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.

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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

Problems in science and engineering have traditionally been solved by a combination of theory and experiment. In many branches of science, the theories are based on mathematical models, usually in the form of equations describing the physical world. By formulating and solving these equations, one can understand and predict the physical world. The theories are constructed from or validated by physical experiments under controlled conditions.

Scientific Computing - Why, What, How and What's Next

Operator splitting is a common method for solving systems of partial differential equations. This is particularly the case if the system is composed of separate equations for which suitable software already exists. In such cases, operator splitting combined with explicit time-stepping is a straight forward approach. However, for some systems, implicit time-stepping is to be preferred because of better stability properties. The problem of combining existing codes in a fully implicit manner is much harder than the explicit case. The purpose of the present chapter is to discuss some examples illustrating the possibilities of combining existing codes in an implicit manner.

Fully Implicit Methods for Systems of PDEs

Many code developers think the implementation of mixed finite elements on general unstructured meshes is challenging. This assertion is reflected in the many efforts on constructing stabilized finite element methods (e.g. for flow problems), where standard elements can be used, and the fact that there are very few libraries offering a mixed finite element programming environment. In this paper we described a software framework for mixed finite element programming, which makes the use of mixed methods as easy as standard finite element methods. The various abstractions used in this framework will be explained. The tools are made available in the generic Diffpack software.

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A Software Framework for Mixed Finite Element Programming

This chapter exemplifies various techniques for reading data into a program. Most emphasis is put on reading data from the command line. We introduce the concept of exceptions for dealing with invalid input or erroneous conditions in a program. The chapter also explains how to organize your Python code in your own modules (libraries).

https://link.springer.com/content/pdf/10.1007%2F978-3-642-30293-0_4.pdf

Input Data and Error Handling

We shall now take the ideas from previous chapters and put together such that we can solve PDEs using the flexible finite element basis functions. This is quite a machinery with many details, but the chapter is mostly an assembly of concepts and details we have already met.

Hans Petter Langtangen

Papers

Scientific Computing - Why, What, How and What's Next

Fully Implicit Methods for Systems of PDEs

A Software Framework for Mixed Finite Element Programming

Input Data and Error Handling

Variational Formulations with Finite Elements