21,363 research outputs found
Applying machine learning to the problem of choosing a heuristic to select the variable ordering for cylindrical algebraic decomposition
Cylindrical algebraic decomposition(CAD) is a key tool in computational
algebraic geometry, particularly for quantifier elimination over real-closed
fields. When using CAD, there is often a choice for the ordering placed on the
variables. This can be important, with some problems infeasible with one
variable ordering but easy with another. Machine learning is the process of
fitting a computer model to a complex function based on properties learned from
measured data. In this paper we use machine learning (specifically a support
vector machine) to select between heuristics for choosing a variable ordering,
outperforming each of the separate heuristics.Comment: 16 page
Cylindrical Algebraic Decomposition Using Local Projections
We present an algorithm which computes a cylindrical algebraic decomposition
of a semialgebraic set using projection sets computed for each cell separately.
Such local projection sets can be significantly smaller than the global
projection set used by the Cylindrical Algebraic Decomposition (CAD) algorithm.
This leads to reduction in the number of cells the algorithm needs to
construct. We give an empirical comparison of our algorithm and the classical
CAD algorithm
An implementation of CAD in Maple utilising problem formulation, equational constraints and truth-table invariance
Cylindrical algebraic decomposition (CAD) is an important tool for the
investigation of semi-algebraic sets, with applications within algebraic
geometry and beyond. We recently reported on a new implementation of CAD in
Maple which implemented the original algorithm of Collins and the subsequent
improvement to projection by McCallum. Our implementation was in contrast to
Maple's in-built CAD command, based on a quite separate theory. Although
initially developed as an investigative tool to compare the algorithms, we
found and reported that our code offered functionality not currently available
in any other existing implementations. One particularly important piece of
functionality is the ability to produce order-invariant CADs. This has allowed
us to extend the implementation to produce CADs invariant with respect to
either equational constraints (ECCADs) or the truth-tables of sequences of
formulae (TTICADs). This new functionality is contained in the second release
of our code, along with commands to consider problem formulation which can be a
major factor in the tractability of a CAD. In the report we describe the new
functionality and some theoretical discoveries it prompted. We describe how the
CADs produced using equational constraints are able to take advantage of not
just improved projection but also improvements in the lifting phase. We also
present an extension to the original TTICAD algorithm which increases both the
applicability of TTICAD and its relative benefit over other algorithms. The
code and an introductory Maple worksheet / pdf demonstrating the full
functionality of the package are freely available online.Comment: 12 pages; University of Bath, Dept. Computer Science Technical Report
Series, 2013-02, 201
Formulating problems for real algebraic geometry
We discuss issues of problem formulation for algorithms in real algebraic
geometry, focussing on quantifier elimination by cylindrical algebraic
decomposition. We recall how the variable ordering used can have a profound
effect on both performance and output and summarise what may be done to assist
with this choice. We then survey other questions of problem formulation and
algorithm optimisation that have become pertinent following advances in CAD
theory, including both work that is already published and work that is
currently underway. With implementations now in reach of real world
applications and new theory meaning algorithms are far more sensitive to the
input, our thesis is that intelligently formulating problems for algorithms,
and indeed choosing the correct algorithm variant for a problem, is key to
improving the practical use of both quantifier elimination and symbolic real
algebraic geometry in general.Comment: To be presented at The "Encuentros de \'Algebra Computacional y
Aplicaciones, EACA 2014" (Meetings on Computer Algebra and Applications) in
Barcelon
Constructing Fewer Open Cells by GCD Computation in CAD Projection
A new projection operator based on cylindrical algebraic decomposition (CAD)
is proposed. The new operator computes the intersection of projection factor
sets produced by different CAD projection orders. In other words, it computes
the gcd of projection polynomials in the same variables produced by different
CAD projection orders. We prove that the new operator still guarantees
obtaining at least one sample point from every connected component of the
highest dimension, and therefore, can be used for testing semi-definiteness of
polynomials. Although the complexity of the new method is still doubly
exponential, in many cases, the new operator does produce smaller projection
factor sets and fewer open cells. Some examples of testing semi-definiteness of
polynomials, which are difficult to be solved by existing tools, have been
worked out efficiently by our program based on the new method.Comment: Accepted by ISSAC 2014 (July 23--25, 2014, Kobe, Japan
Multiscale differential Riccati equations for linear quadratic regulator problems
We consider approximations to the solutions of differential Riccati equations
in the context of linear quadratic regulator problems, where the state equation
is governed by a multiscale operator. Similarly to elliptic and parabolic
problems, standard finite element discretizations perform poorly in this
setting unless the grid resolves the fine-scale features of the problem. This
results in unfeasible amounts of computation and high memory requirements. In
this paper, we demonstrate how the localized orthogonal decomposition method
may be used to acquire accurate results also for coarse discretizations, at the
low cost of solving a series of small, localized elliptic problems. We prove
second-order convergence (except for a logarithmic factor) in the
operator norm, and first-order convergence in the corresponding energy norm.
These results are both independent of the multiscale variations in the state
equation. In addition, we provide a detailed derivation of the fully discrete
matrix-valued equations, and show how they can be handled in a low-rank setting
for large-scale computations. In connection to this, we also show how to
efficiently compute the relevant operator-norm errors. Finally, our theoretical
results are validated by several numerical experiments.Comment: Accepted for publication in SIAM J. Sci. Comput. This version differs
from the previous one only by the addition of Remark 7.2 and minor changes in
formatting. 21 pages, 12 figure
A framework for deflated and augmented Krylov subspace methods
We consider deflation and augmentation techniques for accelerating the
convergence of Krylov subspace methods for the solution of nonsingular linear
algebraic systems. Despite some formal similarity, the two techniques are
conceptually different from preconditioning. Deflation (in the sense the term
is used here) "removes" certain parts from the operator making it singular,
while augmentation adds a subspace to the Krylov subspace (often the one that
is generated by the singular operator); in contrast, preconditioning changes
the spectrum of the operator without making it singular. Deflation and
augmentation have been used in a variety of methods and settings. Typically,
deflation is combined with augmentation to compensate for the singularity of
the operator, but both techniques can be applied separately.
We introduce a framework of Krylov subspace methods that satisfy a Galerkin
condition. It includes the families of orthogonal residual (OR) and minimal
residual (MR) methods. We show that in this framework augmentation can be
achieved either explicitly or, equivalently, implicitly by projecting the
residuals appropriately and correcting the approximate solutions in a final
step. We study conditions for a breakdown of the deflated methods, and we show
several possibilities to avoid such breakdowns for the deflated MINRES method.
Numerical experiments illustrate properties of different variants of deflated
MINRES analyzed in this paper.Comment: 24 pages, 3 figure
- …