Array programming provides a powerful, compact and expressive syntax for accessing, manipulating and operating on data in vectors, matrices and higher-dimensional arrays. NumPy is the primary array programming library for the Python language. It has an essential role in research analysis pipelines in fields as diverse as physics, chemistry, astronomy, geoscience, biology, psychology, materials science, engineering, finance and economics. For example, in astronomy, NumPy was an important part of the software stack used in the discovery of gravitational waves1 and in the first imaging of a black hole2. Here we review how a few fundamental array concepts lead to a simple and powerful programming paradigm for organizing, exploring and analysing scientific data. NumPy is the foundation upon which the scientific Python ecosystem is constructed. It is so pervasive that several projects, targeting audiences with specialized needs, have developed their own NumPy-like interfaces and array objects. Owing to its central position in the ecosystem, NumPy increasingly acts as an interoperability layer between such array computation libraries and, together with its application programming interface (API), provides a flexible framework to support the next decade of scientific and industrial analysis. NumPy is the primary array programming library for Python; here its fundamental concepts are reviewed and its evolution into a flexible interoperability layer between increasingly specialized computational libraries is discussed.

/pdf/array-programming-with-numpy-31s3jl24ra.pdf

Array programming with NumPy

Array programming provides a powerful, compact, expressive syntax for accessing, manipulating, and operating on data in vectors, matrices, and higher-dimensional arrays. NumPy is the primary array programming library for the Python language. It plays an essential role in research analysis pipelines in fields as diverse as physics, chemistry, astronomy, geoscience, biology, psychology, material science, engineering, finance, and economics. For example, in astronomy, NumPy was an important part of the software stack used in the discovery of gravitational waves and the first imaging of a black hole. Here we show how a few fundamental array concepts lead to a simple and powerful programming paradigm for organizing, exploring, and analyzing scientific data. NumPy is the foundation upon which the entire scientific Python universe is constructed. It is so pervasive that several projects, targeting audiences with specialized needs, have developed their own NumPy-like interfaces and array objects. Because of its central position in the ecosystem, NumPy increasingly plays the role of an interoperability layer between these new array computation libraries.

/pdf/array-programming-with-numpy-4gbytt9sw2.pdf

Array Programming with NumPy

Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition)

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

1 Discretization in time by a Forward Euler scheme 2 1.1 Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Space discretization . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Variational forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Simplified notation for the solution at recent time levels . . . . . 4 1.5 Deriving the linear systems . . . . . . . . . . . . . . . . . . . . . 5 1.6 Computational algorithm . . . . . . . . . . . . . . . . . . . . . . 6 1.7 Example using sinusoidal basis functions . . . . . . . . . . . . . . 7 1.8 Comparing P1 elements with the finite difference method . . . . 8

Time-Dependent Variational Forms

A comparison of methods for the statistics of slow-drift oscillations

A successful family of methods, usually referred to as Conjugate Gradient-like algorithms, or Krylov subspace methods, can be viewed as Galerkin or least-squares methods applied to a linear system Ax = b. This view is different from the standard approaches to deriving the classical Conjugate Gradient method in the literature. Nevertheless, the fundamental ideas of least squares and Galerkin approximations from Chap. 2 can be used to derive the most popular and successful methods for linear systems, and this is the topic of the present chapter. Such a view may increase the general understanding of variational methods and their applicability.

Variational Methods for Linear Systems

Many successful numerical solution methods for differential equations, including the finite element method, aim at approximating the unknown function by a sum.

Function Approximation by Global Functions

Many mathematical models involve m + 1 unknown functions governed by a system of m + 1 differential equations.

Hans Petter Langtangen

Papers

Time-Dependent Variational Forms

A comparison of methods for the statistics of slow-drift oscillations

Variational Methods for Linear Systems

Function Approximation by Global Functions

Variational Forms for Systems of PDEs