31,948 research outputs found
Numerical Algorithms for Polynomial Optimisation Problems with Applications
In this thesis, we study tensor eigenvalue problems and polynomial optimization problems. In particular, we present a fast algorithm for computing the spectral radii of symmetric nonnegative tensors without requiring the partition of the tensors. We also propose some polynomial time approximation algorithms with new approximation bounds for nonnegative polynomial optimization problems over unit spheres. Furthermore, we develop an efficient and effective algorithm for the maximum clique problem
Spline-based Rayleigh-Ritz methods for the approximation of the natural modes of vibration for flexible beams with tip bodies
Rayleigh-Ritz methods for the approximation of the natural modes for a class of vibration problems involving flexible beams with tip bodies using subspaces of piecewise polynomial spline functions are developed. An abstract operator theoretic formulation of the eigenvalue problem is derived and spectral properties investigated. The existing theory for spline-based Rayleigh-Ritz methods applied to elliptic differential operators and the approximation properties of interpolatory splines are useed to argue convergence and establish rates of convergence. An example and numerical results are discussed
Polynomial (chaos) approximation of maximum eigenvalue functions: efficiency and limitations
This paper is concerned with polynomial approximations of the spectral
abscissa function (the supremum of the real parts of the eigenvalues) of a
parameterized eigenvalue problem, which are closely related to polynomial chaos
approximations if the parameters correspond to realizations of random
variables.
Unlike in existing works, we highlight the major role of the smoothness
properties of the spectral abscissa function. Even if the matrices of the
eigenvalue problem are analytic functions of the parameters, the spectral
abscissa function may not be everywhere differentiable, even not everywhere
Lipschitz continuous, which is related to multiple rightmost eigenvalues or
rightmost eigenvalues with multiplicity higher than one.
The presented analysis demonstrates that the smoothness properties heavily
affect the approximation errors of the Galerkin and collocation-based
polynomial approximations, and the numerical errors of the evaluation of
coefficients with integration methods. A documentation of the experiments,
conducted on the benchmark problems through the software Chebfun, is publicly
available.Comment: This is a pre-print of an article published in Numerical Algorithms.
The final authenticated version is available online at:
https://doi.org/10.1007/s11075-018-00648-
Factorizing the Stochastic Galerkin System
Recent work has explored solver strategies for the linear system of equations
arising from a spectral Galerkin approximation of the solution of PDEs with
parameterized (or stochastic) inputs. We consider the related problem of a
matrix equation whose matrix and right hand side depend on a set of parameters
(e.g. a PDE with stochastic inputs semidiscretized in space) and examine the
linear system arising from a similar Galerkin approximation of the solution. We
derive a useful factorization of this system of equations, which yields bounds
on the eigenvalues, clues to preconditioning, and a flexible implementation
method for a wide array of problems. We complement this analysis with (i) a
numerical study of preconditioners on a standard elliptic PDE test problem and
(ii) a fluids application using existing CFD codes; the MATLAB codes used in
the numerical studies are available online.Comment: 13 pages, 4 figures, 2 table
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