In this paper, we study ill-posedness concepts of nonlinear and linear inverse problems in a Hilbert space setting. We define local ill-posedness of a nonlinear operator equation $F(x) = y_0$ in a solution point $x_0$ and the interplay between the nonlinear problem and its linearization using the Frechet derivative $F\acent(x_0)$ . To find an appropriate ill-posedness concept for the linarized equation we define intrinsic ill-posedness for linear operator equations $Ax = y$ and compare this approach with the ill-posedness definitions due to Hadamard and Nashed.

The paper is concerned with the finite element resolution of layers appearing in singularly perturbed problems. A special anisotropic grid of Shishkin type is constructed for reaction diffusion problems. Estimates of the finite element error in the energy norm are derived for two methods, namely the standard Galerkin method and a stabilized Galerkin method. The estimates are uniformly valid with respect to the (small) diffusion parameter. One ingredient is a pointwise description of derivatives of the continuous solution. A numerical example supports the result. Another key ingredient for the error analysis is a refined estimate for (higher) derivatives of the interpolation error. The assumptions on admissible anisotropic finite elements are formulated in terms of geometrical conditions for triangles and tetrahedra. The application of these estimates is not restricted to the special problem considered in this paper.

This paper is concerned with the numerical analysis of the autoconvolution equation $x*x=y$ restricted to the interval [0,1]. We present a discrete constrained least squares approach and prove its convergence in $L^p(0,1),1<p<\infinite$ , where the regularization is based on a prescribed bound for the total variation of admissible solutions. This approach includes the case of non-smooth solutions possessing jumps. Moreover, an adaption to the Sobolev space $H^1(0,1)$ and some remarks on monotone functions are added. The paper is completed by a numerical case study concerning the determination of non-monotone smooth and non-smooth functions x from the autoconvolution equation with noisy data y.

The paper deals with tau-extrapolation - a modification of the multigrid method, which leads to solutions with an improved con- vergence order. The number of numerical operations depends linearly on the problem size and is not much higher than for a multigrid method without this modification. The paper starts with a short mathematical foundation of the tau-extrapolation. Then follows a careful tuning of some multigrid components necessary for a successful application of tau-extrapolation. The next part of the paper presents numerical illustrations to the theoretical investigations for one- dimensional test problems. Finally some experience with the use of tau-extrapolation for the Navier-Stokes equations is given.

The numerical solution of the convection-diffusion-reaction problem is considered in two and three dimensions. A stabilized finite element method of Galerkin/Least squares type accomodates diffusion-dominated as well as convection- and/or reaction- dominated situations. The resolution of boundary layers occuring in the singularly perturbed case is accomplished using anisotropic mesh refinement in boundary layer regions. In this paper, the standard analysis of the stabilized Galerkin method on isotropic meshes is extended to more general meshes with boundary layer refinement. Simplicial Lagrangian elements of arbitrary order are used.

The investigation of new parallel algorithms for MIMD computers requires some postprocessing facilities for quickly evaluating the behavior of those algorithms We present two kinds of visualization tool implementations for 2D and 3D finite element applications to be used on a parallel computer and a host workstation.

Some boundary value problems yield anisotropic solutions, e.g. solutions with boundary layers. If such problems are to be solved with the finite element method (FEM), anisotropically refined meshes can be advantageous. In order to construct these meshes or to control the error one aims at reliable error estimators. For \emph{isotropic} meshes many estimators are known, but they either fail when used on \emph{anisotropic} meshes, or they were not applied yet. For rectangular (or cuboidal) anisotropic meshes a modified error estimator had already been found. We are investigating error estimators on anisotropic tetrahedral or triangular meshes because such grids offer greater geometrical flexibility. For the Poisson equation a residual error estimator, a local Dirichlet problem error estimator, and an $L_2$ error estimator are derived, respectively. Additionally a residual error estimator is presented for a singularly perturbed reaction diffusion equation. It is important that the anisotropic mesh corresponds to the anisotropic solution. Provided that a certain condition is satisfied, we have proven that all estimators bound the error reliably.

In this thesis we make several contributions to the theory of moduli spaces of smooth manifolds, especially in dimension two. In Chapter 2 (joint with Soren Galatius) we give a new geometric proof of a generalisation of the Madsen-Weiss theorem, which does not rely on the tangential structure under investigation having homological stability. This allows us to compute the stable homology of moduli spaces of surfaces equipped with many different tangential structures. In Chapter 3 we give a general approach to homological stability problems, especially focused on stability for moduli spaces of surfaces with tangential structure. We give a sufficient condition for a structure to exhibit homological stability, and thus obtain stability ranges for many tangential structures of current interest (orientations, maps to a simply-connected background space, etc.), which match or improve the previously known ranges in all cases. In Chapter 4 we define and study the cobordism category of submanifolds of a fixed background manifold, and extend the work of Galatius-Madsen-Tillmann-Weiss to identify the homotopy type of these categories. We describe several applications of this theory. In Chapter 5 we compute the stable (co)homology of the non-orientable mapping class group, and find a family of geometrically-defined torsion cohomology classes. This is in contrast to the oriented mapping class group, where few are known. In Chapter 6 (joint with Johannes Ebert) we study the divisibility of certain characteristic classes of bundles of unoriented surfaces introduced by Wahl, analogues of the Miller-Morita-Mumford classes for unoriented surfaces. We show them to be indivisible in the free quotient of cohomology.

It is well known that the n-th eigenfunction to one-dimensional Sturm-Liouville eigenvalue problems has exactly n-1 nodes, i.e. non-degenerate zeros. For higher dimensions, it is much more complicated to obtain general statements on the zeros of eigenfunctions. The author states a new conjecture on the number of nodal domains of spherical harmonics, i.e. of connected components of S^2 \ N(u) with the nodal set N(u) = (x in S^2 : u(x) = 0) of the eigenfunction u, and proves it for the first six eigenvalues. It is a sharp upper bound, thus improving known bounds as the Courant nodal domain theorem, see S. Y. Cheng, Comment. Math. Helv. 51, 43-55 (1976; Zbl 334.35022). The proof uses facts on real projective plane algebraic curves (see D. A. Gudkov, Usp. Mat. Nauk 29(4), 3-79, Russian Math. Surveys 29(4), 1-79 (1979; Zbl 316.14018)), because they are the zero sets of homogeneous polynomials, and the spherical harmonics are the restrictions of spherical harmonic homogeneous polynomials in the space to the plane. / Series: Preprint Series / Department of Applied Statistics and Data Processing

INTRODUCTION: Stem cells have become promising treatments for osteoarthritis due to the cells’ ability to regenerate cartilage and availability from bone marrow. Various studies have established the chondrogenic potential of human marrow stromal cells (hMSCs) upon treatment with transforming growth factor β1 (TGF-β1), yet the difference in potential between cells derived from young subjects and those derived from elder subjects has not been confirmed. OBJECTIVES: This study seeks to establish whether the chondrogenic potential of hMSCs changes with age and sex. This study used a high-density 2D model to measure the acute response of hMSCs to chondrogenic induction over a short time course and various treatment levels. The experiments investigated the expression of chondrogenic genes and expression of TGF-β1 receptors (ALK5) in hMSCs after TGF-β1 treatment to determine whether pediatric hMSCs have more potential for chondrocyte differentiation than adult hMSCs. METHODS: With IRB approval, nine bone marrow samples were obtained from discarded tissue of adults undergoing total hip replacement and juveniles requiring bone graft for alveolar cleft repair. Subject ages ranged from age 8 to 66. Low-density mononucleated cells were cultured in plastic tissue culture dishes. Adherent hMSCs were expanded in monolayer culture with phenol red-free α-MEM medium with 10% fetal bovine serum. After 48 hours of treatment with TGF-β1, cells were collected for RNA extraction and RT-PCR analysis of chondrogenic genes and TGF-β1 receptor levels. Alcian blue staining in 24-well plates of hMSCs was performed after 10 days to compare the effects of different concentrations of TGF-β1, and the effects of another inducer of chondrogenesis, kartogenin (KGN) on matrix accumulation. RESULTS: Gel electrophoresis of PCR products revealed no consistent trend in chondrogenic mRNA expression in pediatric cells compared to adult cells, or female cells compared to male cells. The data indicate that the change in chondrogenic potential of hMSCs with age and sex is inconsistent. KGN showed no consistent effect on hMSCs. Cells with high baseline levels of TGF-β1 receptor (ALK5) showed no upregulation of ALK5 after TGF-β1 treatment, while samples with low basal expression of TGF-β1 receptors showed upregulation after TGF-β1 treatment. CONCLUSIONS: There is still much debate in the literature regarding the potential of adult hMSC chondrogenesis compared to juveniles. This study confirms the irreproducibility of displaying differences between young and adult hMSCs. A larger sample size is needed to establish a correlation between age and chondrogenic potential. Further in vitro studies will consider the optimum time course and concentration of TGF-β1 to observe differences in gene expression of cells, and will identify other clinical determinants of differentiation potential.