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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.

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 (41-60 of 146)

  1. 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...

  2. 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...

  3. 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...

  4. A Scalable Algorithm for Inverse Medium Problems with Multiple Sources

    04 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....

  5. 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...

  6. Geometric Multigrid for MHD Simulations with Nedelec Finite Elements on Tetrahedral Grids

    04 Feb 2016 | | Contributor(s):: Chris Hansen

    The Magneto-HydroDynamic (MHD) model is used extensively to simulate macroscopic plasma dynamics in Magnetic Confinement Fusion (MCF) devices. In these simulations, the span of time scales from fast wave dynamics to the desired evolution of equilibrium due to transport processes is large,...

  7. HPGMG: Benchmarking Computers Using Multigrid

    04 Feb 2016 | | Contributor(s):: Jed Brown

    HPGMG (https://hpgmg.org) is a geometric multigrid benchmark designed to measure the performance and versatility of computers. For a benchmark to be representative of applications, good performance on the benchmark should be sufficient to ensure good performance on most important applications and...

  8. Parallel Multigrid Preconditioner Based on Automatic 3D Tetradedric Meshes

    04 Feb 2016 | | Contributor(s):: Frederic Vi

    Multigrid methods are efficient for solving large sparse linear systems. Geometric (GMG) and Algebraic Multigrid (AMG) have both their own benefits and limitations. Combining the simplicity of AMG with the efficiency of GMG lead us to the development of an Hybrid Multigrid preconditionner. From...

  9. Space-time constrained FOSLS with AMGe upscaling

    04 Feb 2016 | | Contributor(s):: Panayot Vassilevski

    We consider time-dependent PDEs discretized in combined space-time domains. We first reduce the PDE to a first order system. Very often in practice, one of the equations of the reduced system involves the divergence operator (in space-time). The popular FOSLS (first order system least-squares)...

  10. Stable Discretizations and Robust Block Preconditioners for Fluid-Structure Interaction Systems

    04 Feb 2016 | | Contributor(s):: Kai Yang

    In our work we develop a family of preconditioners for the linear algebraic systems arising from the arbitrary Lagrangian-Eulerian discretization of some fluid-structure interaction models. After the time discretization, we formulate the fluid-structure interaction equations as saddle point...

  11. Understanding the Propagation of Silent Data Corruption in Algebraic Multigrid

    04 Feb 2016 | | Contributor(s):: Jon Calhoun

    Sparse linear solvers from a fundamental kernel in high performance computing (HPC). Exascale systems are expected to be more complex than systems of today being composed of thousands of heterogeneous processing elements that operate at near-threshold-voltage to meet power constraints. The...

  12. Hao Zhuang

    https://nanohub.org/members/105541

  13. Patrick Mulligan

    https://nanohub.org/members/105370

  14. 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

  15. Integrated Imaging Seminar Series

    30 Apr 2013 | | Contributor(s):: Charles Addison Bouman

    Integrated imaging seminar series is jointly sponsored by the Birck Nanotechnology Center and ECE. Integrated Imaging is defined as a cross-disciplinary field combining sensor science, information processing, and computer systems for the creation of novel imaging and sensing systems. In this...

  16. ECE 595E Lecture 35: MEEP Tutorial I

    18 Apr 2013 | | Contributor(s):: Peter Bermel

    Outline:MEEP InterfacesMEEP ClassesTutorial examples:WaveguideBent waveguide

  17. 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,...

  18. ECE 595 Course Policy - Spring 2013

    03 Jan 2013 | | Contributor(s):: Peter Bermel

    A description of the key policies that will govern the administration of ECE 595 on "Numerical Methods" in Spring 2013.

  19. Sergio Feliciano Mendoza-Barrera

    Industrial and Electronics Engineering. Electronics Department. "Instituto Tecnológico de Puebla", MexicoScience Master in Electronics. Electronics Department. Astrophysics, Optics and Electronics...

    https://nanohub.org/members/62625

  20. The Pioneers of Quantum Computing

    19 Nov 2010 | | Contributor(s):: David P. Di Vincenzo

    This talk profiles the persons whose insights and visions created the subject of quantum information science. Some famous, some not, they all thought deeply about the puzzles and contradictions that were apparent to the founders of quantum theory. After many years of germination, the confluence...