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.
PennyLane - Automatic Differentiation and Machine Learning of Quantum Computations
29 Apr 2020 | | Contributor(s):: Nathan Killoran
PennyLane is a Python-based software framework for optimization and machine learning of quantum and hybrid quantum-classical computations.
Advances in Computational and Quantum Imaging Workshop
28 Jan 2020 |
The purpose of the workshop is to bring different communities together, review recent theoretical and experimental advances and explore synergetic collaborations. The workshop aligns well with the significant investments in quantum technologies through the National Quantum Initiative in the...
ECE 595ML Lecture 1: Linear Regression - Introduction
21 Jan 2020 | | Contributor(s):: Stanley H. Chan
ECE 595ML Lecture 2: Regularized Linear Regression
ECE 595ML: Machine Learning I
17 Jan 2020 | | Contributor(s):: Stanley H. Chan
Spring 2020 - This course is in productionCourse Website: https://engineering.purdue.edu/ChanGroup/ECE595/index.htmlCourse Outline:Part 1: Mathematical BackgroundLinear Regression and OptimizationPart 2: ClassificationMethods to train linear classifiersFeature analysis, Geometry, Bayesian...
Universal Variational Quantum Computation
28 Oct 2019 | | Contributor(s):: Jacob Biamonte
We show that the variational approach to quantum enhanced algorithms admits a universal model of quantum computation.
Quantum Algorithmic Breakeven: on Scaling Up with Noisy Qubits
21 Aug 2019 | | Contributor(s):: Daniel Lidar
In this talk I will argue in favor of a different criterion I call "quantum algorithmic breakeven," which focuses on demonstrating an algorithmic scaling improvement in an error-corrected setting over the uncorrected setting. I will present evidence that current experiments with...
Overview of Computational Methods and Machine Learning: Panel Discussion
14 Jun 2019 | | Contributor(s):: Brett Matthew Savoie, Pradeep Kumar Gurunathan, Peilin Liao, Xiulin Ruan, Guang Lin
The individual Panel Talks which accompanies this discussion can be found here.Why do we need experiments?Are your methods “descriptive” or “predictive”?Do you work with any other theory/simulation groups?On the 5 year timescale: is machine-learning hype or a real...
Overview of Computational Methods and Machine Learning: Panel Talks
The Panel Discussion which follows these individual presentations can be found here.Individucal Presentations:Theory and Machine Learning in the Chemical Sciences, Brett Matthew Savoie;Divide and Conquer with QM/MM Methods, Pradeep Kumar Gurunathan;Computational Chemistry/Materials, Peilin...
Big Data in Reliability and Security: Some Basics
30 May 2019 | | Contributor(s):: Saurabh Bagchi
Big Data in Reliability and Security: Applications
Human-Interpretable Concept Learning via Information Lattices
23 May 2019 | | Contributor(s):: Lav R. Varshney
The basic idea is an iterative discovery algorithm that has a student-teacher architecture and that operates on a generalization of Shannon’s information lattice, which itself encodes a hierarchy of abstractions and is algorithmically constructed from group-theoretic foundations.
Feb 25 2019
Software Productivity and Sustainability for CSE and Data Science
Networked Dynamical Systems for Function and Learning: Paradigms for Data-Driven Control and Learning in Neurosensory Systems
16 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...
Data-Driven Discovery of Governing Equations of Physical Systems
We introduce a number of data-driven strategies for discovering nonlinear multiscale dynamical systems and their embeddings from data. We consider two canonical cases: (i) systems for which we have full measurements of the governing variables, and (ii) systems for which we have incomplete...
Quantifying Uncertainties in Physical Models
28 Aug 2017 | | 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 the number of parameters required to accurately specify its initial/boundary conditions, constitutive...
A Distributed Algorithm for Computing a Common Fixed Point of a Family of Paracontractions
21 Jun 2017 | | 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 simultaneously computed by m agents assuming each agent knows only paracontraction, the current estimates...