CAD Adjacency Computation Using Validated Numerics

We present an algorithm for computation of cell adjacencies for well-based cylindrical algebraic decomposition. Cell adjacency information can be used to compute topological operations e.g. closure, boundary, connected components, and topological properties e.g. homology groups. Other applications include visualization and path planning. Our algorithm determines cell adjacency information using validated numerical methods similar to those used in CAD construction, thus computing CAD with adjacency information in time comparable to that of computing CAD without adjacency information. We report on implementation of the algorithm and present empirical data.Comment: 20 page

Non-constructive interval simulation of dynamic systems

Publisher PD

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/.Comment: 9 page

Computation of Steady Incompressible Flows in Unbounded Domains

In this study we revisit the problem of computing steady Navier-Stokes flows in two-dimensional unbounded domains. Precise quantitative characterization of such flows in the high-Reynolds number limit remains an open problem of theoretical fluid dynamics. Following a review of key mathematical properties of such solutions related to the slow decay of the velocity field at large distances from the obstacle, we develop and carefully validate a spectrally-accurate computational approach which ensures the correct behavior of the solution at infinity. In the proposed method the numerical solution is defined on the entire unbounded domain without the need to truncate this domain to a finite box with some artificial boundary conditions prescribed at its boundaries. Since our approach relies on the streamfunction-vorticity formulation, the main complication is the presence of a discontinuity in the streamfunction field at infinity which is related to the slow decay of this field. We demonstrate how this difficulty can be overcome by reformulating the problem using a suitable background "skeleton" field expressed in terms of the corresponding Oseen flow combined with spectral filtering. The method is thoroughly validated for Reynolds numbers spanning two orders of magnitude with the results comparing favourably against known theoretical predictions and the data available in the literature.Comment: 39 pages, 12 figures, accepted for publication in "Computers and Fluids

Fast Computation of Smith Forms of Sparse Matrices Over Local Rings

We present algorithms to compute the Smith Normal Form of matrices over two families of local rings. The algorithms use the \emph{black-box} model which is suitable for sparse and structured matrices. The algorithms depend on a number of tools, such as matrix rank computation over finite fields, for which the best-known time- and memory-efficient algorithms are probabilistic. For an \nxn matrix $A$ over the ring \Fzfe, where $f^e$ is a power of an irreducible polynomial f \in \Fz of degree $d$, our algorithm requires \bigO(\eta de^2n) operations in \F, where our black-box is assumed to require \bigO(\eta) operations in \F to compute a matrix-vector product by a vector over \Fzfe (and $\eta$ is assumed greater than \Pden). The algorithm only requires additional storage for \bigO(\Pden) elements of \F. In particular, if \eta=\softO(\Pden), then our algorithm requires only \softO(n^2d^2e^3) operations in \F, which is an improvement on known dense methods for small $d$ and $e$. For the ring \ZZ/p^e\ZZ, where $p$ is a prime, we give an algorithm which is time- and memory-efficient when the number of nontrivial invariant factors is small. We describe a method for dimension reduction while preserving the invariant factors. The time complexity is essentially linear in $\mu n r e \log p,$ where $\mu$ is the number of operations in \ZZ/p\ZZ to evaluate the black-box (assumed greater than $n$) and $r$ is the total number of non-zero invariant factors. To avoid the practical cost of conditioning, we give a Monte Carlo certificate, which at low cost, provides either a high probability of success or a proof of failure. The quest for a time- and memory-efficient solution without restrictions on the number of nontrivial invariant factors remains open. We offer a conjecture which may contribute toward that end.Comment: Preliminary version to appear at ISSAC 201

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

Planar Two-Loop Five-Parton Amplitudes from Numerical Unitarity

We compute a complete set of independent leading-color two-loop five-parton amplitudes in QCD. These constitute a fundamental ingredient for the next-to-next-to-leading order QCD corrections to three-jet production at hadron colliders. We show how to consistently consider helicity amplitudes with external fermions in dimensional regularization, allowing the application of a numerical variant of the unitarity approach. Amplitudes are computed by exploiting a decomposition of the integrand into master and surface terms that is independent of the parton type. Master integral coefficients are numerically computed in either finite-field or floating-point arithmetic and combined with known analytic master integrals. We recompute two-loop leading-color four-parton amplitudes as a check of our implementation. Results are presented for all independent four- and five-parton processes including contributions with massless closed fermion loops.Comment: v3: corrected wrong signs for five-gluon $N_f^2$ amplitudes with vanishing tree

An Analysis of Publication Venues for Automatic Differentiation Research

We present the results of our analysis of publication venues for papers on automatic differentiation (AD), covering academic journals and conference proceedings. Our data are collected from the AD publications database maintained by the autodiff.org community website. The database is purpose-built for the AD field and is expanding via submissions by AD researchers. Therefore, it provides a relatively noise-free list of publications relating to the field. However, it does include noise in the form of variant spellings of journal and conference names. We handle this by manually correcting and merging these variants under the official names of corresponding venues. We also share the raw data we get after these corrections.Comment: 6 pages, 3 figure