Gabriel Wittum

On the Robustness of ILU Smoothing

In the present paper, a detailed analysis of a multigrid method with an ILU smoother applied to a singularly perturbed problem is given. Based on the analysis of a simple anisotropic model problem, a variant of the usual incomplete LU factorization is introduced, which is especially suited as robust smoother. For this variant a detailed analysis and a proof of robustness is given. Furthermore, some contradictions between the smoothing rates predicted by local Fourier analysis and the practically observed convergence factors are explained (see [W. Hackbusch, Multi-grid Methods and Applications, Springer-Verlag, Berlin, Heidelberg, 1985; R. Kettler, “Analysis and comparison of relaxation schemes in robust multi-grid and preconditioned conjugate gradient methods,” in Multi-grid Methods, Lecture Notes in Math. 960, Springer-Verlag, Berlin, 1982; C. A. Thole, Beitrage zur Fourieranalyse von Mehrgitterver fahren, Diplomarbeit, Universitat Bonn, 1983]. The theoretical results are confirmed by numerical tests.

The saltpool benchmark problem – numerical simulation of saltwater upconing in a porous medium

Recently, a series of laboratory experiments was carried out in which a typical variable-density flow problem in a porous medium was investigated. A stable layering of saltwater below freshwater is affected by the recharge and discharge of freshwater on the top. The experiments were conducted with 1% and 10% initial salt-mass fraction contrasts. In this paper, we define a mathematical model problem, which is able to reproduce the experimental results within a reasonable accuracy. To this end, the sensitivity of the model with respect to model parameters is investigated. An inverse modelling leads to an appropriate choice of the model parameters and the definition of the mathematical benchmark problem. It will become clear that, in the 10% case, the transversal dispersion coefficient plays an important role. Detailed numerical investigations are carried out and reference solutions are obtained. For the high concentration case a very high spatial grid resolution using up to 16 million grid points is necessary. Error bounds are derived for the solutions without any a priori assumption on regularity and convergence.

Non Steady-state Descriptions of Drug Permeation Through Stratum Corneum. I. The Biphasic Brick-and-Mortar Model

AbstractPurpose. The diffusion equation should be solved for the non-steady-state problem of drug diffusion within a two-dimensional, biphasic stratum corneum membrane having homogeneous lipid and corneocyte phases. Methods. A numerical method was developed for a brick-and-mortar SC-geometry, enabling an explicit solution for time-dependent drug concentration within both phases. The lag time and permeability were calculated. Results. It is shown how the barrier property of this model membrane depends on relative phase permeability, corneocyte alignment, and corneocyte-lipid partition coefficient. Additionally, the time-dependent drug concentration profiles within the membrane can be observed during the lag and steady-state phases. Conclusions. The model SC-membrane predicts, from purely morphological principles, lag times and permeabilities that are in good agreement with experimental values. The long lag times and very small permeabilities reported for human SC can only be predicted for a highly-staggered corneocyte geometry and corneocytes that are 1000 times less permeable than the lipid phase. Although the former conclusion is reasonable, the latter is questionable. The elongated, flattened corneocyte shape renders lag time and permeability insensitive to large changes in their alignment within the SC. Corneocyte/lipid partitioning is found to be fundamentally different to SC/donor partitioning, since increasing drug lipophilicity always reduces both lag time and permeability.

Multi-grid methods for stokes and navier-stokes equations

SummaryIn the present paper we introduce transforming iterations, an approach to construct smoothers for indefinite systems. This turns out to be a convenient tool to classify several well-known smoothing iterations for Stokes and Navier-Stokes equations and to predict their convergence behaviour, epecially in the case of high Reynolds-numbers. Using this approach, we are able to construct a new smoother for the Navier-Stokes equations, based on incomplete LU-decompositions, yielding a highly effective and robust multi-grid method. Besides some qualitative theoretical convergence results, we give large numerical comparisons and tests for the Stokes as well as for the Navier-Stokes equations. For a general convergence theory we refer to [29].

Synaptic Activity Induces Dramatic Changes in the Geometry of the Cell Nucleus: Interplay between Nuclear Structure, Histone H3 Phosphorylation, and Nuclear Calcium Signaling

Synaptic activity initiates many adaptive responses in neurons. Here we report a novel form of structural plasticity in dissociated hippocampal cultures and slice preparations. Using a recently developed algorithm for three-dimensional image reconstruction and quantitative measurements of cell organelles, we found that many nuclei from hippocampal neurons are highly infolded and form unequally sized nuclear compartments. Nuclear infoldings are dynamic structures, which can radically transform the geometry of the nucleus in response to neuronal activity. Action potential bursting causing synaptic NMDA receptor activation dramatically increases the number of infolded nuclei via a process that requires the ERK-MAP kinase pathway and new protein synthesis. In contrast, death-signaling pathways triggered by extrasynaptic NMDA receptors cause a rapid loss of nuclear infoldings. Compared with near-spherical nuclei, infolded nuclei have a larger surface and increased nuclear pore complex immunoreactivity. Nuclear calcium signals evoked by cytosolic calcium transients are larger in small nuclear compartments than in the large compartments of the same nucleus; moreover, small compartments are more efficient in temporally resolving calcium signals induced by trains of action potentials in the theta frequency range (5 Hz). Synaptic activity-induced phosphorylation of histone H3 on serine 10 was more robust in neurons with infolded nuclei compared with neurons with near-spherical nuclei, suggesting a functional link between nuclear geometry and transcriptional regulation. The translation of synaptic activity-induced signaling events into changes in nuclear geometry facilitates the relay of calcium signals to the nucleus, may lead to the formation of nuclear signaling microdomains, and could enhance signal-regulated transcription.

Efficient Hierarchical Approximation of High-Dimensional Option Pricing Problems

A major challenge in computational finance is the pricing of options that depend on a large number of risk factors. Prominent examples are basket or index options where dozens or even hundreds of stocks constitute the underlying asset and determine the dimensionality of the corresponding degenerate parabolic equation. The objective of this article is to show how an efficient discretization can be achieved by hierarchical approximation as well as asymptotic expansions of the underlying continuous problem. The relation to a number of state-of-the-art methods is highlighted.

Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. Part 1: a smooth problem and globally quasi-uniform meshes

A class of a posteriori estimators is studied for the error in the maximum-norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems. The meshes are unstructured and, in particular, it is not assumed that there are any known superconvergent points. The estimators are based on averaging operators which are approximate gradients, recovered gradients, which are then compared to the actual gradient of the approximation on each element. Conditions are given under which they are asympotically exact or equivalent estimators on each single element of the underlying meshes. Asymptotic exactness is accomplished by letting the approximate gradient operator average over domains that are large, in a controlled fashion to be detailed below, compared to the size of the elements.

NeuGen: A tool for the generation of realistic morphology of cortical neurons and neural networks in 3D

Abstract We introduce the software package NeuGen for the efficient generation of anatomically accurate synthetic neurons and neural networks. NeuGen generates non-identical neurons of morphological classes of the cortex, e.g., pyramidal cells and stellate neurons, and synaptically connected neural networks in 3D. It is based on sets of descriptive and iterative rules which represent the axonal and dendritic geometry of neurons by inter-correlating morphological parameters. The generation algorithm stochastically samples parameter values from distribution functions induced by experimental data. The generator is adequate for the geometric modelling and for the construction of the morphology. The generated neurons can be exported into a 3D graphic format for visualization and into multi-compartment files for simulations with the program NEURON. NeuGen is intended for scientists aiming at simulations of realistic networks in 3D. The software includes a graphical user interface and is available at http://neugen.uni-hd.de .

Nonlinear anisotropic diffusion filtering of three-dimensional image data from two-photon microscopy

Two-photon microscopy in combination with novel fluorescent labeling techniques enables imaging of three-dimensional neuronal morphologies in intact brain tissue. In principle it is now possible to automatically reconstruct the dendritic branching patterns of neurons from 3-D fluorescence image stacks. In practice however, the signal-to-noise ratio can be low, in particular in the case of thin dendrites or axons imaged relatively deep in the tissue. Here we present a nonlinear anisotropic diffusion filter that enhances the signal-to-noise ratio while preserving the original dimensions of the structural elements. The key idea is to use structural information in the raw data—the local moments of inertia—to locally control the strength and direction of diffusion filtering. A cylindrical dendrite, for example, is effectively smoothed only parallel to its longitudinal axis, not perpendicular to it. This is demonstrated for artificial data as well as for in vivo two-photon microscopic data from pyramidal neurons of rat neocortex. In both cases noise is averaged out along the dendrites, leading to bridging of apparent gaps, while dendritic diameters are not affected. The filter is a valuable general tool for smoothing cellular processes and is well suited for preparing data for subsequent image segmentation and neuron reconstruction.

On the convergence of multi-grid methods with transforming smoothers

SummaryIn the present paper we give a convergence theory for multi-grid methods with transforming smoothers as introduced in [31] applied to a general system of partial differential equations. The theory follows Hackbuschs approach for scalar pde and allows a convergence proof for some well-known multi-grid methods for Stokes- and Navier-Stokes equations as DGS by Brandt-Dinar, [5], TILU from [31] and the SIMPLE-methods by Patankar-Spalding, [23].