A new adaptive eigendecomposition algorithm that can be used for on-line high-resolution spectral/spatial analysis is presented. The formulation of the algorithm is based on the interpretation of the correction term in the recursive update of the data covariance matrix estimate as a perturbation term, with the forgetting factor playing the role of a perturbation parameter. Following this interpretation, a first-order perturbation analysis is made to obtain a new recursion expressing the eigenstructure estimate of the true data covariance matrix at time k, in terms of the eigenstructure estimate at time k-1. The resulting algorithm can be realized by means of M linear combiners with nonlinear weight-vector adaption equations, where M is the signal-subspace dimensionality. Comparative simulation results for narrowband array data indicate very good performance of the proposed algorithm. >

Perturbation theory of eigenvalue problems

A new adaptive eigendecomposition algorithm based on a first-order perturbation criterion

In this paper we give necessary and sufficient conditions for the equality case in Wielandt's eigenvalue inequality.

Equality in Wielandt's eigenvalue inequality

We study the existence/nonexistence of positive solution to the problem of the type: \begin{equation}\tag{$P_{\lambda}$} \begin{cases} \Delta^2u-\mu a(x)u=f(u)+\lambda b(x)\quad\textrm{in $\Omega$,}\\ u>0 \quad\textrm{in $\Omega$,}\\ u=0=\Delta u \quad\textrm{on $\partial\Omega$,} \end{cases} \end{equation} where $\Omega$ is a smooth bounded domain in $\mathbb R^N$, $N\geq 5$, $a, b, f$ are nonnegaive functions satisfying certain hypothesis which we will specify later. $\mu,\lambda$ are positive constants. Under some suitable conditions on functions $a, b, f$ and the constant $\mu$, we show that there exists $\lambda^*>0$ such that when $0 \lambda^*$, it does not have any solution in $W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)$. Moreover as $\lambda\uparrow\lambda^*$, minimal positive solution of ($P_{\lambda}$) converges in $W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)$ to a solution of ($P_{\lambda^*}$). We also prove that there exists $\tilde{\lambda}^* \tilde{\lambda}^*$, the above problem ($P_{\lambda}$) does not have any solution even in the distributional sense/very weak sense and there is complete {\it blow-up}. Under an additional integrability condition on $b$, we establish the uniqueness of positive solution of ($P_{\lambda^*}$) in $W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)$.

methods

Semilinear elliptic PDE's with biharmonic operator and a singular potential

Non-smoothness at optimal points is a common phenomenon in many eigenvalue optimization problems. We consider two recent algorithms to minimize the largest eigenvalue of a Hermitian matrix dependent on one parameter, both proven to be globally convergent unaffected by non-smoothness. One of these models the eigenvalue function with a piece-wise quadratic function, and effective in dealing with non-convex problems. The other projects the Hermitian matrix into subspaces formed of eigenvectors, and effective in dealing with large-scale problems. We generalize the latter slightly to cope with non-smoothness. For both algorithms, we analyze the rate-of-convergence in the non-smooth setting, when the largest eigenvalue is multiple at the minimizer and zero is strictly in the interior of the generalized Clarke derivative, and prove that both algorithms converge rapidly. The algorithms are applied to, and the deduced results are illustrated on the computation of the inner numerical radius, the modulus of the point on the boundary of the field of values closest to the origin, which carries significance for instance for the numerical solution of a definite generalized symmetric eigenvalue problem.

background

Nonsmooth Rate-of-Convergence Analyses of Algorithms for Eigenvalue Optimization

We analyze the spectrum of the massless Dirac operator on the 3-torus $\mathbb{T}^3$. It is known that it is possible to calculate this spectrum explicitly, that it is symmetric about zero and that each eigenvalue has even multiplicity. However, for a general oriented closed Riemannian 3-manifold $(M,g)$ there is no reason for the spectrum of the massless Dirac operator to be symmetric. Using perturbation theory, we derive the asymptotic formulae for its eigenvalues and prove that by the perturbation of the Euclidean metric on the 3-torus, it is possible to obtain spectral asymmetry of the massless Dirac operator in the axisymmetric case.

/pdf/analyzing-the-spectral-a-symmetry-of-the-massless-dirac-l3sz71ssfq.pdf

Perturbation theory of eigenvalue problems

Citations

A new adaptive eigendecomposition algorithm based on a first-order perturbation criterion

Equality in Wielandt's eigenvalue inequality

Semilinear elliptic PDE's with biharmonic operator and a singular potential

Nonsmooth Rate-of-Convergence Analyses of Algorithms for Eigenvalue Optimization

Analyzing the spectral (a)symmetry of the massless Dirac operator on the 3-torus.

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