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Whether you're simulating the electronic structure of a carbon nanotube or the strain within an automobile part, the
calculations usually boil down to a simple matrix equation,
Ax = f. The faster you can fill the
matrix A with the coefficients for your partial
differential equation (PDE), and the faster you can solve for
the vector x given a forcing function f, the faster you have your overall solution. Things get interesting when the matrix A is too large to fit in the memory available on one machine, or when the coefficients in A cause the matrix to be ill-conditioned.
Ax = f
Many different algorithms have been developed to map a PDE onto a matrix, to pre-condition the matrix to a better form, and to solve the matrix with blinding speed. Different algorithms usually exploit some property of the matrix, such as symmetry, to reduce either memory requirements or solution speed or both.
Learn more about algorithms from the many resources on this site, listed below.
A Distributed Algorithm for Computing a Common Fixed Point of a Family of Paracontractions
21 Jun 2017 | | Contributor(s):: A. Stephen Morse
In this talk a distributed algorithm is described for finding a common fixed point of a family of m paracontractions assuming that such a common fixed point exists. The common fixed point is simultaneously computed by m agents assuming each agent knows only paracontraction, the current estimates...
A Primer on Semiconductor Device Simulation
out of 5 stars
23 Jan 2006 | | Contributor(s):: Mark Lundstrom
Computer simulation is now an essential tool for the research and development of semiconductor processes and devices, but to use a simulation tool intelligently, one must know what's "under the hood." This talk is a tutorial introduction designed for someone using semiconductor device simulation...
A Scalable Algorithm for Inverse Medium Problems with Multiple Sources
02 Feb 2016 | | Contributor(s):: Keith Kelly
We consider the problem of acoustic scattering as described by the free-space, time-harmonic scalar wave equation given by (0.1) along with radiation boundary conditions. Here, is a point in , is the source term, and is the wavenumber. Our formulation is based on potential theory....
An Introduction to Quantum Computing
12 Sep 2008 | | Contributor(s):: Edward Gerjuoy
Quantum mechanics, as formulated more than 80 years ago by Schrodinger, Heisenberg, Dirac and other greats, is a wholly sufficient foundation for its modern interrelated subfields of quantum computation (qc) and quantum information (qi), which generally are lumped together into a single subfield...
Autonomic Adaptation of Virtual Distributed Environments in a Multi-Domain Infrastructure
11 Jul 2006 | | Contributor(s):: Ryan Riley, Dongyan Xu
By federating resources from multiple domains, a shared infrastructure provides aggregated computation resources to a large number of users. With rapid advances in virtualization technologies, we propose the concept of virtual distributed environments as a new sharing paradigm for a multi-domain...
Bandstructure in Nanoelectronics
01 Nov 2005 | | Contributor(s):: Gerhard Klimeck
This presentation will highlight, for nanoelectronic device examples, how the effective mass approximation breaks down and why the quantum mechanical nature of the atomically resolved material needs to be included in the device modeling. Atomistic bandstructure effects in resonant tunneling...
BNC Annual Research Review: An Introduction to PRISM and MEMS Simulation
04 Jun 2008 | | Contributor(s):: Jayathi Murthy
This presentation is part of a collection of presentations describing the projects, people, and capabilities enhanced by research performed in the Birck Center, and a look at plans for the upcoming year.
Calculating Resonances Using a Complex Absorbing Potential
13 Mar 2008 | | Contributor(s):: Robin Santra
The Siegert (or Gamow) wave function associated with a resonance state is exponentially divergent at large distances from the scattering target. A complex absorbing potential (CAP) provides a computationally simple and efficient technique for calculating the complex Siegert energy of a resonance...
Challenges and Strategies for High End Computing
20 Dec 2007 | | Contributor(s):: Katherine A. Yelick
This presentation was one of 13 presentations in the one-day forum, "Excellence in Computer Simulation," which brought together a broad set of experts to reflect on the future of computational science and engineering.
Computational Mathematics: Role, Impact, Challenges
20 Dec 2007 | | Contributor(s):: Juan C. Meza
Computational Methods for NEMS
16 Jun 2004 | | Contributor(s):: Narayan Aluru
Computational Methods for NEMS
Computing the Horribleness of Soft Condensed Matter
19 Oct 2007 | | Contributor(s):: Eric Jakobsson
A great triumph of computer simulations 40 years ago was to make the liquid state of matter understandable in terms of physical interactions between individual molecules. Prior to the first simulations of liquid argon and liquid water in the 1960's, there was no quantitatively rigorous molecular...
Data-adaptive Filtering and the State of the Art in Image Processing
15 Apr 2013 | | Contributor(s):: Peyman Milanfar
In this talk, I will present a practical and unified framework for understanding some common underpinnings of these methods. This leads to new insights and a broad understanding of how these diverse methods interrelate. I will also discuss the statistical performance of the resulting algorithms,...
Data-Centric Models for Multilevel Algorithms
07 Feb 2016 | | Contributor(s):: Samuel Guiterrez
Today, computational scientists must contend with a diverse set of supercomputer architectures that are capable of exposing unprecedented levels of parallelism and complexity. Effectively placing, moving, and operating on data residing in complex distributed memory hierarchies is quickly...
ECE 595E Lecture 35: MEEP Tutorial I
18 Apr 2013 | | Contributor(s):: Peter Bermel
Outline:MEEP InterfacesMEEP ClassesTutorial examples:WaveguideBent waveguide
ECE 595E Lecture 36: MEEP Tutorial II
30 Apr 2013 | | Contributor(s):: Peter Bermel
Outline:Recap from MondayExamplesMultimode ring resonatorsIsolating individual resonancesKerr nonlinearitiesQuantifying third-harmonic generation
ECE 695NS Lecture 2: Computability and NP-hardness
13 Jan 2017 | | Contributor(s):: Peter Bermel
Outline:OverviewDefinitionsComputing MachinesChurch-Turing ThesisPolynomial Time (Class P)Class NPNon-deterministic Turing machinesReducibilityCook-Levin theoremCoping with NP Hardness
ECE 695NS Lecture 3: Practical Assessment of Code Performance
25 Jan 2017 | | Contributor(s):: Peter Bermel
Outline:Time ScalingExamplesGeneral performance strategiesComputer architecturesMeasuring code speedReduce strengthMinimize array writesProfiling
Experiences with nonintrusive polynomial Chaos and stochastic collocation methods for uncertainty analysis and design
13 Mar 2009 | | Contributor(s):: Michael S. Eldred
Non—intrusive polynomial chaos expansion (PCE) and stochastic collocation (SC) methods are attractive techniques for uncertainty quantification due to their abilities to produce functional representations of stochastic variability and to achieve exponential convergence rates in statistics of...
Experiment vs. Modelling: What's the problem?
10 Aug 2009 | | Contributor(s):: William L. Barnes
Progress in plasmonics has been greatly assisted by developments in experimental techniques and in numerical modelling. This talk will look at some of the difficulties that emerge when comparisons are made between experiment and theory. Through the use of four examples I will illustrate what...