Rob H. De Staelen

First order partial differential equations with time delay and retardation of a state variable

We construct a finite difference scheme for the numerical solution of a first order partial differential equation with a time delay and retardation of a state variable. Such equations are used to model the dynamics of structured cell populations when age and maturity level are taken into account. For the supplied difference schemes the order of approximation, stability and convergence order are studied. We illustrate the obtained results with a test example.

On a Finite Difference Scheme for an Inverse Integro-Differential Problem Using Semigroup Theory: A Functional Analytic Approach

ABSTRACT In this article, the problem of reconstructing an unknown memory kernel from an integral overdetermination in an abstract linear (of convolution type) evolution equation of parabolic type is considered. After illustrating some results of the existence and uniqueness of a solution for the differential problem, we study its approximation by Rothes method. We prove a result of stability and another concerning the order of approximation of the solution in dependence of its regularity. The main tool is a maximal regularity result for solutions to abstract parabolic finite difference schemes. Two model problems to which the results are applicable are illustrated.

Characteristic times for multiscale diffusion of active ingredients in coated textiles

A three-scale approach for textile models was given in 5]: a one-dimensional fiber model and a room model, with a meso-level in between, which is the yarn scale. To analyze and simplify the model, its characteristic times are investigated here. At these times the fiber and yarn model and the yarn and room model, respectively, tend to reach a partial equilibrium concentration. The identification of these characteristic times is key to reducing the model to its variously scaled components when simplifying it.

Optimal Portfolio Choice with Benchmarks

We construct an algorithm that makes it possible to numerically obtain an investor’s optimal portfolio under general preferences. In particular, the objective function and risks constraints may be driven by benchmarks (reflecting state-dependent preferences). We apply the algorithm to various classic optimal portfolio problems for which explicit solutions are available and show that our numerical solutions are compatible with them. This observation allows us to conclude that the algorithm can be trusted as a viable way to deal with portfolio optimization problems for which explicit solutions are not in reach.

A difference scheme for multidimensional transfer equations with time delay

This paper continues research initiated in Solodushkin etźal. (2015). We develop a finite difference scheme for a first order multidimensional partial differential equation including a time delay. This class of equations is used to model different time lapse phenomena, e.g. birds migration, proliferation of viruses or bacteria and transfer of nuclear particles. For the constructed difference schemes the order of approximation, stability and convergence order are substantiated. To conclude we support the obtained results with some test examples.

On cost function transformations for the reduction of uncertain model parameters' impact towards the optimal solutions

Uncertainties affect the accuracy of nonlinear static or dynamic optimization and inverse problems. The propagation of uncertain model parameters towards the optimal problem solutions can be assessed in a deterministic or stochastic way using Monte Carlo based techniques and efficient spectral collocation and Galerkin projection methods. This paper presents cost function transformations for reducing the impact of uncertain model parameters towards the optimal solutions. We assess the consistency of the methodology by determining sufficient conditions on the cost function transformations and apply the methodology on several test functions.