A Brief Review on Mathematical Tools Applicable to Quantum Computing for Modelling and Optimization Problems in Engineering

Since its emergence, quantum computing has enabled a wide spectrum of new possibilities and advantages, including its efficiency in accelerating computational processes exponentially. This has directed much research towards completely novel ways of solving a wide variety of engineering problems, especially through describing quantum versions of many mathematical tools such as Fourier and Laplace transforms, differential equations, systems of linear equations, and optimization techniques, among others. Exploration and development in this direction will revolutionize the world of engineering. In this manuscript, we review the state of the art of these emerging techniques from the perspective of quantum computer development and performance optimization, with a focus on the most common mathematical tools that support engineering applications. This review focuses on the application of these mathematical tools to quantum computer development and performance improvement/optimization. It also identifies the challenges and limitations related to the exploitation of quantum computing and outlines the main opportunities for future contributions. This review aims at offering a valuable reference for researchers in fields of engineering that are likely to turn to quantum computing for solutions. Doi: 10.28991/ESJ-2023-07-01-020 Full Text: PD

Preconditioning polynomial systems using Macaulay dual spaces

Includes bibliographical references.2015 Summer.Polynomial systems arise in many applications across a diverse landscape of subjects. Solving these systems has been an area of intense research for many years. Methods for solving these systems numerically fit into the field of numerical algebraic geometry. Many of these methods rely on an idea called homotopy continuation. This method is very effective for solving systems of polynomials in many variables. However, in the case of zero-dimensional systems, we may end up tracking many more solutions than actually exist, leading to excess computation. This project preconditions these systems in order to reduce computation. We present the background on homotopy continuation and numerical algebraic geometry as well as the theory of Macaulay dual spaces. We show how to turn an algebraic geometric preconditioning problem into one of numerical linear algebra. Algorithms for computing an H-basis and thereby preconditioning the original system to remove extraneous calculation are presented. The concept of the Closedness Subspace is introduced and used to replace a bottleneck computation. A novel algorithm employing this method is introduced and discussed

Topics in exact precision mathematical programming

The focus of this dissertation is the advancement of theory and computation related to exact precision mathematical programming. Optimization software based on floating-point arithmetic can return suboptimal or incorrect resulting because of round-off errors or the use of numerical tolerances. Exact or correct results are necessary for some applications. Implementing software entirely in rational arithmetic can be prohibitively slow. A viable alternative is the use of hybrid methods that use fast numerical computation to obtain approximate results that are then verified or corrected with safe or exact computation. We study fast methods for sparse exact rational linear algebra, which arises as a bottleneck when solving linear programming problems exactly. Output sensitive methods for exact linear algebra are studied. Finally, a new method for computing valid linear programming bounds is introduced and proven effective as a subroutine for solving mixed-integer linear programming problems exactly. Extensive computational results are presented for each topic.Ph.D.Committee Chair: Dr. William J. Cook; Committee Member: Dr. George Nemhauser; Committee Member: Dr. Robin Thomas; Committee Member: Dr. Santanu Dey; Committee Member: Dr. Shabbir Ahmed; Committee Member: Dr. Zonghao G

Cache-Friendly, Modular and Parallel Schemes For Computing Subresultant Chains

The RegularChains library in Maple offers a collection of commands for solving polynomial systems symbolically with taking advantage of the theory of regular chains. The primary goal of this thesis is algorithmic contributions, in particular, to high-performance computational schemes for subresultant chains and underlying routines to extend that of RegularChains in a C/C++ open-source library. Subresultants are one of the most fundamental tools in computer algebra. They are at the core of numerous algorithms including, but not limited to, polynomial GCD computations, polynomial system solving, and symbolic integration. When the subresultant chain of two polynomials is involved in a client procedure, not all polynomials of the chain, or not all coefficients of a given subresultant, may be needed. Based on that observation, we design so-called speculative and caching strategies which yield great performance improvements within our polynomial system solver. Our implementation of these techniques has been highly optimized. We have implemented optimized core arithmetic routines and multithreaded subresultant algorithms for univariate, bivariate and multivariate polynomials. We further examine memory access patterns and data locality for computing subresultants of multivariate polynomials, and study different optimization techniques for the fraction-free LU decomposition algorithm to compute subresultants based on determinant of Bezout matrices. Our code is publicly available at www.bpaslib.org as part of the Basic Polynomial Algebra Subprograms (BPAS) library that is mainly written in C, with concurrency support and user interfaces written in C++