219 research outputs found
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 when the
right-hand side is a (1D) line source . 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 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 in the neighbourhood of , but exhibits
piecewise -regularity parallel to . The splitting theorem can
further be used to formulate a numerical method in which the solution is
approximated via its correction function . This approach has several
benefits. Firstly, it recasts the problem as a 3D elliptic problem with a 3D
right-hand side belonging to , a problem for which the discretizations and
solvers are readily available. Secondly, it makes the numerical approximation
independent of the discretization of ; 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
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 UnterstĂŒtzung der Artenvielfalt am Land
Das EU-Projekt FRAMEwork fördert bottom-up Innovationen getragen von Landwirt*innen via elf "Farmer Cluster" in neun LĂ€ndern Europas. Anhand zweier Farmer Cluster im Mostviertel und im Burgenland in Ăsterreich veranschaulicht das Poster zwei komplementĂ€re Citizen Science Praktiken, die einen evidenzbasierten, lokal eingebetteten, gemeinschaftlichen Ansatz zum Schutz und zur Verbesserung der biologischen Vielfalt unterstĂŒtzen und strukturiertes Monitoring mit adaptiven Landmanagementpraktiken kombinieren
MIXED FINITE ELEMENT METHODS: IMPLEMENTATION WITH ONE UNKNOWN PER ELEMENT, LOCAL FLUX EXPRESSIONS, POSITIVITY, POLYGONAL MESHES, AND RELATIONS TO OTHER METHODS
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
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