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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.

Resources (41-60 of 135)

  1. HPGMG: Benchmarking Computers Using Multigrid

    Online Presentations | 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...

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

    Online Presentations | 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...

  3. Compatible Relaxation Based Geometric-Algebraic Multigrid

    Online Presentations | 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...

  4. Space-time constrained FOSLS with AMGe upscaling

    Online Presentations | 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)...

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

    Online Presentations | 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...

  6. ECE 595E Lecture 36: MEEP Tutorial II

    Online Presentations | 30 Apr 2013 | Contributor(s):: Peter Bermel

    Outline:Recap from MondayExamplesMultimode ring resonatorsIsolating individual resonancesKerr nonlinearitiesQuantifying third-harmonic generation

  7. Integrated Imaging Seminar Series

    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...

  8. ECE 595E Lecture 35: MEEP Tutorial I

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

    Outline:MEEP InterfacesMEEP ClassesTutorial examples:WaveguideBent waveguide

  9. Data-adaptive Filtering and the State of the Art in Image Processing

    Online Presentations | 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,...

  10. ECE 595 Course Policy - Spring 2013

    Teaching Materials | 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.

  11. The Pioneers of Quantum Computing

    Online Presentations | 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...

  12. Nanoelectronic Modeling Lecture 29: Introduction to the NEMO3D Tool

    Online Presentations | 04 Aug 2010 | Contributor(s):: Gerhard Klimeck

    This presentation provides a very high level software overview of NEMO3D. The items discussed are:Modeling Agenda and MotivationTight-Binding Motivation and basic formula expressionsTight binding representation of strainSoftware structureNEMO3D algorithm flow NEMO3D parallelization scheme –...

  13. Nanoelectronic Modeling Lecture 28: Introduction to Quantum Dots and Modeling Needs/Requirements

    Online Presentations | 20 Jul 2010 | Contributor(s):: Gerhard Klimeck

    This presentation provides a very high level software overview of NEMO1D.Learning Objectives:This lecture provides a very high level overview of quantum dots. The main issues and questions that are addressed are:Length scale of quantum dotsDefinition of a quantum dotQuantum dot examples and...

  14. Nanoelectronic Modeling Lecture 26: NEMO1D -

    Online Presentations | 09 Mar 2010 | Contributor(s):: Gerhard Klimeck

    NEMO1D demonstrated the first industrial strength implementation of NEGF into a simulator that quantitatively simulated resonant tunneling diodes. The development of efficient algorithms that simulate scattering from polar optical phonons, acoustic phonons, alloy disorder, and interface roughness...

  15. Nanoelectronic Modeling Lecture 27: NEMO1D -

    Online Presentations | 09 Mar 2010 | Contributor(s):: Gerhard Klimeck

    This presentation provides a very high level software overview of NEMO1D. The items discussed are:User requirementsGraphical user interfaceSoftware structureProgram developer requirementsDynamic I/O design for batch and GUIResonance finding algorithmInhomogeneous energy meshingInformation flow,...

  16. Nanoelectronic Modeling Lecture 21: Recursive Green Function Algorithm

    Online Presentations | 07 Feb 2010 | Contributor(s):: Gerhard Klimeck

    The Recursive Green Function (RGF) algorithms is the primary workhorse for the numerical solution of NEGF equations in quasi-1D systems. It is particularly efficient in cases where the device is partitioned into reservoirs which may be characterized by a non-Hermitian Hamiltonian and a central...

  17. Lecture 6: Neighbor Lists

    Presentation Materials | 05 Jan 2010 | Contributor(s):: Ashlie Martini

    Topics:Saving simulation timeVerlet listsCell lists

  18. Lecture 3: Integration Algorithms

    Presentation Materials | 05 Jan 2010 | Contributor(s):: Ashlie Martini

    Topics:General guidelinesVerlet algorithmPredictor-corrector methods

  19. Short Course on Molecular Dynamics Simulation

    Courses | 13 Oct 2009 | Contributor(s):: Ashlie Martini

    This set of ten presentations accompanied a graduate level course on Molecular Dynamics simulation. The specific objective of the course (and the presentations) is to provide: 1. Awareness of the opportunities and limitations of Molecular Dynamics as a tool for scientific and engineering...

  20. Illinois ECE 498AL: Programming Massively Parallel Processors, Lecture 15: Kernel and Algorithm Patterns for CUDA

    Online Presentations | 30 Sep 2009 | Contributor(s):: Wen-Mei W Hwu

    Kernel and Algorithm Patterns for CUDATopics: Reductions and Memory Patterns Reduction Patterns in CUDA Mapping Data into CUDA's Memories Input/Output Convolution Generic Algorithm Description What could each thread be assigned? Thread Assignment Trade-offs What memory Space does the Data use?...