An analysis of a class of variational multiscale methods based on subspace decomposition

Numerical homogenization tries to approximate the solutions of elliptic partial differential equations with strongly oscillating coefficients by functions from modified finite element spaces. We present in this paper a class of such methods that are very closely related to the method of M{\aa}lqvist and Peterseim [Math. Comp. 83, 2014]. Like the method of M{\aa}lqvist and Peterseim, these methods do not make explicit or implicit use of a scale separation. Their compared to that in the work of M{\aa}lqvist and Peterseim strongly simplified analysis is based on a reformulation of their method in terms of variational multiscale methods and on the theory of iterative methods, more precisely, of additive Schwarz or subspace decomposition methods.Comment: published electronically in Mathematics of Computation on January 19, 201

Fractal homogenization of multiscale interface problems

Inspired by continuum mechanical contact problems with geological fault networks, we consider elliptic second order differential equations with jump conditions on a sequence of multiscale networks of interfaces with a finite number of non-separating scales. Our aim is to derive and analyze a description of the asymptotic limit of infinitely many scales in order to quantify the effect of resolving the network only up to some finite number of interfaces and to consider all further effects as homogeneous. As classical homogenization techniques are not suited for this kind of geometrical setting, we suggest a new concept, called fractal homogenization, to derive and analyze an asymptotic limit problem from a corresponding sequence of finite-scale interface problems. We provide an intuitive characterization of the corresponding fractal solution space in terms of generalized jumps and gradients together with continuous embeddings into L2 and Hs, s<1/2. We show existence and uniqueness of the solution of the asymptotic limit problem and exponential convergence of the approximating finite-scale solutions. Computational experiments involving a related numerical homogenization technique illustrate our theoretical findings

Glutamatergic-dopaminergic balance in the brain. Its importance in motor disorders and schizophrenia

Dopamine appears to be of less importance in the regulation of psychomotor functions than was previously thought. A central dopaminergic-glutamatergic balance may be important for both akinetic motor disorders and psychosis. In Parkinson's disease glutamate antagonists may counteract central glutamatergic hyperactivity and may be of value as anti-parkinsonian drugs. An increase of dopaminergic activity and/or a reduction of glutamatergic activity may contribute to the development of paranoid hallucinatory psychosis in schizophrenic patients and of pharmacotoxic psychosis in Parkinson's disease. Because of possibly severe side-effects of glutamatergic antagonists and agonists in the treatment of akinesia and psychosis, the development of partial glutamate agonists/antagonists could be an alternative strategy capable of producing antipsychotic or anti-kinetic effects with only mild adverse reaction

Monotone iterations for elliptic variational inequalities

A wide range of free boundary problems occurring in engineering and industry can be rewritten as a minimization problem for a strictly convex, piecewise smooth but non–differentiable energy functional. The fast solution of related discretized problems is a very delicate question, because usual Newton techniques cannot be applied. We propose a new approach based on convex minimization and constrained Newton type linearization. While convex min- imization provides global convergence of the overall iteration, the subsequent constrained Newton type linearization is intended to accelerate the conver- gence speed. We present a general convergence theory and discuss several applications

On constrained Newton linearization and multigrid for variational inequalities

We consider the fast solution of a class of large, piecewise smooth minimization problems. For lack of smoothness, usual Newton multigrid methods cannot be applied. We propose a new approach based on a combination of convex minization with constrained Newton linearization. No regularization is involved. We show global convergence of the resulting monotone multigrid methods and give polylogarithmic upper bounds for the asymptotic convergence rates. Efficiency is illustrated by numerical experiments

Adaptive monotone multigrid methods for some non-smooth optimization problems

We consider the fast solution of non-smooth optimization problems as resulting for example from the approximation of elliptic free boundary problems of obstacle or Stefan type. Combining well-known concepts of successive subspace correction methods with convex analysis, we derive a new class of multigrid methods which are globally convergent and have logarithmic bounds of the asymptotic convergence rates. The theoretical considerations are illustrated by numerical experiments

Development and Characterization of a Predictive Cell-based Test Method for the Identification of Substances with Estrogenic Properties

The endocrine system is an integral part during development and the regulation of various physiological processes in the human body. Dysregulation of the endocrine system has frequently been linked to the development of adverse health effects including cancer. Man-made chemicals that have the capacity to interfere with the endocrine system thereby eliciting adverse health effects (termed endocrine disrupting chemicals (EDC)) are therefore of high concern. Available in vitro assays that allow the identification and characterization of EDCs mainly provide information on mechanisms and pathways of endocrine activity. However, these assays usually do not cover functional endpoints that are predictive for adversity such as hormone-related tumor formation and progression, necessitating the use of complex in vivo studies that require high numbers of test animals. This thesis introduces the E-Morph Assay: a novel robust and predictive in vitro test method that allows the identification and characterization of chemicals that interfere with the estrogen signaling pathway. The development of this assay is based on the finding that estrogen signaling modulates the organization of adherens junctions (AJ) in the human MCF7 breast cancer cell line. The specificity of this effect to the estrogen receptor α (ERα) signaling pathway could be verified by inhibition and knock down studies targeting ERα, while modulation of the G-protein-coupled estrogen receptor 1 (GPER1) did not have any influence. It could further be shown that AJ reorganization is mediated by the ERα target gene Amphiregulin (AREG) involving a crosstalk with the epidermal growth factor receptor (EGFR) and the downstream RhoA and Src family kinase signaling pathways. These cancer-related signaling pathways support the mechanistic and clinical relevance of AJ organization to be used as a novel functional endpoint in the E-Morph Assay for high-content/high-throughput phenotypic screening. The development of a 96 well plate assay set-up and a pipeline for automated image acquisition and quantitative image analysis allows the rapid testing of chemicals at multiple concentrations. Pilot screening using a test set of 17 reference chemicals with known estrogenic properties demonstrated a high predictive capacity of the E-Morph Assay. In conclusion, the E-Morph Assay will provide a valuable in vitro screening method to identify and characterize chemicals with estrogenic activity using estrogen dependent changes in AJ organization as a functional readout.Das endokrine System ist ein zentraler Bestandteil während der Entwicklung und sowie der Regulierung physiologischer Prozesse im menschlichen Körper. Eine Dysregulation des endokrinen Systems steht in Verbindung mit der Entwicklung verschiedener Krankheiten einschließlich Krebs. Vom Menschen hergestellte Chemikalien, die in der Lage sind, das endokrine System zu stören und dadurch nachteilige Auswirkungen auf die Gesundheit haben (als endokrin wirkende Chemikalien (EDC) bezeichnet), geben daher Anlass zur Sorge. Verfügbare in-vitro-Assays, mit denen EDCs identifiziert und charakterisiert werden können, liefern hauptsächlich Informationen über Mechanismen und Wege der endokrinen Aktivität. Diese Assays decken jedoch in der Regel keine funktionellen Endpunkte ab, die eine Vorhersage zur Krankheitsentwicklung wie hormonbedingte Tumorbildung ermöglichen. Daher müssen komplexe in-vivo-Studien durchgeführt werden, für die eine große Anzahl von Testtieren erforderlich ist. In dieser Arbeit wird der E-Morph-Assay vorgestellt: eine neue robuste und prädiktive in-vitro-Testmethode, mit der Chemikalien identifiziert werden können, die den Östrogensignalweg stören. Die Entwicklung dieses Assays basiert auf der Feststellung, dass der Östrogensignalweg die Organisation von Adherens Junctions (AJ) in der menschlichen MCF7 Brustkrebszelllinie verändert. Die Spezifität dieses Effekts für den Östrogenrezeptor α (ERα) Signalweg konnte durch Inhibitions- und Knock-Down-Studien verifiziert werden, während die Modulation des G-Protein-gekoppelten Östrogenrezeptors 1 (GPER1) keinen Einfluss hatte. Es konnte gezeigt werden, dass die AJ-Reorganisation durch das ERα-Zielgen Amphiregulin (AREG) vermittelt wird. Zusätzlich sind der epidermalen Wachstumsfaktor Rezeptor (EGFR) und nachgeschaltete Kinase-Signalwege der RhoA- und Src-Familie beteiligt. Diese krebsrelevanten Signalwege unterstützen die mechanistische und klinische Relevanz der AJ-Organisation als neuen funktionellen Endpunkt im E-Morph-Assay. Die Entwicklung einer Pipeline für die automatisierte Bilderfassung und quantitative Bildanalyse ermöglicht das schnelle Analysieren von Chemikalien. Das Testen von 17 Referenzchemikalien mit bekannten östrogenen Eigenschaften zeigte eine hohe Vorhersagekapazität des E-Morph-Assays

Adaptive monotone multigrid methods for nonlinear variational problems

A wide range of problems occurring in engineering and industry is characterized by the presence of a free (i.e. a priori unknown) boundary where the underlying physical situation is changing in a discontinuous way. Mathematically, such phenomena can be often reformulated as variational inequalities or related non–smooth minimization problems. In these research notes, we will describe a new and promising way of constructing fast solvers for the corresponding discretized problems providing globally convergent iterative schemes with (asymptotic) multigrid convergence speed. The presentation covers physical modelling, existence and uniqueness results, finite element approximation and adaptive mesh–refinement based on a posteriori error estimation. The numerical properties of the resulting adaptive multilevel algorithm are illustrated by typical applications, such as semiconductor device simulation or continuous casting

Monotone multigrid methods for elliptic variational inequalities II

We derive globally convergent multigrid methods for discrete elliptic variational inequalities of the second kind as obtained from the approximation of related continuous problems by piecewise linear finite elements. The coarse grid corrections are computed from certain obstacle problems. The actual constraints are fixed by the preceding nonlinear fine grid smoothing. This new approach allows the implementation as a classical V-cycle and preserves the usual multigrid efficiency. We give 1−O(j−3) estimates for the asymptotic convergence rates. The numerical results indicate a significant improvement as compared with previous multigrid approaches