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Periodic orbits in Hamiltonian systems with involutory symmetries
Get PDFWe study the existence of families of periodic solutions in a neighbourhood of a symmetric equilibrium point in two classes of Hamiltonian systems with involutory symmetry. In both classes, the involution reverses the sign of the Hamiltonian function, and the system is in 1:-1 resonance. In the first class we study a Hamiltonian system with a reversing involution R acting symplectically. We first recover a result of Buzzi and Lamb showing that the equilibrium point is contained in a three dimensional conical subspace which consists of a two parameter family of periodic solutions with symmetry R, and furthermore that there may or may not exist two families of non-symmetric periodic solutions, depending on the coefficients of the Hamiltonian (correcting a minor error in their paper).
In the second problem we study an equivariant Hamiltonian system with a symmetry S that acts anti-symplectically. Generically, there is no S-symmetric solution in a neighbourhood of the equilibrium point. Moreover, we prove the existence of at least 2 and at most 12 families of non-symmetric periodic solutions. We conclude with a brief study of systems with both forms of symmetry, showing they have very similar structure to the system with symmetry R
A rational deferred correction approach to PDE-constrained optimization
Get PDFThe accurate and efficient solution of time-dependent PDE-constrained optimization problems is a challenging task, in large part due to the very high dimension of the matrix systems that need to be solved. We devise a new deferred correction method for coupled systems of time-dependent PDEs, allowing one to iteratively improve the accuracy of low-order time stepping schemes. We consider two variants of our method, a splitting and a coupling version, and analyze their convergence properties. We then test our approach on a number of PDE-constrained optimization problems. We obtain solution accuracies far superior to that achieved when solving a single discretized problem, in particular in cases where the accuracy is limited by the time discretization. Our approach allows for the direct reuse of existing solvers for the resulting matrix systems, as well as state-of-the-art preconditioning strategies
UManSysProp V1.0: An Online and Open-Source Facility for Molecular Property Prediction and Atmospheric Aerosol Calculations
Get PDFIn this paper we describe the development and application of a new web-based facility, UManSysProp (http://umansysprop.seaes.manchester.ac.uk), for automating predictions of molecular and atmospheric aerosol properties. Current facilities include pure component vapour pressures, critical properties, and sub-cooled densities of organic molecules; activity coefficient predictions for mixed inorganic�organic liquid systems; hygroscopic growth factors and CCN (cloud condensation nuclei) activation potential of mixed inorganic�organic aerosol particles; and absorptive partitioning calculations with/without a treatment of non-ideality. The aim of this new facility is to provide a single point of reference for all properties relevant to atmospheric aerosol that have been checked for applicability to atmospheric compounds where possible. The group contribution approach allows users to upload molecular information in the form of SMILES (Simplified Molecular Input Line Entry System) strings and UManSysProp will automatically extract the relevant information for calculations. Built using open-source chemical informatics, and hosted at the University of Manchester, the facilities are provided via a browser and device-friendly web interface, or can be accessed using the user�s own code via a JSON API (application program interface). We also provide the source code for all predictive techniques provided on the site, covered by the GNU GPL (General Public License) license to encourage development of a user community. We have released this via a Github repository (doi:10.5281/zenodo.45143). In this paper we demonstrate its use with specific examples that can be simulated using the web-browser interface
Matrix Depot: An Extensible Test Matrix Collection for Julia
Get PDFMatrix Depot is a Julia software package that provides easy access to a large and diverse collection of test matrices. Its novelty is threefold. First, it is extensible by the user, and so can be adapted to include the user's own test problems. In doing so it facilitates experimentation and makes it easier to carry out reproducible research. Second, it amalgamates in a single framework two different types of existing matrix collections, comprising parametrized test matrices (including Hansen's set of regularization test problems and Higham's Test Matrix Toolbox) and real-life sparse matrix data (giving access to the University of Florida Sparse Matrix Collection). Third, it fully exploits the Julia language. It uses multiple dispatch to help provide a simple interface and, in particular, to allow matrices to be generated in any of the numeric data types supported by the language
Polynomial eigenvalue solver based on tropically scaled Lagrange linearization
Get PDFWe propose a novel approach to solve polynomial eigenvalue problems via linearization. The novelty lies in (a) our choice of linearization, which is constructed using input from tropical algebra and the notion of well-separated tropical roots, (b) an appropriate scaling applied to the linearization and (c) a modified stopping criterion for the QZ iterations that takes advantage of the properties of our
scaled linearization. Numerical experiments show that our polynomial eigensolver computes all the finite and well-conditioned eigenvalues to high relative accuracy even
when they are very different in magnitude
Computing the weighted geometric mean of two large-scale matrices and its inverse times a vector
Get PDFWe investigate different approaches for computing the action of the weighted geometric mean of two large-scale positive definite matrices on a vector. We derive and analyze several algorithms, based on numerical quadrature and on the Krylov subspace, and compare them in terms of convergence speed and execution time. By exploiting an algebraic relation between the weighted geometric mean and its inverse, we show how these methods can be used to efficiently solve large linear systems whose coefficient matrix is a weighted geometric mean. According to our experiments, some of the algorithms proposed in both families are suitable choices for black-box implementations
Constructing strong linearizations of matrix polynomials expressed in the Chebyshev bases
Get PDFThe need to solve polynomial eigenvalue problems for matrix polynomials expressed
in nonmonomial bases has become a very important problem. Among the most important bases in
numerical applications are the Chebyshev polynomials of the first and second kind. In this work,
we introduce a new approach for constructing strong linearizations for matrix polynomials expressed
in Chebyshev bases, generalizing the classical colleague pencil, and expanding the arena in which
to look for linearizations of matrix polynomials expressed in Chebyshev bases. We show that any
of these linearizations is a strong linearization regardless whether the matrix polynomial is regular
or singular. In addition, we show how to recover eigenvectors, minimal indices, and minimal bases
of the polynomial from those of any of the new linearizations. As an example, we also construct
strong linearizations for matrix polynomials of odd degree that are symmetric whenever the matrix
polynomials are symmetric
On the symmetric square of quaternionic projective space
Get PDFThe main purpose of this thesis is to calculate the integral cohomology ring of the symmetric square of quaternionic projective space, which has been an open problem since computations with symmetric squares were first proposed in the 1930's. The geometry of this particular case forms an essential part of the thesis, and unexpected results concerning two universal Pin(4) bundles are also included. The cohomological computations involve a commutative ladder of long exact sequences, which arise by decomposing the symmetric square and the corresponding Borel space in compatible ways. The geometry and the cohomology of the configuration space of unordered pairs of distinct points in quaternionic projective space, and of the Thom space MPin(4), also feature, and seem to be of independent interest
Parallelization of the rational Arnoldi algorithm
Get PDFRational Krylov methods are applicable to a wide range of scientific computing problems, and the rational Arnoldi algorithm is a commonly used procedure for computing an orthonormal basis of a rational Krylov space. Typically, the computationally most expensive component of this algorithm is the solution of a large linear system of equations at each iteration. We explore the option of solving several linear systems simultaneously, thus constructing the rational Krylov basis in parallel. If this is not done carefully, the basis being orthogonalized may become badly conditioned, leading to numerical instabilities in the orthogonalization process. We introduce the new concept of continuation pairs which gives rise to a near-optimal parallelization strategy that allows to control the growth of the condition number of this nonorthogonal basis. As a consequence we obtain a significantly more accurate and reliable parallel rational Arnoldi algorithm. The computational benefits are illustrated using several numerical examples from different application areas
Workshop on Batched, Reproducible, and Reduced Precision BLAS
Get PDFThis report summarises the main points raised on a recent workshop discussing various extensions to the BLAS standard, held at the University of Tennessee in May 2016. In particular the discussions focused on batched, reproducible, and reduced precision BLAS extensions. Various members of the linear algebra community and representatives from industry were present to generate and evaluate ideas in each of these areas