equations. The user has to specify only the matrix and the right-hand-side, and remains unaware of the underlying multigrid method. We find that a large class of equations can be solved efficiently in this way. The equation may be non-self-adjoint, and its coefficients are arbitrary. Special attention is given to the treatment of the convection-diffusion equation at high Peclet number. Details are given of an available portable FORTRAN code, which vectorizes satisfactorily on vector machines. CP time statistics are given for a CYBER-205.

A trust-region strategy for manifold-mapping optimization

Studying the space-mapping technique by Bandler et al. [J. Bandler, R. Biernacki, S. Chen, P. Grobelny, R.H. Hemmers, Space mapping technique for electromagnetic optimization, IEEE Trans. Microwave Theory Tech. 42 (1994) 2536-2544] for the solution of optimization problems, we observe the possible difference between the solution of the optimization problem and the computed space-mapping solution. We repair this discrepancy by exploiting the correspondence with defect-correction iteration and we construct the manifold-mapping algorithm, which is as efficient as the space-mapping algorithm but converges to the exact solution. To increase the robustness of the algorithm we introduce a trust-region strategy (a regularization technique) based on the generalized singular value decomposition of the linearized fine and coarse manifold representations. The effect of this strategy is shown by the solution of a variety of small non-linear least squares problems. Finally we show the use of the technique for a more challenging engineering problem.

Optimisation in electromagnetics with the space-mapping technique

Purpose – Optimisation in electromagnetics, based on finite element models, is often very time‐consuming. In this paper, we present the space‐mapping (SM) technique which aims at speeding up such procedures by exploiting auxiliary models that are less accurate but much cheaper to compute.Design/methodology/approach – The key element in this technique is the SM function. Its purpose is to relate the two models. The SM function, combined with the low accuracy model, makes a surrogate model that can be optimised more efficiently.Findings – By two examples we show that the SM technique is effective. Further we show how the choice of the low accuracy model can influence the acceleration process. On one hand, taking into account more essential features of the problem helps speeding up the whole procedure. On the other hand, extremely simple auxiliary models can already yield a significant acceleration.Research limitations/implications – Obtaining the low accuracy model is not always straightforward. Some resear...