Stefan Vandewalle

Analysis of the Parareal Time-Parallel Time-Integration Method

The parareal algorithm is a method to solve time-dependent problems parallel in time: it approximates parts of the solution later in time simultaneously to parts of the solution earlier in time. In this paper the relation of the parareal algorithm to space-time multigrid and multiple shooting methods is first briefly discussed. The focus of the paper is on new convergence results that show superlinear convergence of the algorithm when used on bounded time intervals, and linear convergence for unbounded intervals.

On the Evaluation of Highly Oscillatory Integrals by Analytic Continuation

We consider the integration of one-dimensional highly oscillatory functions. Based on analytic continuation, rapidly converging quadrature rules are derived for a general class of oscillatory integrals with an analytic integrand. The accuracy of the quadrature increases both for the case of a fixed number of points and increasing frequency, and for the case of an increasing number of points and fixed frequency. These results are then used to obtain quadrature rules for more general oscillatory integrals, i.e., for functions that exhibit some smoothness but that are not analytic. The approach described in this paper is related to the steepest descent method, but it does not employ asymptotic expansions. It can be used for small or moderate frequencies as well as for very high frequencies. The approach is compared with the oscillatory integration techniques recently developed by Iserles and Norsett.

A space-time multigrid method for parabolic partial differential equations

We consider the solution of parabolic partial differential equations (PDEs). In standard time-stepping techniques multigrid can be used as an iterative solver for the elliptic equations arising at each discrete time step. By contrast, the method presented in this paper treats the whole of the space-time problem simultaneously. Thus the multigrid operations of smoothing and coarse-grid correction are defined on all of the space-time variables of a given grid level. The method is characterized by a coarsening strategy with prolongation and restriction operators which depend at each grid level on the degree of anisotropy of the discretization stencil. Numerical results for the one- and two-dimensional heat equations are presented and are shown to agree closely with predictions from Fourier mode analysis.

A Sparse Discretization for Integral Equation Formulations of High Frequency Scattering Problems

We consider two-dimensional scattering problems, formulated as an integral equation defined on the boundary of the scattering obstacle. The oscillatory nature of high-frequency scattering problems necessitates a large number of unknowns in classical boundary element methods. In addition, the corresponding discretization matrix of the integral equation is dense. We formulate a boundary element method with basis functions that incorporate the asymptotic behavior of the solution at high frequencies. The method exhibits the effectiveness of asymptotic methods at high frequencies with only few unknowns, but retains accuracy for lower frequencies. New in our approach is that we combine this hybrid method with very effective quadrature rules for oscillatory integrals. As a result, we obtain a sparse discretization matrix for the oscillatory problem. Moreover, numerical experiments indicate that the accuracy of the solution actually increases with increasing frequency. The sparse discretization applies to problems where the phase of the solution can be predicted a priori, for example in the case of smooth and convex scatterers.

An Analysis of Delay-Dependent Stability for Ordinary and Partial Differential Equations with Fixed and Distributed Delays

This paper is concerned with the study of the stability of ordinary and partial differential equations with both fixed and distributed delays, and with the study of the stability of discretizations of such differential equations. We start with a delay-dependent asymptotic stability analysis of scalar ordinary differential equations with real coefficients. We study the exact stability region of the continuous problem as a function of the parameters of the model. Next, it is proved that a time discretization based on the trapezium rule can preserve the asymptotic stability for the considered set of test problems. In the second part of the paper, we study delay partial differential equations. The stability region of the fully continuous problem is analyzed first. Then a semidiscretization in space is applied. It is shown that the spatial discretization leads to a reduction of the stability region when the standard second-order central difference operator is employed to approximate the diffusion operator. Finally we consider the delay-dependent stability of the fully discrete problem, where the partial differential equation is discretized both in space and in time. Some numerical examples and further discussions are given.

Solution strategies for transient, field-circuit coupled systems

Transient simulation time for field-circuit coupled models of realistic electromagnetic devices becomes unacceptably high. A magnetodynamic formulation is coupled to an electric circuit analysis, yielding a sparse, symmetric and indefinite matrix. The unknown circuit currents correspond to negative eigenvalues in the matrix spectrum. The Quasi-Minimal Residual method performs better than the Minimal Residual approach that is restricted to positive definite preconditioners. The positive definite variant is solved by the Conjugate Gradient method without explicitly building the dense coupled matrix. As an example, both approaches are applied to an induction motor.

A continuum model for metabolic gas exchange in pear fruit.

Exchange of O2 and CO2 of plants with their environment is essential for metabolic processes such as photosynthesis and respiration. In some fruits such as pears, which are typically stored under a controlled atmosphere with reduced O2 and increased CO2 levels to extend their commercial storage life, anoxia may occur, eventually leading to physiological disorders. In this manuscript we have developed a mathematical model to predict the internal gas concentrations, including permeation, diffusion, and respiration and fermentation kinetics. Pear fruit has been selected as a case study. The model has been used to perform in silico experiments to evaluate the effect of, for example, fruit size or ambient gas concentration on internal O2 and CO2 levels. The model incorporates the actual shape of the fruit and was solved using fluid dynamics software. Environmental conditions such as temperature and gas composition have a large effect on the internal distribution of oxygen and carbon dioxide in fruit. Also, the fruit size has a considerable effect on local metabolic gas concentrations; hence, depending on the size, local anaerobic conditions may result, which eventually may lead to physiological disorders. The model developed in this manuscript is to our knowledge the most comprehensive model to date to simulate gas exchange in plant tissue. It can be used to evaluate the effect of environmental stresses on fruit via in silico experiments and may lead to commercial applications involving long-term storage of fruit under controlled atmospheres.

A Riemannian Optimization Approach for Computing Low-Rank Solutions of Lyapunov Equations

We propose a new framework based on optimization on manifolds to approximate the solution of a Lyapunov matrix equation by a low-rank matrix. The method minimizes the error on the Riemannian manifold of symmetric positive semidefinite matrices of fixed rank. We detail how objects from differential geometry, like the Riemannian gradient and Hessian, can be efficiently computed for this manifold. As a minimization algorithm we use the Riemannian trust-region method of [P.-A. Absil, C. Baker, and K. Gallivan, Found. Comput. Math., 7 (2007), pp. 303-330] based on a second-order model of the objective function on the manifold. Together with an efficient preconditioner, this method can find low-rank solutions with very little memory. We illustrate our results with numerical examples.

General Linear Methods for Volterra Integro-differential Equations with Memory

A new class of numerical methods for Volterra integro-differential equations with memory is developed. The methods are based on the combination of general linear methods with compound quadrature rules. Sufficient conditions that guarantee global and asymptotic stability of the solution of the differential equation and its numerical approximation are established. Numerical examples illustrate the convergence and effectiveness of the numerical methods.

Microscale mechanisms of gas exchange in fruit tissue

* Gas-filled intercellular spaces are considered the predominant pathways for gas transport through bulky plant organs such as fruit. Here, we introduce a methodology that combines a geometrical model of the tissue microstructure with mathematical equations to describe gas exchange mechanisms involved in fruit respiration. * Pear (Pyrus communis) was chosen as a model system. The two-dimensional microstructure of cortex tissue was modelled based on light microscopy images. The transport of O(2) and CO(2) in the intercellular space, cell wall network and cytoplasm was modelled using diffusion laws, irreversible thermodynamics and enzyme kinetics. * In silico analysis showed that O(2) transport mainly occurred through intercellular spaces and less through the intracellular liquid, while CO(2) was transported at equal rates in both phases. Simulations indicated that biological variation of the apparent diffusivity appears to be caused by the random distribution of cells and intercellular spaces in tissue. Temperature does not affect modelled gas exchange properties; it rather acts on the respiration metabolism. * This modelling approach provides, for the first time, detailed information about gas exchange mechanisms at the microscopic scale in bulky plant organs, such as fruit, and can be used to study conditions of anoxia.