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Near-optimal perfectly matched layers for indefinite Helmholtz problems
Get PDFA new construction of an absorbing boundary condition for indefinite Helmholtz problems on unbounded domains is presented. This construction is based on a near-best uniform rational interpolant of the inverse square root function on the union of a negative and positive real interval, designed with the help of a classical result by Zolotarev. Using Krein's interpretation of a Stieltjes continued fraction, this interpolant can be converted into a three-term finite difference discretization of a perfectly matched layer (PML) which converges exponentially fast in the number of grid points. The convergence rate is asymptotically optimal for both propagative and evanescent wave modes. Several numerical experiments and illustrations are included
ON THE DIMENSION OF PRODUCTS OF HOMOGENEOUS SUBSPACES IN FREE LIE ALGEBRAS
Get PDFLet be a free Lie algebra of finite rank over a field and let denote the degree homogeneous component of .
Formulae for the dimension of the subspaces
for all and were obtained by the second author and Michael Vaughan-Lee. In this note we consider subspaces of the form . Surprisingly, in contrast to the case of a product of two homogeneous components, the dimension of such products may depend on the characteristic of the field . For example, the dimension of over fields of characteristic is different from the dimension over fields of characteristic other than . Our main result are formulae for the dimension of . Under certain conditions on , and they lead to explicit formulae that do not depend on the characteristic of , and express the dimension of in terms of Witt's dimension function
A Preconditioner for Fictitious Domain Formulations of Elliptic PDEs on Uncertain Parameterized Domains
No full textWe consider the numerical solution of elliptic boundary-value problems on uncertain two-dimensional domains via the fictitious domain method. This leads to variational problems of saddle point form. Working under the standard assumption that the domain can be described by a finite number of independent random variables, discretization is achieved by a stochastic collocation mixed finite element method. We focus on the efficient iterative solution of the resulting sequence of indefinite linear systems and introduce a novel and efficient preconditioner for use with the minimal residual method. The challenging task is to construct a matrix that provides a robust approximation to a discrete representation of a trace space norm on a parameterized boundary
NLEIGS: A class of robust fully rational Krylov methods for nonlinear eigenvalue problems
Get PDFA new rational Krylov method for the efficient solution of nonlinear eigenvalue problems is proposed. This iterative method, called fully rational Krylov method for nonlinear eigenvalue problems (abbreviated as NLEIGS), is based on linear rational interpolation and generalizes the Newton rational Krylov method proposed in [R. Van Beeumen, K. Meerbergen, and W. Michiels, SIAM J. Sci. Comput., 35 (2013), pp. A327-A350]. NLEIGS utilizes a dynamically constructed rational interpolant of the nonlinear operator and a new companion-type linearization for obtaining a generalized eigenvalue problem with special structure. This structure is particularly suited for the rational Krylov method. A new approach for the computation of rational divided differences using matrix functions is presented. It is shown that NLEIGS has a computational cost comparable to the Newton rational Krylov method but converges more reliably, in particular, if the nonlinear operator has singularities nearby the target set. Moreover, NLEIGS implements an automatic scaling procedure which makes it work robustly independent of the location and shape of the target set, and it also features low-rank approximation techniques for increased computational efficiency. Small- and large-scale numerical examples are included
Matrix Functions: A Short Course
No full textA summary is given of a course on functions of matrices delivered
by the first author (lecturer) and second author (teaching assistant) at the
Gene Golub SIAM Summer School 2013 at Fudan University, Shanghai, China,
July 22--26 2013. This article covers some essential features of the theory
and computation of matrix functions
Problems of bounding the -length and Fitting height of finite soluble groups
No full textThis paper is a survey of some open problems and recent results about bounding the Fitting height and -length of finite soluble groups. In many problems of finite group theory, nowadays the classification greatly facilitates reduction to soluble groups. Bounding their Fitting height or -length can be regarded as further reduction to nilpotent groups. This is usually achieved by methods of representation theory, such as Clifford's theorem or theorems of Hall--Higman type. In some problems, it is the case of nilpotent groups where open questions remain, in spite of great successes achieved, in particular, by using Lie ring methods. But there are also important questions that still require reduction to nilpotent groups; the present survey is focused on reduction problems of this type. As examples, we discuss finite groups with fixed-point-free and almost fixed-point-free automorphisms, as well as generalizations of the Restricted Burnside Problem. We also discuss results on coset identities, which have applications in the study of profinite groups. Finally, we mention the open problem of bounding the Fitting height in the study of the analogue of the Restricted Burnside Problem for Moufang loops
Torsional wave propagation in a pre-stressed hyperelastic annular circular cylinder
Get PDFWe consider torsional wave propagation in a pre-stressed annular cylinder. Hydrostatic pressure is applied to the inner and outer surfaces of an incompressible hyperelastic annular cylinder, of circular cross-section, whose constitutive behaviour is governed by a Mooney�Rivlin strain
energy function. The pressure difference creates an inhomogeneous deformation field, which modifies the inner and outer radii of the annular cylinder.We wish to determine the effect that this pre-stress, and a given axial stretch, has on the propagation of small-amplitude torsional waves through the medium. We use the theory of small-on-large to deduce the linear wave equation
that governs incremental torsional waves and then determine the dispersion relation for the pre-stressed annulus by using an approximation procedure (the Liouville�Green transformation). We show that this scheme compares well to numerical solutions except in regions very close to
turning points (of the transformed ordinary differential equation). In particular, we find that the inhomogeneous deformation makes the coefficients of the governing ordinary differential equation spatially dependent and affects the location of the roots of the dispersion relation.We observe that, if the pressure on the outer surface of the annular cylinder is greater (smaller) than that on the inner, then the cut-on frequencies are spaced further apart (closer) than they would be in the stress-free
case. This result could potentially be used to tune the propagation characteristics of the cylinder over a range of frequencies
Systematic construction of kinetic models from genome-scale metabolic networks
Get PDFThe quantitative effects of environmental and genetic perturbations on metabolism can be studied in silico using kinetic models. We present a strategy for large-scale model construction based on a logical layering of data such as reaction fluxes, metabolite concentrations, and kinetic constants. The resulting models contain realistic standard rate laws and plausible parameters, adhere to the laws of thermodynamics, and reproduce a predefined steady state. These features have not been simultaneously achieved by previous workflows. We demonstrate the advantages and limitations of the workflow by translating the yeast consensus metabolic network into a kinetic model. Despite crudely selected data, the model shows realistic control behaviour, a stable dynamic, and realistic response to perturbations in extracellular glucose concentrations. The paper concludes by outlining how new data can continuously be fed into the workflow and how iterative model building can assist in directing experiments
Deformation of geometry and bifurcations of vortex rings
Get PDFWe construct a smooth family of Hamiltonian systems, together with a family of group symmetries and momentum maps, for the dynamics of point vortices on surfaces parametrized by the curvature of the surface. Equivariant bifurcations in this family are characterized, whence the stability of the Thomson heptagon is deduced without recourse to the Birkhoff normal form, which has hitherto been a necessary tool
A Note on Computing Involution Centralizers
Get PDFFor a black box group G and t an involution of G we describe a computational procedure which produces elements of C_G(t) by making use of the local fusion graph F(G,X), where X is the G-conjugacy class of t