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We present a technique for proving convergence of h and hp adaptive finite element methods through comparison with certain reference refinement schemes based on interpolation error. We then construct a testing environment where properties of different adaptive approaches can be evaluated and improved.

One of the biggest challenges in high-energy physics is to analyze a complex mix of experimental and simulation data to gain new insights into the underlying physics. Currently, this analysis relies primarily on the intuition of trained experts often using nothing more sophisticated than default scatter plots. Many advanced analysis techniques are not easily accessible to scientists and not flexible enough to explore the potentially interesting hypotheses in an intuitive manner. Furthermore, results from individual techniques are often difficult to integrate, leading to a confusing patchwork of analysis snippets too cumbersome for data exploration. This paper presents a case study on how a combination of techniques from statistics, machine learning, topology, and visualization can have a significant impact in the field of inertial confinement fusion. We present the $$\mathrm{ND}^2\mathrm{AV}$$ : N-dimensional data analysis and visualization framework, a user-friendly tool aimed at exploiting the intuition and current workflow of the target users. The system integrates traditional analysis approaches such as dimension reduction and clustering with state-of-the-art techniques such as neighborhood graphs and topological analysis, and custom capabilities such as defining combined metrics on the fly. All components are linked into an interactive environment that enables an intuitive exploration of a wide variety of hypotheses while relating the results to concepts familiar to the users, such as scatter plots. $$\mathrm{ND}^2\mathrm{AV}$$ uses a modular design providing easy extensibility and customization for different applications. $$\mathrm{ND}^2\mathrm{AV}$$ is being actively used in the National Ignition Campaign and has already led to a number of unexpected discoveries.

We present a Lagrangian data assimilation experiment in an open channel flow above a broad-crested weir. The observations consist of trajectories of particles transported by the flow and extracted from a video film, in addition to classical water level measurements. However, the presence of vertical recirculations on both sides of the weir actually conducts to the identification of an equivalent topography corresponding to the lower limit of a surface jet. In addition, results on the identification of the Manning coefficient may allow to detect the presence of bottom recirculations.

We consider the problem of uncertainty quantification for extreme scale parameter dependent problems where an underlying low rank property of the parameter dependency is assumed. For this type of dependency the hierarchical Tucker format offers a suitable framework to approximate a given output function of the solutions of the parameter dependent problem from a number of samples that is linear in the number of parameters. In particular we can a posteriori compute the mean, variance or other interesting statistical quantities of interest. In the extreme scale setting it is already assumed that the underlying fixed-parameter problem is distributed and solved for in parallel. We provide in addition a parallel evaluation scheme for the sampling phase that allows us on the one hand to combine several solves and on the other hand parallelise the sampling.

In this paper a model for the electrical activity of the heart will be presented and an approach to solve the resulting numerical problem will be suggested. The Bidomain Model is used to compute the spatial distribution of the electrical potential. The partial differential equations are discretized with the finite element method and the multigrid method is used to solve the corresponding linear equations. Adaptivity is applied to resolve the steep gradients in the solution.

We propose a linear finite-element discretization of Dirichlet problems for static Hamilton–Jacobi equations on unstructured triangulations. The discretization is based on simplified localized Dirichlet problems that are solved by a local variational principle. It generalizes several approaches known in the literature and allows for a simple and transparent convergence theory. In this paper the resulting system of nonlinear equations is solved by an adaptive Gauss–Seidel iteration that is easily implemented and quite effective as a couple of numerical experiments show.

In this paper, we will present efficient strategies how composite finite elements can be realized for the discretization of PDEs on domains containing small geometric details. In contrast to standard finite elements, the minimal dimension of this new class of finite element spaces is completely independent of the number of geometric details of the physical domains. Hence, it allows coarse level discretization of PDEs which can be used, e.g., preferably for multi-grid methods and homogenization of PDEs in non-periodic situations.