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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.

All Categories (21-40 of 146)

  1. 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.

  2. Feb 25 2019

    Software Productivity and Sustainability for CSE and Data Science

    The SIAM CSE conference seeks to enable in-depth technical discussions on a wide variety of major computational efforts on large-scale problems in science and engineering, foster the...

    https://nanohub.org/events/details/1738

  3. 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...

  4. Data-Driven Discovery of Governing Equations of Physical Systems

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

    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...

  5. Myron DSilva

    https://nanohub.org/members/182103

  6. Purdue ME 581-Numerical Methods in Engineering Using Jupyter Notebooks

    06 Sep 2017 | | Contributor(s):: Marisol Koslowski, Peter Kolis, Meredith Michele Meyer, Akshay Vivek Dandekar, Camilo Alberto Duarte-Cordon

    Jupyter notebooks with an introduction to python and examples for Numerical Methods in Engineering.

  7. 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...

  8. 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...

  9. ECE 695NS Lecture 3: Practical Assessment of Code Performance

    25 Jan 2017 | | Contributor(s):: Peter Bermel

    Outline:Time ScalingExamplesGeneral performance strategiesComputer architecturesMeasuring code speedReduce strengthMinimize array writesProfiling

  10. ECE 695NS Lecture 2: Computability and NP-hardness

    13 Jan 2017 | | Contributor(s):: Peter Bermel

    Outline:OverviewDefinitionsComputing MachinesChurch-Turing ThesisPolynomial Time (Class P)Class NPNon-deterministic Turing machinesReducibilityCook-Levin theoremCoping with NP Hardness

  11. Jupyter Notebooks for Scientific Programming

    06 Jan 2017 | | Contributor(s):: Martin Hunt

    An overview of using Jupyter Notebooks for conveying scientific information.

  12. Machine learned approximations to Density Functional Theory Hamiltonians - Towards High-Throughput Screening of Electronic Structure and Transport in Materials

    13 Dec 2016 | | 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 existing semi-empirical approaches to the problem. The technique we have proposed requires little...

  13. Nikhil Chand Kashyap Chitta

    https://nanohub.org/members/155131

  14. High Accuracy Atomic Force Microscope with Self-Optimizing Scan Control

    19 Sep 2016 | | 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 behavior, inadequate to address the metrology needs of industrial applications: accuracy,...

  15. Tian Qiu

    Tian Qiu is a senior student at Purdue University, who is major in computer engineering and mathematics with computer science. He is from Wuhan China. Currently he is working with Prof. Lin,...

    https://nanohub.org/members/145718

  16. Memory-Efficient Particle Annihilation Algorithm for Wigner Monte Carlo Simulations

    10 Feb 2016 | | Contributor(s):: Paul Ellinghaus

    IWCE 2015 presentation. The Wigner Monte Carlo solver, using the signed-particle method, is based on the generation and annihilation of numerical particles. The memory demands of the annihilation algorithm can become exorbitant, if a high spatial resolution is used, because the entire discretized...

  17. Data-Centric Models for Multilevel Algorithms

    07 Feb 2016 | | 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, moving, and operating on data residing in complex distributed memory hierarchies is quickly becoming...

  18. New FOSLS Formulation of Nonlinear Stokes Flow for Glaciers

    07 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...

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

    07 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...

  20. Range Decomposition: A Low Communication Algorithm for Solving PDEs on Massively Parallel Machines

    07 Feb 2016 | | 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 necessary before reaching a solution within a small multiple of discretization error. The target application...