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This paper outlines our ideas on how to teach linear algebra in a mechanized mathematical environment, and discusses some of our reasons for thinking that this is a better way to teach linear algebra than the ``old fashioned way''. We…
Presented in this paper is a new sparse linear solver methodology motivated by multigrid principles and based around general local transformations that diagonalize a matrix while maintaining its sparsity. These transformations are…
We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for…
This Ph.D. thesis contains original contributions to several areas within the disciplines of disordered systems, numerical linear algebra, and scientific computing: (1) Theoretical and numerical study of the errors caused by using certain…
This book provides an introduction to the mathematical analysis of deep learning. It covers fundamental results in approximation theory, optimization theory, and statistical learning theory, which are the three main pillars of deep neural…
We consider the problem of efficiently solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. Whilst some classical approaches are theoretically well founded, they can face…
Recent work in deep learning has opened new possibilities for solving classical algorithmic tasks using end-to-end learned models. In this work, we investigate the fundamental task of solving linear systems, particularly those that are…
We suggest that straight-line programs designed for algebraic computations should be accompanied by a comprehensive complexity analysis that takes into account both the number of fundamental algebraic operations needed, as well as memory…
We review the basic outline of the highly successful diffusion Monte Carlo technique commonly used in contexts ranging from electronic structure calculations to rare event simulation and data assimilation, and propose a new class of…
Born from years of teaching undergraduate and graduate algebra courses at Chongqing University, this text is designed to introduce Galois theory while minimizing prerequisites. It seeks to reconnect the abstract machinery of modern algeba:…
There exist several methods for computing exact solutions of algebraic differential equations. Most of the methods, however, do not ensure existence and uniqueness of the solutions and might fail after several steps, or are restricted to…
Computer Algebra systems are widely spread because of some of their remarkable features such as their ease of use and performance. Nonetheless, this focus on performance sometimes leads to unwanted consequences: algorithms and computations…
Linear algebra's main concerns are sets of vectors, linear functions, subspaces, linear systems, matrices and concepts about those, such as whether the solution of linear system exists or is unique; a set of vectors is linearly independent…
This manuscript synthesizes almost fifteen years of research in algebraic combinatorics, in order to highlight, theme by theme, its perspectives. In part one, building on my thesis work, I use tools from commutative algebra, and in…
Recent advancements in quantum computing and quantum-inspired algorithms have sparked renewed interest in binary optimization. These hardware and software innovations promise to revolutionize solution times for complex problems. In this…
Matrix multiplication is a fundamental computation in many scientific disciplines. In this paper, we show that novel fast matrix multiplication algorithms can significantly outperform vendor implementations of the classical algorithm and…
The works presented in this habilitation concern the algorithmics of polynomials. This is a central topic in computer algebra, with numerous applications both within and outside the field - cryptography, error-correcting codes, etc. For…
We develop a linear-algebraic framework for dimensional analysis in systems with constraints, particularly when variables are numerous or related by implicit relations so that direct elimination is impractical. By expressing both…
With the advent of computers, one needs algebraic structures that can simultaneously work with bulk data. One such algebraic structure, namely, n-linear algebras of type I are introduced in this book and its applications to n-Markov chains…
One of the grand challenges of Mathematics instruction is to provide students with problems that are both accessible and have a reasonably elegant solution. Instructors commonly resort to resources like course textbooks, online-learning…