460 research outputs found
Resultants and subresultants of p-adic polynomials
We address the problem of the stability of the computations of resultants and
subresultants of polynomials defined over complete discrete valuation rings
(e.g. Zp or k[[t]] where k is a field). We prove that Euclide-like algorithms
are highly unstable on average and we explain, in many cases, how one can
stabilize them without sacrifying the complexity. On the way, we completely
determine the distribution of the valuation of the principal subresultants of
two random monic p-adic polynomials having the same degree
Computation of Discrete Logarithms in GF(2^607)
International audienceWe describe in this article how we have been able to extend the record for computations of discrete logarithms in characteristic 2 from the previous record over GF(2^503) to a newer mark of GF(2^607), using Coppersmith's algorithm. This has been made possible by several practical improvements to the algorithm. Although the computations have been carried out on fairly standard hardware, our opinion is that we are nearing the current limits of the manageable sizes for this algorithm, and that going substantially further will require deeper improvements to the method
On the analysis of mixed-index time fractional differential equation systems
In this paper we study the class of mixed-index time fractional differential
equations in which different components of the problem have different time
fractional derivatives on the left hand side. We prove a theorem on the
solution of the linear system of equations, which collapses to the well-known
Mittag-Leffler solution in the case the indices are the same, and also
generalises the solution of the so-called linear sequential class of time
fractional problems. We also investigate the asymptotic stability properties of
this class of problems using Laplace transforms and show how Laplace transforms
can be used to write solutions as linear combinations of generalised
Mittag-Leffler functions in some cases. Finally we illustrate our results with
some numerical simulations.Comment: 21 pages, 6 figures (some are made up of sub-figures - there are 15
figures or sub-figures
A finite element method for fully nonlinear elliptic problems
We present a continuous finite element method for some examples of fully
nonlinear elliptic equation. A key tool is the discretisation proposed in
Lakkis & Pryer (2011, SISC) allowing us to work directly on the strong form of
a linear PDE. An added benefit to making use of this discretisation method is
that a recovered (finite element) Hessian is a biproduct of the solution
process. We build on the linear basis and ultimately construct two different
methodologies for the solution of second order fully nonlinear PDEs. Benchmark
numerical results illustrate the convergence properties of the scheme for some
test problems including the Monge-Amp\`ere equation and Pucci's equation.Comment: 22 pages, 31 figure
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 beschĂ€ftigt sich mit neuen parallelen Algorithmen, die das Leistungspotenzial der Grafik-Prozessoren (GPUs) zur exakten Berechnungen mit ganzzahlige Polynomen nutzen. Solche symbolische Berechnungen sind von groĂer Bedeutung zur Lösung vieler Probleme aus der reellen algebraischen Geometrie. FĂŒr die effziente Implementierung eines Algorithmus auf massiv-parallelen Hardwarearchitekturen, wie z.B. GPU, ist vor allem auf eine hohe DatenparallelitĂ€t zu achten. Unter Verwendung von Ergebnissen aus der strukturierten Matrix-Theorie konnten wir die entsprechenden Operationen mit Polynomen auf der Grafikkarte leicht ĂŒbertragen. AuĂerdem zeigt eine KomplexitĂ€tanalyse im PRAM-Rechenmodell, dass die von uns entwickelten Verfahren eine deutlich bessere KomplexitĂ€t aufweisen als dies fĂŒr 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 fĂŒr andere GPU Anwendungen, insbesondere der Public-Key-Kryptographie, als nĂŒtzlich erweisen wird. DarĂŒber hinaus betrachten wir Algorithmen fĂŒr 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 VerfĂŒgung stellen. Diese Dissertation ist in englischer Sprache verfasst
Robust isogeometric preconditioners for the Stokes system based on the Fast Diagonalization method
In this paper we propose a new class of preconditioners for the isogeometric
discretization of the Stokes system. Their application involves the solution of
a Sylvester-like equation, which can be done efficiently thanks to the Fast
Diagonalization method. These preconditioners are robust with respect to both
the spline degree and mesh size. By incorporating information on the geometry
parametrization and equation coefficients, we maintain efficiency on
non-trivial computational domains and for variable kinematic viscosity. In our
numerical tests we compare to a standard approach, showing that the overall
iterative solver based on our preconditioners is significantly faster.Comment: 31 pages, 4 figure
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