Splitting method for elliptic equations with line sources

In this paper, we study the mathematical structure and numerical approximation of elliptic problems posed in a (3D) domain $\Omega$ when the right-hand side is a (1D) line source $\Lambda$. The analysis and approximation of such problems is known to be non-standard as the line source causes the solution to be singular. Our main result is a splitting theorem for the solution; we show that the solution admits a split into an explicit, low regularity term capturing the singularity, and a high-regularity correction term $w$ being the solution of a suitable elliptic equation. The splitting theorem states the mathematical structure of the solution; in particular, we find that the solution has anisotropic regularity. More precisely, the solution fails to belong to $H^1$ in the neighbourhood of $\Lambda$, but exhibits piecewise $H^2$-regularity parallel to $\Lambda$. The splitting theorem can further be used to formulate a numerical method in which the solution is approximated via its correction function $w$. This approach has several benefits. Firstly, it recasts the problem as a 3D elliptic problem with a 3D right-hand side belonging to $L^2$, a problem for which the discretizations and solvers are readily available. Secondly, it makes the numerical approximation independent of the discretization of $\Lambda$; thirdly, it improves the approximation properties of the numerical method. We consider here the Galerkin finite element method, and show that the singularity subtraction then recovers optimal convergence rates on uniform meshes, i.e., without needing to refine the mesh around each line segment. The numerical method presented in this paper is therefore well-suited for applications involving a large number of line segments. We illustrate this by treating a dataset (consisting of $\sim 3000$ line segments) describing the vascular system of the brain

Research and Education in Computational Science and Engineering

Over the past two decades the field of computational science and engineering (CSE) has penetrated both basic and applied research in academia, industry, and laboratories to advance discovery, optimize systems, support decision-makers, and educate the scientific and engineering workforce. Informed by centuries of theory and experiment, CSE performs computational experiments to answer questions that neither theory nor experiment alone is equipped to answer. CSE provides scientists and engineers of all persuasions with algorithmic inventions and software systems that transcend disciplines and scales. Carried on a wave of digital technology, CSE brings the power of parallelism to bear on troves of data. Mathematics-based advanced computing has become a prevalent means of discovery and innovation in essentially all areas of science, engineering, technology, and society; and the CSE community is at the core of this transformation. However, a combination of disruptive developments---including the architectural complexity of extreme-scale computing, the data revolution that engulfs the planet, and the specialization required to follow the applications to new frontiers---is redefining the scope and reach of the CSE endeavor. This report describes the rapid expansion of CSE and the challenges to sustaining its bold advances. The report also presents strategies and directions for CSE research and education for the next decade.Comment: Major revision, to appear in SIAM Revie

Calibration to American options: numerical investigation of the de-Americanization method.

American options are the reference instruments for the model calibration of a large and important class of single stocks. For this task, a fast and accurate pricing algorithm is indispensable. The literature mainly discusses pricing methods for American options that are based on Monte Carlo, tree and partial differential equation methods. We present an alternative approach that has become popular under the name de-Americanization in the financial industry. The method is easy to implement and enjoys fast run-times (compared to a direct calibration to American options). Since it is based on ad hoc simplifications, however, theoretical results guaranteeing reliability are not available. To quantify the resulting methodological risk, we empirically test the performance of the de-Americanization method for calibration. We classify the scenarios in which de-Americanization performs very well. However, we also identify the cases where de-Americanization oversimplifies and can result in large errors

Landwirt*innen nutzen Citizen Science Praktiken zur Unterstützung der Artenvielfalt am Land

Das EU-Projekt FRAMEwork fördert bottom-up Innovationen getragen von Landwirt*innen via elf "Farmer Cluster" in neun Ländern Europas. Anhand zweier Farmer Cluster im Mostviertel und im Burgenland in Österreich veranschaulicht das Poster zwei komplementäre Citizen Science Praktiken, die einen evidenzbasierten, lokal eingebetteten, gemeinschaftlichen Ansatz zum Schutz und zur Verbesserung der biologischen Vielfalt unterstützen und strukturiertes Monitoring mit adaptiven Landmanagementpraktiken kombinieren

A monotone multigrid solver for two body contact problems in biomechanics

The purpose of the paper is to apply monotone multigrid methods to static and dynamic biomechanical contact problems. In space, a finite element method involving a mortar discretization of the contact conditions is used. In time, a new contact-stabilized Newmark scheme is presented. Numerical experiments for a two body Hertzian contact problem and a biomechanical application are reported

An X-Ray Absorption Spectroscopy Study of Ball-Milled Lithium Tantalate and Lithium Titanate Nanocrystals

Previous work has shown that nanocrystalline samples of lithium tantalate and titanate prepared by high-energy milling show unusually high lithium ion conductivity. Here, we report an X-ray absorption spectroscopy (XAS) study at the Ti K-edge and the Ta L3 edge of samples that have been milled for various lengths of time. For both systems the results show that milling creates amorphous material whose quantity increases with the milling time. The more extensive data for the tantalate shows that milling for only 30 minutes generates ~25% amorphous content in the sample. The content rises to ~60% after 16 hours. It is suggested that it is the motion of the lithium ions through the amorphous content that provides the mechanism for the high ionic conductivity