arXiv subjects

David Wilson

Publications and source records attributed to David Wilson.

31 records · Page 2Linked to original sources

arXiv2015

An implementation of Sub-CAD in Maple

Cylindrical algebraic decomposition (CAD) is an important tool for the investigation of semi-algebraic sets, with applications in algebraic geometry and beyond. We have previously reported on an implementation of CAD in Maple which offers the original projection and lifting algorithm of Collins along with subsequent improvements. Here we report on new functionality: specifically the ability to build cylindrical algebraic sub-decompositions (sub-CADs) where only certain cells are returned. We have implemented algorithms to return cells of a prescribed dimensions or higher (layered {\scad}s), and an algorithm to return only those cells on which given polynomials are zero (variety {\scad}s). These offer substantial savings in output size and computation time. The code described and an introductory Maple worksheet / pdf demonstrating the full functionality of the package are freely available online at http://opus.bath.ac.uk/43911/.

cs.SC↗

arXiv2014

Using the distribution of cells by dimension in a cylindrical algebraic decomposition

We investigate the distribution of cells by dimension in cylindrical algebraic decompositions (CADs). We find that they follow a standard distribution which seems largely independent of the underlying problem or CAD algorithm used. Rather, the distribution is inherent to the cylindrical structure and determined mostly by the number of variables. This insight is then combined with an algorithm that produces only full-dimensional cells to give an accurate method of predicting the number of cells in a complete CAD. Since constructing only full-dimensional cells is relatively inexpensive (involving no costly algebraic number calculations) this leads to heuristics for helping with various questions of problem formulation for CAD, such as choosing an optimal variable ordering. Our experiments demonstrate that this approach can be highly effective.

cs.SC↗

arXiv2014

Choosing a variable ordering for truth-table invariant cylindrical algebraic decomposition by incremental triangular decomposition

Cylindrical algebraic decomposition (CAD) is a key tool for solving problems in real algebraic geometry and beyond. In recent years a new approach has been developed, where regular chains technology is used to first build a decomposition in complex space. We consider the latest variant of this which builds the complex decomposition incrementally by polynomial and produces CADs on whose cells a sequence of formulae are truth-invariant. Like all CAD algorithms the user must provide a variable ordering which can have a profound impact on the tractability of a problem. We evaluate existing heuristics to help with the choice for this algorithm, suggest improvements and then derive a new heuristic more closely aligned with the mechanics of the new algorithm.

cs.SC↗

arXiv2014

Using the Regular Chains Library to build cylindrical algebraic decompositions by projecting and lifting

Cylindrical algebraic decomposition (CAD) is an important tool, both for quantifier elimination over the reals and a range of other applications. Traditionally, a CAD is built through a process of projection and lifting to move the problem within Euclidean spaces of changing dimension. Recently, an alternative approach which first decomposes complex space using triangular decomposition before refining to real space has been introduced and implemented within the RegularChains Library of Maple. We here describe a freely available package ProjectionCAD which utilises the routines within the RegularChains Library to build CADs by projection and lifting. We detail how the projection and lifting algorithms were modified to allow this, discuss the motivation and survey the functionality of the package.

cs.SC↗

arXiv2014

A comparison of three heuristics to choose the variable ordering for CAD

Cylindrical algebraic decomposition (CAD) is a key tool for problems in real algebraic geometry and beyond. When using CAD there is often a choice over the variable ordering to use, with some problems infeasible in one ordering but simple in another. Here we discuss a recent experiment comparing three heuristics for making this choice on thousands of examples.

cs.SC↗

arXiv2014

Problem formulation for truth-table invariant cylindrical algebraic decomposition by incremental triangular decomposition

Cylindrical algebraic decompositions (CADs) are a key tool for solving problems in real algebraic geometry and beyond. We recently presented a new CAD algorithm combining two advances: truth-table invariance, making the CAD invariant with respect to the truth of logical formulae rather than the signs of polynomials; and CAD construction by regular chains technology, where first a complex decomposition is constructed by refining a tree incrementally by constraint. We here consider how best to formulate problems for input to this algorithm. We focus on a choice (not relevant for other CAD algorithms) about the order in which constraints are presented. We develop new heuristics to help make this choice and thus allow the best use of the algorithm in practice. We also consider other choices of problem formulation for CAD, as discussed in CICM 2013, revisiting these in the context of the new algorithm.

cs.SC↗

arXiv2014

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.

cs.SC↗

arXiv2014

A "Piano Movers" Problem Reformulated

It has long been known that cylindrical algebraic decompositions (CADs) can in theory be used for robot motion planning. However, in practice even the simplest examples can be too complicated to tackle. We consider in detail a "Piano Mover's Problem" which considers moving an infinitesimally thin piano (or ladder) through a right-angled corridor. Producing a CAD for the original formulation of this problem is still infeasible after 25 years of improvements in both CAD theory and computer hardware. We review some alternative formulations in the literature which use differing levels of geometric analysis before input to a CAD algorithm. Simpler formulations allow CAD to easily address the question of the existence of a path. We provide a new formulation for which both a CAD can be constructed and from which an actual path could be determined if one exists, and analyse the CADs produced using this approach for variations of the problem. This emphasises the importance of the precise formulation of such problems for CAD. We analyse the formulations and their CADs considering a variety of heuristics and general criteria, leading to conclusions about tackling other problems of this form.

cs.CG↗

arXiv2014

Sliding window and compressive sensing for low-field dynamic magnetic resonance imaging

We describe an acquisition/processing procedure for image reconstruction in dynamic Magnetic Resonance Imaging (MRI). The approach requires sliding window to record a set of trajectories in the k-space, standard regularization to reconstruct an estimate of the object and compressed sensing to recover image residuals. We validated this approach in the case of specific simulated experiments and, in the case of real measurements, we showed that the procedure is reliable even in the case of data acquired by means of a low-field scanner.

cs.CE↗

arXiv2013

Optimising Problem Formulation for Cylindrical Algebraic Decomposition

Cylindrical algebraic decomposition (CAD) is an important tool for the study of real algebraic geometry with many applications both within mathematics and elsewhere. It is known to have doubly exponential complexity in the number of variables in the worst case, but the actual computation time can vary greatly. It is possible to offer different formulations for a given problem leading to great differences in tractability. In this paper we suggest a new measure for CAD complexity which takes into account the real geometry of the problem. This leads to new heuristics for choosing: the variable ordering for a CAD problem, a designated equational constraint, and formulations for truth-table invariant CADs (TTICADs). We then consider the possibility of using Groebner bases to precondition TTICAD and when such formulations constitute the creation of a new problem.

cs.SC↗

arXiv2013

Understanding Branch Cuts of Expressions

We assume some standard choices for the branch cuts of a group of functions and consider the problem of then calculating the branch cuts of expressions involving those functions. Typical examples include the addition formulae for inverse trigonometric functions. Understanding these cuts is essential for working with the single-valued counterparts, the common approach to encoding multi-valued functions in computer algebra systems. While the defining choices are usually simple (typically portions of either the real or imaginary axes) the cuts induced by the expression may be surprisingly complicated. We have made explicit and implemented techniques for calculating the cuts in the computer algebra programme Maple. We discuss the issues raised, classifying the different cuts produced. The techniques have been gathered in the BranchCuts package, along with tools for visualising the cuts. The package is included in Maple 17 as part of the FunctionAdvisor tool.

cs.MS↗

arXiv2013

Cylindrical Algebraic Decompositions for Boolean Combinations

This article makes the key observation that when using cylindrical algebraic decomposition (CAD) to solve a problem with respect to a set of polynomials, it is not always the signs of those polynomials that are of paramount importance but rather the truth values of certain quantifier free formulae involving them. This motivates our definition of a Truth Table Invariant CAD (TTICAD). We generalise the theory of equational constraints to design an algorithm which will efficiently construct a TTICAD for a wide class of problems, producing stronger results than when using equational constraints alone. The algorithm is implemented fully in Maple and we present promising results from experimentation.

cs.SC↗

arXiv2012

Program Verification in the presence of complex numbers, functions with branch cuts etc

In considering the reliability of numerical programs, it is normal to "limit our study to the semantics dealing with numerical precision" (Martel, 2005). On the other hand, there is a great deal of work on the reliability of programs that essentially ignores the numerics. The thesis of this paper is that there is a class of problems that fall between these two, which could be described as "does the low-level arithmetic implement the high-level mathematics". Many of these problems arise because mathematics, particularly the mathematics of the complex numbers, is more difficult than expected: for example the complex function log is not continuous, writing down a program to compute an inverse function is more complicated than just solving an equation, and many algebraic simplification rules are not universally valid. The good news is that these problems are theoretically capable of being solved, and are practically close to being solved, but not yet solved, in several real-world examples. However, there is still a long way to go before implementations match the theoretical possibilities.

cs.SC↗