Tags: algorithms

Description

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.

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.

All Categories (1-20 of 146)

  1. 2004 Computational Materials Science Summer School

    29 Aug 2005

    This short course will explore a range of computational approaches relevant for nanotechnology.

  2. How to ensure that the stiffness matrix is square?

    Q&A|Closed | Responses: 1

    Every time I run the solver, it stops giving an error that the stiffness matrix is not square (though it is symmetric). What should I do to ensure that the stiffness matrix is square so as to...

    https://nanohub.org/answers/question/49

  3. 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...

  4. A Fast Multigrid Approach for Solving the Helmholtz Equation with a Point Source

    04 Feb 2016 | | Contributor(s):: Eran Treister

    Solving the discretized Helmholtz equations with high wave numbers in large dimensions is a challenging task. However, in many scenarios, the solution of these equations is required for a point source. In this case, the problem can be be reformulated and split into two parts: one in a solution of...

  5. A Massively Parallel Semicoarsening Multigrid for 3D Reservoir Simulation on Multi-core and Multi-GPU Architectures

    04 Feb 2016 | | Contributor(s):: Abdulrahman Manea

    In this work, we have designed and implemented a massively parallel version of the Semicoarsening Black Box Multigrid Solver [1], which is capable of handling highly heterogeneous and anisotropic 3D reservoirs, on a parallel architecture with multiple GPU’s. For comparison purposes, the...

  6. A Performance Comparison of Algebraic Multigrid Preconditioners on GPUs and MIC

    04 Feb 2016 | | Contributor(s):: Karl Rupp

    Algebraic multigrid (AMG) preconditioners for accelerators such as graphics processing units (GPUs) and Intel's many-integrated core (MIC) architecture typically require a careful, problem-dependent trade-off between efficient hardware use, robustness, and convergence rate in order to...

  7. A Primer on Semiconductor Device Simulation

    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...

  8. 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....

  9. Advances in Computational and Quantum Imaging Workshop

    08 Jan 2020 |

    The purpose of the workshop is to bring different communities together, review recent theoretical and experimental advances and explore synergetic collaborations. The workshop aligns well with the significant investments in quantum technologies through the National Quantum Initiative in the...

  10. 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...

  11. 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...

  12. 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...

  13. Big Data in Reliability and Security: Applications

    29 May 2019 | | Contributor(s):: Saurabh Bagchi

  14. Big Data in Reliability and Security: Some Basics

    29 May 2019 | | Contributor(s):: Saurabh Bagchi

  15. 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.

  16. 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...

  17. 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.

  18. Compatible Relaxation Based Geometric-Algebraic Multigrid

    04 Feb 2016 | | Contributor(s):: Fei Cao

    We develop compatible relaxation algorithms for smoothed aggregation-based multigrid coarsening. In the proposed method, we use the geometry of the given discrete problem on the finest level to coarsen the system together with compatible relaxation to from the sparsity structure of the...

  19. Computational Mathematics: Role, Impact, Challenges

    20 Dec 2007 | | Contributor(s):: Juan C. Meza

    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.

  20. Computational Methods for NEMS

    16 Jun 2004 | | Contributor(s):: Narayan Aluru

    Computational Methods for NEMS