06271 Abstracts Collection -- Challenges in Symbolic Computation Software

From 02.07.06 to 07.07.06, the Dagstuhl Seminar 06271 ``Challenges in Symbolic Computation Software\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

New Acceleration of Nearly Optimal Univariate Polynomial Root-findERS

Univariate polynomial root-finding has been studied for four millennia and is still the subject of intensive research. Hundreds of efficient algorithms for this task have been proposed. Two of them are nearly optimal. The first one, proposed in 1995, relies on recursive factorization of a polynomial, is quite involved, and has never been implemented. The second one, proposed in 2016, relies on subdivision iterations, was implemented in 2018, and promises to be practically competitive, although user's current choice for univariate polynomial root-finding is the package MPSolve, proposed in 2000, revised in 2014, and based on Ehrlich's functional iterations. By proposing and incorporating some novel techniques we significantly accelerate both subdivision and Ehrlich's iterations. Moreover our acceleration of the known subdivision root-finders is dramatic in the case of sparse input polynomials. Our techniques can be of some independent interest for the design and analysis of polynomial root-finders.Comment: 89 pages, 5 figures, 2 table

Harnessing the power of GPUs for problems in real algebraic geometry

This thesis presents novel parallel algorithms to leverage the power of GPUs (Graphics Processing Units) for exact computations with polynomials having large integer coefficients. The significance of such computations, especially in real algebraic geometry, is hard to undermine. On massively-parallel architectures such as GPU, the degree of datalevel parallelism exposed by an algorithm is the main performance factor. We attain high efficiency through the use of structured matrix theory to assist the realization of relevant operations on polynomials on the graphics hardware. A detailed complexity analysis, assuming the PRAM model, also confirms that our approach achieves a substantially better parallel complexity in comparison to classical algorithms used for symbolic computations. Aside from the theoretical considerations, a large portion of this work is dedicated to the actual algorithm development and optimization techniques where we pay close attention to the specifics of the graphics hardware. As a byproduct of this work, we have developed high-throughput modular arithmetic which we expect to be useful for other GPU applications, in particular, open-key cryptography. We further discuss the algorithms for the solution of a system of polynomial equations, topology computation of algebraic curves and curve visualization which can profit to the full extent from the GPU acceleration. Extensive benchmarking on a real data demonstrates the superiority of our algorithms over several state-of-the-art approaches available to date. This thesis is written in English.Diese Arbeit beschäftigt sich mit neuen parallelen Algorithmen, die das Leistungspotenzial der Grafik-Prozessoren (GPUs) zur exakten Berechnungen mit ganzzahlige Polynomen nutzen. Solche symbolische Berechnungen sind von großer Bedeutung zur Lösung vieler Probleme aus der reellen algebraischen Geometrie. Für die effziente Implementierung eines Algorithmus auf massiv-parallelen Hardwarearchitekturen, wie z.B. GPU, ist vor allem auf eine hohe Datenparallelität zu achten. Unter Verwendung von Ergebnissen aus der strukturierten Matrix-Theorie konnten wir die entsprechenden Operationen mit Polynomen auf der Grafikkarte leicht übertragen. Außerdem zeigt eine Komplexitätanalyse im PRAM-Rechenmodell, dass die von uns entwickelten Verfahren eine deutlich bessere Komplexität aufweisen als dies für die klassischen Verfahren der Fall ist. Neben dem theoretischen Ergebnis liegt ein weiterer Schwerpunkt dieser Arbeit in der praktischen Implementierung der betrachteten Algorithmen, wobei wir auf der Besonderheiten der Grafikhardware achten. Im Rahmen dieser Arbeit haben wir hocheffiziente modulare Arithmetik entwickelt, von der wir erwarten, dass sie sich für andere GPU Anwendungen, insbesondere der Public-Key-Kryptographie, als nützlich erweisen wird. Darüber hinaus betrachten wir Algorithmen für die Lösung eines Systems von Polynomgleichungen, Topologie Berechnung der algebraischen Kurven und deren Visualisierung welche in vollem Umfang von der GPU-Leistung profitieren können. Zahlreiche Experimente belegen dass wir zur Zeit die beste Verfahren zur Verfügung stellen. Diese Dissertation ist in englischer Sprache verfasst

The Acceleration of Polynomial Methods for Blind Image Deconvolution Using Graphical Processing Units (GPUs)

Image processing has become an integral part of many areas of study. Unfortunately, the process of capturing images can often result in undesirable blurring and noise, and thus can make processing the resulting images problematic. Methods are therefore required that attempt to remove blurring. The main body of work in this field is in Bayesian methods for image deblurring, with many algorithms aimed at solving this problem relying on the Fourier transform. The Fourier transform results in the amplification of noise in the image, which can lead to many of the same problems as blurring. Winkler presented a method of blind image deconvolution (BID) without the Fourier transform, which treated the rows and columns of the blurred image as the coefficients of univariate polynomials. By treating the rows and columns of the image in this way, the problem of computing the blurring function becomes a problem of computing the greatest common divisor (GCD) of these polynomials. The computation of the GCD of two polynomials is ill posed, as any noise in the polynomials causes them to be coprime. Thus an approximate GCD (AGCD) must be computed instead. The computation of an AGCD is a computationally expensive process, resulting in the BID algorithm being expensive. The research presented in this thesis investigates the fundamental mathematical processes underpinning such an algorithm, and presents multiple methods through which this algorithm can be accelerated using a GPU. This acceleration results in an implementation that is 30 times faster than a CPU parallel approach. The process of accelerating the BID algorithm in this way required a first of its kind GPU accelerated algorithm for the computation of an AGCD, with multiple novel techniques utilised to achieve this acceleration

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics