H-matrix accelerated second moment analysis for potentials with rough correlation

We consider the efficient solution of partial differential equationsfor strongly elliptic operators with constant coefficients and stochastic Dirichlet data by the boundary integral equation method. The computation of the solution's two-point correlation is well understood if the two-point correlation of the Dirichlet data is known and sufficiently smooth.Unfortunately, the problem becomes much more involved in case of rough data. We will show that the concept of the H-matrix arithmetic provides a powerful tool to cope with this problem. By employing a parametric surface representation, we end up with an H-matrix arithmetic based on balanced cluster trees. This considerably simplifies the implementation and improves the performance of the H-matrix arithmetic. Numerical experiments are provided to validate and quantify the presented methods and algorithms

Isogeometric Boundary Elements in Electromagnetism: Rigorous Analysis, Fast Methods, and Examples

We present a new approach to three-dimensional electromagnetic scattering problems via fast isogeometric boundary element methods. Starting with an investigation of the theoretical setting around the electric field integral equation within the isogeometric framework, we show existence, uniqueness, and quasi-optimality of the isogeometric approach. For a fast and efficient computation, we then introduce and analyze an interpolation-based fast multipole method tailored to the isogeometric setting, which admits competitive algorithmic and complexity properties. This is followed by a series of numerical examples of industrial scope, together with a detailed presentation and interpretation of the results

The importance of miR-17~92 during CD28 co-stimulation of murine CD4+ T cells

The prototypic costimulatory molecule CD28 is essential for proper CD4+ T cell activation and initiation of clonal expansion. CD28 ligation regulates metabolic adaptation, the production of cytokines, survival, differentiation but also T follicular helper cell generation and germinal center response. Moreover, CD28 signaling induces the expression of microRNA cluster miR-17~92 during CD4+ T cell activation. However, despite the importance of this receptor, the molecular understanding of how CD28 exerts its function remains incomplete. In this thesis, we extend previous reports by showing that miR-17~92 expression directly correlates with CD4+ T cell activation, and miR-17~92-deficiency phenocopies CD28-deficiency in mice. We therefore hypothesized that transgenic miR-17~92 expression could substitute for the loss of CD28. Using a B6.CD4cre.R26floxstopfloxmiR1792tg.CD28ko mouse model, we demonstrate that transgenic miR-17~92 expression compensates for CD28 expression during CD4+ T cell activation and differentiation in vitro, but also in vivo during acute LCMV infection. Even though many targets of miR-17~92 have been identified so far, the mechanisms by which miR-17~92 contributes to CD4+ T cell activation have not yet been fully explained. We generate transcriptomic datasets from activated CD4+ T cells with distinct amounts of miR-17~92 expression, with which we identify a new list of bona fide canonical miR-17~92 target genes. Furthermore, we demonstrate with a second dataset that these genes are not only regulated by miR-17~92 but also by CD28 expression. This shows that in addition to the activation of transcription during CD28 dependent CD4+ T cell activation, also the repression of genes which is mediated by miR-17~92 is essential. Moreover, the identified target genes mediate a rescue of the CD28ko transcriptome in rescue cells. We furthermore identify a new miR-17 target regulator of calcineurin 3 (RCAN3) among the list of target genes. Our data strongly support a model in which miR-17~92, in addition to known pathways like PI3K, also regulates the NFAT pathway. This qualifies this miRNA cluster as an important regulator of CD28 co-stimulation, which could have broad implications for a better understanding of T cell activation and immunotherapy

Multipatch Approximation of the de Rham Sequence and its Traces in Isogeometric Analysis

We define a conforming B-spline discretisation of the de Rham complex on multipatch geometries. We introduce and analyse the properties of interpolation operators onto these spaces which commute w.r.t. the surface differential operators. Using these results as a basis, we derive new convergence results of optimal order w.r.t. the respective energy spaces and provide approximation properties of the spline discretisations of trace spaces for application in the theory of isogeometric boundary element methods. Our analysis allows for a straightforward generalisation to finite element methods

On uncertainty quantification of eigenvalues and eigenspaces with higher multiplicity

We consider generalized operator eigenvalue problems in variational form with random perturbations in the bilinear forms. This setting is motivated by variational forms of partial differential equations with random input data. The considered eigenpairs can be of higher but finite multiplicity. We investigate stochastic quantities of interest of the eigenpairs and discuss why, for multiplicity greater than 1, only the stochastic properties of the eigenspaces are meaningful, but not the ones of individual eigenpairs. To that end, we characterize the Fr\'echet derivatives of the eigenpairs with respect to the perturbation and provide a new linear characterization for eigenpairs of higher multiplicity. As a side result, we prove local analyticity of the eigenspaces. Based on the Fr\'echet derivatives of the eigenpairs we discuss a meaningful Monte Carlo sampling strategy for multiple eigenvalues and develop an uncertainty quantification perturbation approach. Numerical examples are presented to illustrate the theoretical results

Hierarchical matrix approximation for the uncertainty quantification of potentials on random domains

Computing statistical quantities of interest of the solution of PDE on random domains is an important and challenging task in engineering. We consider the computation of these quantities by the perturbation approach. Especially, we discuss how third order accurate expansions of the mean and the correlation can numerically be computed. These expansions become even fourth order accurate for certain types of boundary variations. The correction terms are given by the solution of correlation equations in the tensor product domain, which can efficiently be computed by means of H -matrices. They have recently been shown to be an efficient tool to solve correlation equations with rough data correlations, that is, with low Sobolev smoothness or small correlation length, in almost linear time. Numerical experiments in three dimensions for higher order ansatz spaces show the feasibility of the proposed algorithm. The application to a non-smooth domain is also included