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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.
Quantifying Uncertainties in Physical Models
28 Aug 2017 | Online Presentations | Contributor(s): Ilias Bilionis
Increasing modeling detail is not necessarily correlated with increasing predictive ability. Setting modeling and numerical discretization errors aside, the more detailed a model gets, the larger...
A Distributed Algorithm for Computing a Common Fixed Point of a Family of Paracontractions
21 Jun 2017 | Online Presentations | 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...
ECE 695NS Lecture 3: Practical Assessment of Code Performance
25 Jan 2017 | Online Presentations | Contributor(s): Peter Bermel
General performance strategies
Measuring code speed
Minimize array writes
ECE 695NS Lecture 2: Computability and NP-hardness
13 Jan 2017 | Online Presentations | Contributor(s): Peter Bermel
Polynomial Time (Class P)
Non-deterministic Turing machines
Jupyter Notebooks for Scientific Programming
06 Jan 2017 | Online Presentations | Contributor(s): Martin Hunt
An overview of using Jupyter Notebooks for conveying scientific information.
Machine learned approximations to Density Functional Theory Hamiltonians - Towards High-Throughput Screening of Electronic Structure and Transport in Materials
13 Dec 2016 | Online Presentations | Contributor(s): Ganesh Krishna Hegde
We present results from our recent work on direct machine learning of DFT Hamiltonians. We show that approximating DFT Hamiltonians accurately by direct learning is feasible and compare them to...
Nikhil Chand Kashyap Chitta
High Accuracy Atomic Force Microscope with Self-Optimizing Scan Control
19 Sep 2016 | Online Presentations | Contributor(s): Ryan (Young-kook) Yoo
Atomic force microscope (AFM) is a very useful instrument in characterizing nanoscale features, However, the original AFM design, based on piezo-tube scanner, had slow response and non-orthogonal...
Data-Centric Models for Multilevel Algorithms
07 Feb 2016 | Online Presentations | Contributor(s): Samuel Guiterrez
Today, computational scientists must contend with a diverse set of supercomputer architectures that are capable of exposing unprecedented levels of parallelism and complexity. Effectively placing,...
Range Decomposition: A Low Communication Algorithm for Solving PDEs on Massively Parallel Machines
07 Feb 2016 | Online Presentations | Contributor(s): Tom Manteuffel
The Range Decomposition (RD) algorithm uses nested iteration and adaptive mesh refinement locally before performing a global communication step. Only several such steps are observed to be...
A Scalable Algorithm for Inverse Medium Problems with Multiple Sources
04 Feb 2016 | Online Presentations | 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...
ECE 595E Lecture 36: MEEP Tutorial II
30 Apr 2013 | Online Presentations | Contributor(s): Peter Bermel
Recap from Monday
Multimode ring resonators
Isolating individual resonances
Quantifying third-harmonic generation
Integrated Imaging Seminar Series
30 Apr 2013 | Series | 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...
ECE 595E Lecture 35: MEEP Tutorial I
18 Apr 2013 | Online Presentations | Contributor(s): Peter Bermel
Data-adaptive Filtering and the State of the Art in Image Processing
15 Apr 2013 | Online Presentations | 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...
ECE 595 Course Policy - Spring 2013
03 Jan 2013 | Teaching Materials | Contributor(s): Peter Bermel
A description of the key policies that will govern the administration of ECE 595 on "Numerical Methods" in Spring 2013.