There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization. It responds to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems. For this new edition the book has been thoroughly updated throughout. There are new chapters on nonlinear interior methods and derivative-free methods for optimization, both of which are used widely in practice and the focus of much current research. Because of the emphasis on practical methods, as well as the extensive illustrations and exercises, the book is accessible to a wide audience. It can be used as a graduate text in engineering, operations research, mathematics, computer science, and business. It also serves as a handbook for researchers and practitioners in the field. The authors have strived to produce a text that is pleasant to read, informative, and rigorous - one that reveals both the beautiful nature of the discipline and its practical side.

Numerical Optimization

Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

/pdf/iterative-methods-for-sparse-linear-systems-34ozhpwq89.pdf

Iterative Methods for Sparse Linear Systems

SciPy is an open source scientific computing library for the Python programming language. SciPy 1.0 was released in late 2017, about 16 years after the original version 0.1 release. SciPy has become a de facto standard for leveraging scientific algorithms in the Python programming language, with more than 600 unique code contributors, thousands of dependent packages, over 100,000 dependent repositories, and millions of downloads per year. This includes usage of SciPy in almost half of all machine learning projects on GitHub, and usage by high profile projects including LIGO gravitational wave analysis and creation of the first-ever image of a black hole (M87). The library includes functionality spanning clustering, Fourier transforms, integration, interpolation, file I/O, linear algebra, image processing, orthogonal distance regression, minimization algorithms, signal processing, sparse matrix handling, computational geometry, and statistics. In this work, we provide an overview of the capabilities and development practices of the SciPy library and highlight some recent technical developments.

/pdf/scipy-1-0-fundamental-algorithms-for-scientific-computing-in-111lr790ol.pdf

SciPy 1.0--Fundamental Algorithms for Scientific Computing in Python

TensorFlow is an interface for expressing machine learning algorithms, and an
implementation for executing such algorithms. A computation expressed using
TensorFlow can be executed with little or no change on a wide variety of
heterogeneous systems, ranging from mobile devices such as phones and tablets
up to large-scale distributed systems of hundreds of machines and thousands of
computational devices such as GPU cards. The system is flexible and can be used
to express a wide variety of algorithms, including training and inference
algorithms for deep neural network models, and it has been used for conducting
research and for deploying machine learning systems into production across more
than a dozen areas of computer science and other fields, including speech
recognition, computer vision, robotics, information retrieval, natural language
processing, geographic information extraction, and computational drug
discovery. This paper describes the TensorFlow interface and an implementation
of that interface that we have built at Google. The TensorFlow API and a
reference implementation were released as an open-source package under the
Apache 2.0 license in November, 2015 and are available at www.tensorflow.org.

/pdf/tensorflow-large-scale-machine-learning-on-heterogeneous-2yk4fgchw6.pdf

TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems

In this paper, we analyze the main features and discuss the tuning of the algorithms for the direct solution of sparse linear systems on distributed memory computers developed in the context of a long term European research project. The algorithms use a multifrontal approach and are especially designed to cover a large class of problems. The problems can be symmetric positive definite, general symmetric, or unsymmetric matrices, both possibly rank deficient, and they can be provided by the user in several formats. The algorithms achieve high performance by exploiting parallelism coming from the sparsity in the problem and that available for dense matrices. The algorithms use a dynamic distributed task scheduling technique to accommodate numerical pivoting and to allow the migration of computational tasks to lightly loaded processors. Large computational tasks are divided into subtasks to enhance parallelism. Asynchronous communication is used throughout the solution process to efficiently overlap communication with computation.
We illustrate our design choices by experimental results obtained on an SGI Origin 2000 and an IBM SP2 for test matrices provided by industrial partners in the PARASOL project.

/pdf/a-fully-asynchronous-multifrontal-solver-using-distributed-1xlwbmzqj5.pdf

A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling

This paper describes an extension to the set of Basic Linear Algebra Subprograms. The extensions are targeted at matrix-vector operations that should provide for efficient and portable implementations of algorithms for high-performance computers

/pdf/a-set-of-level-3-basic-linear-algebra-subprograms-1ovuidkmqo.pdf

A set of level 3 basic linear algebra subprograms

The subject of sparse matrices has its roots in such diverse fields as management science, power systems analysis, surveying, circuit theory, and structural analysis. Mathematical models in all these areas give rise to very large systems of linear equations which could not be solved were it not for the fact that the matrices contain relatively few non-zero entries. Only comparatively recently, in the last fifteen years or so, has it become apparent that the equations can be solved even when the pattern is irregular, and it is primarily the solution of such problems that is considered in this book. The subject is intensely practical and this book is written with practicalities ever in mind. Whenever two methods are applicable, their rival merits are considered, and conclusions are based on practical experience where theoretical comparison is not possible. Non-numeric computing techniques have been included, as well as frequent illustrations, in an attempt to bridge the usually wide gap between the printed page and the working computer code. Despite this practical bias, it is recognized that many aspects of the subject are of interest in their own right, and the book aims to be suitable also for a student course, probably at MSc level. Exercises have been included to illustrate and strengthen understanding of the material, as well as to extend it. Efficient use of sparsity is a key to solving large problems in many fields. This book will supply both insight and answers for those attempting to solve these problems.

Direct methods for sparse matrices

On etend la methode frontale pour resoudre des systemes lineaires d'equations en permettant a plus d'un front d'apparaitre en meme temps

Iain S. Duff

Papers

A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling

A set of level 3 basic linear algebra subprograms

Direct methods for sparse matrices

The Multifrontal Solution of Indefinite Sparse Symmetric Linear

Multifrontal parallel distributed symmetric and unsymmetric solvers