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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.
Short Course on Molecular Dynamics Simulation
13 Oct 2009 | Courses | 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...
Illinois ECE 498AL: Programming Massively Parallel Processors, Lecture 15: Kernel and Algorithm Patterns for CUDA
30 Sep 2009 | Online Presentations | Contributor(s): Wen-Mei W Hwu
Kernel and Algorithm Patterns for CUDA
Reductions and Memory Patterns
Reduction Patterns in CUDA
Mapping Data into CUDA's Memories
Illinois ECE 498AL: Programming Massively Parallel Processors, Lecture 13: Reductions and their Implementation
15 Sep 2009 | Online Presentations | Contributor(s): Wen-Mei W Hwu
Structuring Parallel Algorithms
Parallel Prefix Sum
Relevance of Scan
Application of Scan
Scan on the CPU
First attempt Parallel Scan Algorithm
Illinois ECE 498AL: Programming Massively Parallel Processors, Lecture 12: Structuring Parallel Algorithms
Structuring Parallel Algorithms
Key Parallel Programming Steps
Choosing Algorithm Structure
Mapping a Divide and Conquer algorithm
Experiment vs. Modelling: What's the problem?
10 Aug 2009 | Online Presentations | 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...
Quantitative, Kinetic Models of Cellular Circuits
04 Apr 2009 | Online Presentations | Contributor(s): Michael R. Brent
Living cells contain complex, analog circuits that regulate the rate at which each gene produces its product. The kinetic properties of these circuits enable cells to respond to changes in their...
The Multicore Era: Crisis or (and?) Opportunity?
27 Mar 2009 | Online Presentations | Contributor(s): Mithuna Thottethodi
This talk will provide a brief overview of how we got to the multicore era, the implications and challenges for hardware/software developers and users, and some informed speculation on where the...
Experiences with nonintrusive polynomial Chaos and stochastic collocation methods for uncertainty analysis and design
0.0 out of 5 stars
13 Mar 2009 | Online Presentations | 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...
Nanoparticle and Colloidal Simulations with Molecular Dynamics
05 Dec 2008 | Online Presentations | 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...
25 Nov 2008 | Online Presentations | Contributor(s): Seung-Jai Min
This tutorial consists of three parts. First, we will discuss about
how OpenMP is typically used and explain OpenMP programming model. Second, we will describe important OpenMP constructs and...
Purdue School on High Performance and Parallel Computing
24 Nov 2008 | Workshops | Contributor(s): Alejandro Strachan, Faisal Saied
The goal of this workshop is to provide training in the area of high performance scientific computing for graduate students and researchers interested in scientific computing. The School will...
Introduction to Parallel Programming with MPI
5.0 out of 5 stars
24 Nov 2008 | Online Presentations | Contributor(s): David Seaman
Single-session course illustrating message-passing techniques. The examples include point-to-point and collective communication using blocking and nonblocking transmission. One application...
Software Productivity Tools
This presentation briefly describes the use of tar(1), make(1), the
Portable Batch System (PBS), and two version control systems: CVS and subversion.
Introduction to TotalView
This single-session course presents an introduction to the use of the TotalView parallel debugger available on Purdue's Linux systems.
Quantum and Thermal Effects in Nanoscale Devices
4.5 out of 5 stars
18 Sep 2008 | Online Presentations | Contributor(s): Dragica Vasileska
To investigate lattice heating within a Monte Carlo device simulation framework, we simultaneously solve the Boltzmann transport equation for the electrons, the 2D Poisson equation to get the...
An Introduction to Quantum Computing
12 Sep 2008 | Online Presentations | 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...
BNC Annual Research Review: An Introduction to PRISM and MEMS Simulation
04 Jun 2008 | Online Presentations | 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...
Calculating Resonances Using a Complex Absorbing Potential
14 Mar 2008 | Online Presentations | 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...
Nanoelectronic Modeling: Multimillion Atom Simulations, Transport, and HPC Scaling to 23,000 Processors
07 Mar 2008 | Online Presentations | 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...
How to ensure that the stiffness matrix is square?
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...