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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.

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  1. Nano-Scale Device Simulations Using PROPHET-Part I: Basics

    out of 5 stars 

    20 Jan 2006 | | Contributor(s):: Yang Liu, Robert Dutton

    Part I covers the basics of PROPHET,including the set-up of simulation structures and parameters based onpre-defined PDE systems.

  2. Nano-Scale Device Simulations Using PROPHET-Part II: PDE Systems

    out of 5 stars 

    20 Jan 2006 | | Contributor(s):: Yang Liu, Robert Dutton

    Part II uses examples toillustrate how to build user-defined PDE systems in PROPHET.

  3. Nanoelectronic Modeling Lecture 21: Recursive Green Function Algorithm

    out of 5 stars 

    05 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...

  4. Nanoelectronic Modeling Lecture 26: NEMO1D -

    out of 5 stars 

    02 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...

  5. Nanoelectronic Modeling Lecture 27: NEMO1D -

    out of 5 stars 

    02 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,...

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

    out of 5 stars 

    02 Mar 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...

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

    out of 5 stars 

    02 Mar 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 –...

  8. Nanoelectronic Modeling: Multimillion Atom Simulations, Transport, and HPC Scaling to 23,000 Processors

    out of 5 stars 

    07 Mar 2008 | | Contributor(s):: Gerhard Klimeck

    Future field effect transistors will be on the same length scales as “esoteric” devices such as quantum dots, nanowires, ultra-scaled quantum wells, and resonant tunneling diodes. In those structures the behavior of carriers and their interaction with their environment need to be fundamentally...

  9. NanoElectronic MOdeling: NEMO

    out of 5 stars 

    20 Dec 2007 | | Contributor(s):: Gerhard Klimeck

    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.Novel nanoelectronic devices such as quantum dots, nanowires, and ultra-scaled...

  10. Nanoparticle and Colloidal Simulations with Molecular Dynamics

    out of 5 stars 

    05 Dec 2008 | | Contributor(s):: Steve Plimpton

    Modeling nanoparticle or colloidal systems in a molecular dynamics (MD) code requires coarse-graining on several levels to achieve meaningful simulation times for study of rheological and other manufacturing properties. These include treating colloids as single particles, moving from explicit to...

  11. NCN Cyberinfrastructure Overview

    out of 5 stars 

    21 Jun 2005 | | Contributor(s):: Gerhard Klimeck

    Presentation of the NCN cyberinfrastructure to the June 2005 NSF review team. The nanoHUB development over 12 months will be presented in a broad overview.

  12. NEMO 1-D: The First NEGF-based TCAD Tool and Network for Computational Nanotechnology

    out of 5 stars 

    28 Dec 2004 | | Contributor(s):: Gerhard Klimeck

    Nanotechnology has received a lot of public attention since U.S. President Clinton announced the U.S.National Nanotechnology Initiative. New approaches to applications in electronics, materials,medicine, biology and a variety of other areas will be developed in this new multi-disciplinary...

  13. NEMO 3D: Intel optimizations and Multiple Quantum Dot Simulations

    out of 5 stars 

    03 Aug 2006 | | Contributor(s):: Anish Dhanekula, Gerhard Klimeck

    NEMO-3D is a nanoelectronic modeling tool that analyzes the electronic structure of nanoscopic devices. Nanoelectronic devices such as Quantum Dots (QDs) can contain millions of atoms,. Therefore, simulating their electronic structure, can take up to several days. In order to simulate and analyze...

  14. Networked Dynamical Systems for Function and Learning: Paradigms for Data-Driven Control and Learning in Neurosensory Systems

    out of 5 stars 

    15 Jan 2019 | | Contributor(s):: J. Nathan Kutz

    Our objective is to use emerging data-driven methods to extract the underlying engineering principles of cognitive capability, namely those that allow complex networks to learn and enact control and functionality in the robust manner observed in neurosensory systems. Mathematically, the...

  15. New FOSLS Formulation of Nonlinear Stokes Flow for Glaciers

    out of 5 stars 

    05 Feb 2016 | | Contributor(s):: Jeffrey Allen

    This talk describes two First-order System Least-squares (FOSLS) formulations of the nonlinear Stokes flow used to model glaciers and ice sheets. The first is a Stress formulation and the second a Stress-Vorticity formulation. Both use fluidity, which is the reciprocal of viscosity and avoid the...

  16. Nikhil Chand Kashyap Chitta

    https://nanohub.org/members/155131

  17. Non-Blocking Conjugate Gradient Methods for Extreme Scale Computing

    out of 5 stars 

    05 Feb 2016 | | Contributor(s):: Paul Eller

    Many scientific and engineering applications use Krylov subspace methods to solve large systems of linear equations. For extreme scale parallel computing systems, the dot products in these methods (implemented using allreduce operations in MPI) can limit performance because they are a...

  18. Numerical Analysis

    out of 5 stars 

    02 Jun 2006 | | Contributor(s):: Dragica Vasileska

  19. Numerical Aspects of NEGF: The Recursive Green Function Algorithm

    out of 5 stars 

    14 Jun 2004 | | Contributor(s):: Gerhard Klimeck

    Numerical Aspects of NEGF: The Recursive Green Function Algorithm

  20. OpenMP Tutorial

    out of 5 stars 

    25 Nov 2008 | | Contributor(s):: Seung-Jai Min

    This tutorial consists of three parts. First, we will discuss abouthow OpenMP is typically used and explain OpenMP programming model. Second, we will describe important OpenMP constructs and data enviroments. Finally, we will show a simple example to illustrate how OpenMP APIs are used to program...