A port-reduced static condensation reduced basis element method for large component-synthesized structures: approximation and A Posteriori error estimation

Background: We consider a static condensation reduced basis element framework for efficient approximation of parameter-dependent linear elliptic partial differential equations in large three-dimensional component-based domains. The approach features an offline computational stage in which a library of interoperable parametrized components is prepared; and an online computational stage in which these component archetypes may be instantiated and connected through predefined ports to form a global synthesized system. Thanks to the component-interior reduced basis approximations, the online computation time is often relatively small compared to a classical finite element calculation. Methods: In addition to reduced basis approximation in the component interiors, we employ in this paper port reduction with empirical port modes to reduce the number of degrees of freedom on the ports and thus the size of the Schur complement system. The framework is equipped with efficiently computable a posteriori error estimators that provide asymptotically rigorous bounds on the error in the approximation with respect to the underlying finite element discretization. We extend our earlier approach for two-dimensional scalar problems to the more demanding three-dimensional vector-field case. Results and Conclusions: This paper focuses on linear elasticity analysis for large structures with tens of millions of finite element degrees of freedom. Through our procedure we effectively reduce the number of degrees of freedom to a few thousand, and we demonstrate through extensive numerical results for a microtruss structure that our approach provides an accurate, rapid, and a posteriori verifiable approximation for relevant large-scale engineering problems.Research Council of NorwayUnited States. Office of Naval Research (ONR Grant N00014-11-0713

Port reduction in parametrized component static condensation: approximation and a posteriori error estimation

We introduce a port (interface) approximation and a posteriori error bound framework for a general component-based static condensation method in the context of parameter-dependent linear elliptic partial differential equations. The key ingredients are as follows: (i) efficient empirical port approximation spaces—the dimensions of these spaces may be chosen small to reduce the computational cost associated with formation and solution of the static condensation system; and (ii) a computationally tractable a posteriori error bound realized through a non-conforming approximation and associated conditioner—the error in the global system approximation, or in a scalar output quantity, may be bounded relatively sharply with respect to the underlying finite element discretization. Our approximation and a posteriori error bound framework is of particular computational relevance for the static condensation reduced basis element (SCRBE) method. We provide several numerical examples within the SCRBE context, which serve to demonstrate the convergence rate of our port approximation procedure as well as the efficacy of our port reduction error bounds.Research Council of NorwayUnited States. Office of Naval Research (Grant N00014-11-0713

Approximation of Parametric Derivatives by the Empirical Interpolation Method

We introduce a general a priori convergence result for the approximation of parametric derivatives of parametrized functions. We consider the best approximations to parametric derivatives in a sequence of approximation spaces generated by a general approximation scheme, and we show that these approximations are convergent provided that the best approximation to the function itself is convergent. We also provide estimates for the convergence rates. We present numerical results with spaces generated by a particular approximation scheme—the Empirical Interpolation Method—to confirm the validity of the general theory

Reduced Basis Methods for Partial Differential Equations: Evaluation of multiple non-compliant flux-type output functionals for a non-affine electrostatics problem

A method for rapid evaluation of flux-type outputs of interest from solutions to partial differential equations (PDEs) is presented within the reduced basis framework for linear, elliptic PDEs. The central point is a Neumann-Dirichlet equivalence that allows for evaluation of the output through the bilinear form of the weak formulation of the PDE. Through a comprehensive example related to electrostatics, we consider multiple outputs, a posteriori error estimators and empirical interpolation treatment of the non-affine terms in the bilinear form. Together with the considered Neumann-Dirichlet equivalence, these methods allow for efficient and accurate numerical evaluation of a relationship mu->s(mu), where mu is a parameter vector that determines the geometry of the physical domain and s(mu) is the corresponding flux-type output matrix of interest. As a practical application, we lastly employ the rapid evaluation of s-> s(mu) in solving an inverse (parameter-estimation) problem

Reduced basis methods for parametrized partial differential equations

PhD i matematikkPhD in Mathematic

A Two-Step Certified Reduced Basis Method

In this paper we introduce a two-step Certified Reduced Basis (RB) method. In the first step we construct from an expensive finite element “truth” discretization of dimension N an intermediate RB model of dimension N≪N . In the second step we construct from this intermediate RB model a derived RB (DRB) model of dimension M≤N. The construction of the DRB model is effected at cost O(N) and in particular at cost independent of N ; subsequent evaluation of the DRB model may then be effected at cost O(M) . The DRB model comprises both the DRB output and a rigorous a posteriori error bound for the error in the DRB output with respect to the truth discretization. The new approach is of particular interest in two contexts: focus calculations and hp-RB approximations. In the former the new approach serves to reduce online cost, M≪N: the DRB model is restricted to a slice or subregion of a larger parameter domain associated with the intermediate RB model. In the latter the new approach enlarges the class of problems amenable to hp-RB treatment by a significant reduction in offline (precomputation) cost: in the development of the hp parameter domain partition and associated “local” (now derived) RB models the finite element truth is replaced by the intermediate RB model. We present numerical results to illustrate the new approach.United States. Air Force Office of Scientific Research (AFOSR Grant number FA9550-07-1-0425)United States. Department of Defense. Office of the Secretary of Defense (OSD/AFOSR Grant number FA9550-09-1-0613)Norwegian University of Science and Technolog

Comparison of some Reduced Representation Approximations

In the field of numerical approximation, specialists considering highly complex problems have recently proposed various ways to simplify their underlying problems. In this field, depending on the problem they were tackling and the community that are at work, different approaches have been developed with some success and have even gained some maturity, the applications can now be applied to information analysis or for numerical simulation of PDE's. At this point, a crossed analysis and effort for understanding the similarities and the differences between these approaches that found their starting points in different backgrounds is of interest. It is the purpose of this paper to contribute to this effort by comparing some constructive reduced representations of complex functions. We present here in full details the Adaptive Cross Approximation (ACA) and the Empirical Interpolation Method (EIM) together with other approaches that enter in the same category

Heat flow and calculus on metric measure spaces with Ricci curvature bounded below - the compact case

We provide a quick overview of various calculus tools and of the main results concerning the heat flow on compact metric measure spaces, with applications to spaces with lower Ricci curvature bounds. Topics include the Hopf-Lax semigroup and the Hamilton-Jacobi equation in metric spaces, a new approach to differentiation and to the theory of Sobolev spaces over metric measure spaces, the equivalence of the L^2-gradient flow of a suitably defined "Dirichlet energy" and the Wasserstein gradient flow of the relative entropy functional, a metric version of Brenier's Theorem, and a new (stronger) definition of Ricci curvature bound from below for metric measure spaces. This new notion is stable w.r.t. measured Gromov-Hausdorff convergence and it is strictly connected with the linearity of the heat flow.Comment: To the memory of Enrico Magenes, whose exemplar life, research and teaching shaped generations of mathematician

Goats are able to adapt to virtual fencing; A field study in commercial goat herds on Norwegian farms

publishedVersio

Interleukin-8 is the single most up-regulated gene in whole genome profiling of H. pylori exposed gastric epithelial cells

<p>Abstract</p> <p>Background</p> <p>The association between <it>Helicobacter pylori </it>infection and upper gastrointestinal disease is well established. However, only a small fraction of <it>H. pylori </it>carriers develop disease, and there are great geographical differences in disease penetrance. The explanation to this enigma lies in the interaction between the bacterium and the host. <it>H. pylori </it>Outer Membrane Phospholipase A (OMPLA) has been suggested to play a role in the virulence of this bacterium. The aim of this study was to profile the most significant cellular pathways and biological processes affected in gastric epithelial cells during 24 h of <it>H. pylori </it>exposure, and to study the inflammatory response to OMPLA⁺and OMPLA^-<it>H. pylori </it>variants.</p> <p>Results</p> <p>Interleukin-8 was the most significantly up-regulated gene and appears to play a paramount role in the epithelial cell response to <it>H. pylori </it>infection and in the pathological processes leading to gastric disease. MAPK and NF-kappaB cellular pathways were powerfully activated, but did not seem to explain the impressive <it>IL-8 </it>response. There was marked up-regulation of <it>TP53BP2</it>, whose corresponding protein ASPP2 may interact with <it>H. pylori </it>CagA and cause marked p53 suppression of apoptosis. Other regulators of apoptosis also showed abberant regulation. We also identified up-regulation of several oncogenes and down-regulation of tumor suppressor genes as early as during the first 24 h of infection. <it>H. pylori </it>OMPLA phase variation did not seem to influence the inflammatory epithelial cell gene response in this experiment.</p> <p>Conclusion</p> <p>In whole genome analysis of the epithelial response to <it>H. pylori </it>exposure, <it>IL-8 </it>demonstrated the most marked up-regulation, and was involved in many of the most important cellular response processes to the infection. There was dysregulation of apoptosis, tumor suppressor genes and oncogenes as early as in the first 24 h of <it>H. pylori </it>infection, which may represent early signs of gastric tumorigenesis. OMPLA^+/-did not affect the acute inflammatory response to <it>H. pylori</it>.</p