Pieter Wilhelm Hemker

• Prof. dr
• Fellow at Centrum Wiskunde & Informatica

Binding of macromolecules to surfaces, or to surface-attached binding partners, is usually described by the classical Langmuir model, which does not include interaction between incoming and adsorbed molecules or between adsorbed molecules. The present study introduces the “Surfint” model, including such interactions. Instead of the exponential bind...

A method of determining the course of enzyme activity that is variable in time, wherein the activity is probed by conversion of a substrate of the enzyme, includes, in a selected test set up and for a determined substrate of the enzyme, determining the velocity of signal production (dFdiag/dt) resulting from a time curve of the signal (Fdiao=f(A))...

Assessment of high-affinity antibody-antigen binding parameters is important in such diverse areas as selection of therapeutic antibodies, detection of unwanted hormones in cattle and sensitive immunoassays in clinical chemistry. Label-free assessment of binding affinities is often carried out by immobilization of one of the binding partners on a b...

Positioning a vessel at a fixed position in deep water is of great importance when working offshore. In recent years a Dynamical Positioning (DP) system was developed at Marin [2]. After the measurement of the current position and external forces (like waves, wind etc.), each thruster of the vessel is actively controlled to hold the desired locatio...

In fluorogenic thrombin generation (TG) experiments, thrombin concentrations cannot be easily calculated from the rate of the fluorescent signal increase, because the calibration coefficient increases during the experiment, due to substrate consumption and quenching of the fluorescent signal by the product. Continuous, external calibration via an i...

In this paper we introduce an adaptive method for the numerical solu-tion of the Pocklington integro-differential equation with exact kernel for the current induced in a smoothly curved thin wire antenna. The hp-adaptive technique is based on the representation of the discrete solution, which is expanded in a piecewise p-hierarchical basis. The key...

In this chapter we present the principles of the space-mapping iteration techniques for the efficient solution of optimization problems. We also show how space-mapping optimization can be understood in the framework of defect correction. We observe the difference between the solution of the optimization problem and the computed space-mapping solut...

Engineering optimization procedures employ highly accurate numerical models that typically have an excessive computational cost, e.g., finite elements (FE). The space mapping (SM) technique speeds up the minimization procedure by exploiting simplified (less accurate) models. We will use the SM terminology of fine and coarse to refer to the accurate...

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...

We first show the idea behind a space-mapping iteration technique for the effi- cient solution of optimization problems. Then we show how space-mapping optimization can be understood in the framework of defect correction. We observe a difference between the solution of the optimization problem and the computed space-mapping solutions. We repair thi...

The mixed defect correction iteration method is considered. The iteration method is described for application to the convection diffusion equation in two dimensions. The convergence of the iteration process is studied. The downstream boundary layer behavior is analyzed. The model problems are discretized by the finite element method and the solutio...

To investigate in how far successful simulation of a thrombin generation (TG) curve gives information about the underlying biochemical reaction mechanism. 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...

Optimization procedures, in practice, are based on highly accurate models that typically have an excessive computational cost. By exploiting auxiliary models that are less accurate, but much cheaper to compute, space mapping (SM) has been reported to accelerate such procedures. However, the SM solution does not always coincide with the accurate mod...

An efficient iterative method has been developed for the accurate solution of the non-isenthalpic steady Euler equations for inviscid flow. First, the system of conservation laws is space-discretized by a first order finite-volume Osher-discretization. Without time stepping, the steady equations are solved by iteration with nonlinear multiple grid...

In this paper, we analyze in some detail the manifold-mapping optimization technique introduced recently [Echeverría and Hemker in space mapping and defect correction. Comput Methods Appl Math 5(2): 107—136, 2005]. Manifold mapping aims at accelerating optimal design procedures that otherwise require many evaluations of time-expensive cost function...

See also Orfeo T, Mann KG. Mathematical and biological models of blood coagulation. This issue, pp 2397–8.

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...

For the solution of convection-diffusion problems we present a multilevel self-adaptive mesh-refinement algorithm to resolve locally strong varying behavior, like boundary and interior layers. The method is based on discontinuous Galerkin (Baumann-Oden DG) discretization. The recursive mesh-adaptation is interwoven with the multigrid solver. The so...

In this paper, we study a multigrid (MG) method for the solution of a linear one-dimensional convection–diffusion equation that is discretized by a discontinuous Galerkin method. In particular we study the convection-dominated case when the perturbation parameter, i.e. the inverse cell-Reynolds-number, is smaller than the finest mesh size. We show...

In this paper we show that space-mapping optimization can be under- stood 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 ∞exibility of the underlying...

Optimization procedures in practice are based on high accuracy models that typically have an excessive computational cost. Within SM terminology, these models are called fine models and we will denote them as f (x), x X being the design parameters. A finite element solution of Maxwell's equations is an example of a model of this type. SM needs a se...

In this paper we study the convergence of a multigrid method for the solution of a two-dimensional linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods. For the Baumann–Oden and for the symmetric DG method, we give a detailed analysis of the convergence for cell- and point-wise block-relaxation strategies. We sh...

In this paper we study the convergence of a multigrid 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. To complement an earlier paper where higher-order methods were studied, here we re...

The Dirichlet boundary value problem for a singularly perturbed elliptic reaction-diffusion equation is considered in a strip. For this problem, special difference schemes are available that converge ε-uniformly with up to the second order of accuracy. Special schemes on piecewise uniform meshes and Richardson's technique are used to construct a sc...

We study numerical approximations for a class of singularly perturbed problems of convection-diffusion type with a moving interior layer. In a domain (a segment) with a moving interface between two subdomains, we consider an initial boundary value problem for a singularly perturbed parabolic convection-diffusion equation. Convection fluxes on the s...

A finite-volume method is considered for the computation of flows of two compressible, immiscible fluids at very different densities. A level-set technique is employed to distinguish between the two fluids. A simple ghost-fluid method is presented as a fix for the solution errors ('pressure oscillations') that may occur near two-fluid interfaces wh...

We consider an initial boundary value problem on an interval for singularly perturbed parabolic PDEs with convection. The highest space derivative in the equation is multiplied by the perturbation parameter ", " 2 (0; 1]. Solutions of well-known classical numerical schemes for such problems do not converge "-uniformly (the errors of such schemes de...

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 p...

The purpose of this paper is to introduce discretization methods of discontinuous Galerkin type for solving second-order elliptic PDEs on a structured, regular rectangular grid, while the problem is defined on a curved boundary. The methods aim at high-order accuracy and the difficulty arises since the regular grid cannot follow the curved boundary...

this paper converge "-uniformly at the rate of O(N 0 ), where N and N 0 denote respectively the number of mesh intervals in the space and time discretizations

A finite-volume method is considered for the computation of flows of two compressible. immiscible fluids at very different densities. A level-set technique is employed to distinguish between the two fluids. A simple ghost-fluid method is presented as a fix for the solution errors ( pressure oscillations) that, may occur near two-fluid interfaces wh...

New high-order accurate finite difference schemes based on defect correction are considered for an initial boundary-value problem on an interval for singularly perturbed parabolic PDEs with convection; the highest space derivative in the equation is multiplied by the perturbation parameter ε, ε ∈ (0, 1]. Solutions of the well-known classical numeri...

C1-inhibitor protein (C1-INH) purified from pooled human plasma is used for the treatment of patients with hereditary angioedema. Recently, the beneficial effects of high-dose C1-INH treatment on myocardial ischemia or reperfusion injury have been reported in various animal models and in humans. We investigated the pharmacokinetic behavior of C1-IN...

We consider the Dirichlet problem on a strip for a singularly perturbed elliptic equation of reaction-diffusion type. For such a problem well-known finite difference schemes converge ...-uniformly with the order of accuracy not higher than second. This can imply some restrictions for practical use of these schemes in applications. Based on special...

In this paper we study a multigrid 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.

The boundary-value problem for a singularly perturbed parabolic PDE with convection is considered on an interval in the case of the singularly perturbed Robin boundary condition; the highest space derivatives in the equation and in the boundary condition are multiplied by the perturbation parameter #. In contrast to the Dirichlet boundary-value pro...

The boundary-value problem for a singularly perturbed parabolic PDE with convection is considered on an interval in the case of the singularly perturbed Robin boundary condition; the highest space derivatives in the equation and in the boundary condition are multiplied by the perturbation parameter ε. The order of convergence for the known ε-unifor...

In this paper we study a multigrid 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 di#erent block-relaxation strategies.

The first boundary value problem for a singularly perturbed parabolic PDE with convection is considered on an interval. For the case of sufficiently smooth data, it is easy to construct a standard finite difference operator and a piecewise uniform mesh condensing in the boundary layer, which gives an epsilon-unifomily convergent difference scheme....

sulting discrete large linear systems. A signicant number (about 40%) of new people were invited, compared with the previous meeting organized by the same group at Oberwolfach in 1998. This resulted in valuable new viewpoints. The number of proposed presentations was much larger than the number than could be accommodated. To promote the disseminati...

The first boundary value problem for a singularly perturbed parabolic equation of convection-diffusion type on an interval is studied. For the approximation of the boundary value problem we use earlier developed finite difference schemes, ɛ-uniformly of a high order of accuracy with respect to time, based on defect correction. New in this paper is...

Contents Chapter 1. Physical Background 5 1. Introduction 5 Chapter 2. Equations of fluid flow 7 1. Conservation of mass 7 2. Conservation of momentum 7 3. Conservation of energy 8 4. The system of equations 9 5. Special cases 9 Chapter 2. Basic equations 13 1. Conservation laws 13 2. Non-ideal fluids 14 3. Navier-Stokes equations 15 Chapter 3. Num...

this paper we study the discrete approximation of a Dirichlet problem on an interval for a singularly perturbed parabolic PDE. The highest derivative in the equation is multiplied by an arbitrarily small parameter #. If the parameter vanishes, the parabolic equation degenerates to a first-order equation, in which only the time derivative remains. F...

We consider the first boundary value problem for a singularly perturbed parabolic PDE with convection on an interval. For the case of sufficiently smooth data, it is easy to construct a standard finite difference operator and a piecewise uniform mesh, condensing in the boundary layer, which gives an #-uniformly convergent difference scheme. The ord...

In this paper we describe methods to approximate functions and differential operators on adaptive sparse (dyadic) grids. We distinguish between several representations of a function on the sparse grid and we describe how finite difference (FD) operators can be applied to these representations. For general variable coefficient equations on sparse gr...

In this paper we show how, under minimal conditions, a combination extrapolation can be introduced for an adaptive sparse grid. We apply this technique for the solution of a two-dimensional model singular perturbation problem, defined on the domain exterior of a circle.

The numerical solution of a singularly perturbed problem, in the form of a two-dimensional convection-diffusion equation, is studied by using the technique of over-set grids. For this purpose the Overture software library is used. The selection of component grids is made on basis of asymptotic analysis. The behavior of the solution is studied for a...

For singularly perturbed convection-diffusion problems with the perturbation parameter ∈ multiplying the highest derivatives, we construct a scheme based on the defect correction method and its parallel variant that converge ∈ -uniformly with second-order accuracy in the time variable.We also give the conditions under which the parallel computation...

In a recent paper [10],we described and analyzed a finite difference discretization on adaptive sparse grids in three space dimensions. In this paper,we show how the discrete equations can be efficiently solved in an iterative process.Several alternatives have been studied before in Sprengel [16],where multigrid algorithms were used. Here,we report...

In this paper we study the discrete approximation of a Dirichlet problem on an interval for a singularly perturbed parabolic PDE. The highest derivative in the equation is multiplied by an arbitrarily small parameter epsilon. If the parameter vanishes, the parabolic equation degenerates to a first-order equation, in which only the time derivative r...

this paper we study such "-uniformly convergent schemes, which combine a standard finite difference operator (see, e.g. [1]) and a grid selection criterion for the space discretization (see, e.g., [2]).

The Dirichlet problem for a singularly perturbed parabolic equation with convective terms is considered on an interval. For the sufficiently smooth data, it is easy to construct a piecewise uniform mesh condensing in the boundary layer and a standard finite difference operator, which give an ε-uniformly convergent difference scheme with the accurac...

We study the discrete approximation of a Neumann problem on an interval for a singularly perturbed parabolic partial differential equation. For this boundary value problem we construct a special piecewise-uniform mesh on which the discretization, based on the classical finite difference approximation, converges ε-uniformly with the order O(N -2 ln...

this paper we study the discrete approximation of a Dirichlet problem on an interval for a singularly perturbed parabolic PDE. The highest derivative in the equation is multiplied by an arbitrarily small parameter ". If the parameter vanishes, the parabolic equation degenerates to a first-order equation, in which only the time derivative remains. F...

In this paper we describe methods to approximate functions and differential operators on adaptive sparse grids. We distinguish between several representations of a function on the sparse grid, and we describe how finite difference (FD) operators can be applied to these representations. For general variable coefficient equations on sparse grids, FD...

In this paper we discuss different possibilities of using partially ordered sets of grids in multigrid algorithms. Because, for a classical sequence of regular grids the number of degrees of freedom grows much faster with the refinement level for 3D than for 2D, it is more difficult to find sufficiently effective relaxation procedures. Therefore, w...

We consider the parabolic Dirichlet problem on an interval for a singularly perturbed parabolic equation. The initial conditions of the problem have a local disturbance over a small area of width 2δ. The perturbation parameters ε 2 , that is the coefficient multiplying the highest derivative, and δ can take arbitrary values from the half-open inter...

In this paper we analyze the approximation of functions on partially ordered sequences of regular grids. We start with the formulation of minimal requirements for useful grid transfer operators. We introduce the notions of nested and of commutative transfer operators. We define mutual coherence for representations on grids that are not related by c...

A significant difficulty of standard multigrid methods for 3D problems, when compared to application to 2D problems, is that the requirements to be imposed on the smoother are much more severe. As a remedy, we investigate three different possibilities of multiple semi-coarsening: full, sparse and semi-sparse. Numerical results are presented for a s...

The convergence behaviour is investigated of solution algorithms for the anisotropic Poisson problem on partially ordered, sparse families of regular grids in 3D. In order to study multilevel techniques on sparse families of grids, first we consider the convergence of a twolevel algorithm that applies semi-coarsening successively in each of the coo...

We construct discrete approximations for a class of singularly perturbed boundary value problems, such as the Dirichlet problem for a parabolic differential equation, for which the coefficient multiplying the highest derivatives can take an arbitrarily small value from the interval (0, 1]. Discretisation errors for classical discrete methods depend...

A multiple semi-coarsened multigrid method for solving discretized, steady three-dimensional Euler equations of gas dynamics, is described and applied. Convergence results are presented for the case of the ONERA-M6 wing under transonic conditions. Comparisons are made with an optimal standard multigrid method, as well as with a single-grid method....

This paper describes the mathematical modelling of a part of the blood coagulation mechanism. The model includes the activation of factor X by a purified enzyme from Russel's Viper Venom (RVV), factor V and prothrombin, and also comprises the inactivation of the products formed. In this study we assume that in principle the mechanism of the process...

This paper consists of two parts. In the first part we give a review of a good multigrid method for solving the steady Euler equations of gas dynamics on a locally refined mesh. The method is self-adaptive and makes use of unstructured grids that can be considered as parts of a nested sequence of structured grids. It is briefly described and applie...

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...

In his series of three papers we study singularly perturbed (SP) boundary value problems for equations of elliptic and parabolic type. For small values of the perturbation parameter parabolic boundary and interior layers appear in these problems. If classical discretisation methods are used, the solution of the finite difference scheme and the appr...

We introduce a multigrid algorithm for the solution of a second order elliptic equation in three dimensions. For the approximation of the solution we use a partially ordered hierarchy of finite-volume discretisations. We show that there is a relation with semicoarsening and approximation by more-dimensional Haar wavelets. By taking a proper subset...

For a system of hyperbolic conservation laws, such as the Euler equations of compressible flow, in this paper we give an outline of the theory necessary to derive first and second-order accurate discretisations on a structured, adaptive finite-volume mesh. The mesh is constructed so that the equations can be defined on a rather arbitrary domain, an...