26,885 research outputs found
Guaranteed Lower Eigenvalue Bound of Steklov Operator with Conforming Finite Element Methods
For the eigenvalue problem of the Steklov differential operator, by following
Liu's approach, an algorithm utilizing the conforming finite element method
(FEM) is proposed to provide guaranteed lower bounds for the eigenvalues. The
proposed method requires the a priori error estimation for FEM solution to
nonhomogeneous Neumann problems, which is solved by constructing the
hypercircle for the corresponding FEM spaces and boundary conditions. Numerical
examples are also shown to confirm the efficiency of our proposed method.Comment: 21 pages, 4 figures, 4 table
Fully computable a posteriori error bounds for eigenfunctions
Fully computable a posteriori error estimates for eigenfunctions of compact
self-adjoint operators in Hilbert spaces are derived. The problem of
ill-conditioning of eigenfunctions in case of tight clusters and multiple
eigenvalues is solved by estimating the directed distance between the spaces of
exact and approximate eigenfunctions. Derived upper bounds apply to various
types of eigenvalue problems, e.g. to the (generalized) matrix, Laplace, and
Steklov eigenvalue problems. These bounds are suitable for arbitrary conforming
approximations of eigenfunctions, and they are fully computable in terms of
approximate eigenfunctions and two-sided bounds of eigenvalues. Numerical
examples illustrate the efficiency of the derived error bounds for
eigenfunctions.Comment: 27 pages, 8 tables, 9 figure
Eigenvalue enclosures and exclosures for non-self-adjoint problems in hydrodynamics
In this paper we present computer-assisted proofs of a number of results in theoretical fluid dynamics and in quantum mechanics. An algorithm based on interval arithmetic yields provably correct eigenvalue enclosures and exclosures for non-self-adjoint boundary eigenvalue problems, the eigenvalues of which are highly sensitive to perturbations. We apply the algorithm to: the Orr-Sommerfeld equation with Poiseuille profile to prove the existence of an eigenvalue in the classically unstable region for Reynolds number R=5772.221818; the Orr-Sommerfeld equation with Couette profile to prove upper bounds for the imaginary parts of all eigenvalues for fixed R and wave number α; the problem of natural oscillations of an incompressible inviscid fluid in the neighbourhood of an elliptical flow to obtain information about the unstable part of the spectrum off the imaginary axis; Squire's problem from hydrodynamics; and resonances of one-dimensional Schrödinger operators
Limits on Sparse Data Acquisition: RIC Analysis of Finite Gaussian Matrices
One of the key issues in the acquisition of sparse data by means of
compressed sensing (CS) is the design of the measurement matrix. Gaussian
matrices have been proven to be information-theoretically optimal in terms of
minimizing the required number of measurements for sparse recovery. In this
paper we provide a new approach for the analysis of the restricted isometry
constant (RIC) of finite dimensional Gaussian measurement matrices. The
proposed method relies on the exact distributions of the extreme eigenvalues
for Wishart matrices. First, we derive the probability that the restricted
isometry property is satisfied for a given sufficient recovery condition on the
RIC, and propose a probabilistic framework to study both the symmetric and
asymmetric RICs. Then, we analyze the recovery of compressible signals in noise
through the statistical characterization of stability and robustness. The
presented framework determines limits on various sparse recovery algorithms for
finite size problems. In particular, it provides a tight lower bound on the
maximum sparsity order of the acquired data allowing signal recovery with a
given target probability. Also, we derive simple approximations for the RICs
based on the Tracy-Widom distribution.Comment: 11 pages, 6 figures, accepted for publication in IEEE transactions on
information theor
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