Luka Grubišić

On estimators for eigenvalue/eigenvector approximations

We consider a large class of residuum based a posteriori eigen-value/eigenvector estimates and present an abstract framework for proving their asymptotic exactness. Equivalence of the estimator and the error is also established. To demonstrate the strength of our abstract approach we present a detailed study of hierarchical error estimators for Laplace eigenvalue problems in planar polygonal regions. To this end we develop new error analysis for the Galerkin approximation which avoids the use of the strengthened Cauchy-Schwarz inequality and the saturation assumption, and gives reasonable and explicitly computable upper bounds on the discretization error. A brief discussion is also given concerning the design of estimators which are in the same spirit, but are based on different a posteriori techniques-notably, those of gradient recovery type.

Representation Theorems for Indefinite Quadratic Forms Revisited

The first and second representation theorems for sign-inde finite, not necessarily semi-bounded quadratic forms are revisited. New straightforward proofs of these theorems are given. A number of necessary and sufficient conditions ensur ing the second representation the- orem to hold is proved. A new simple and explicit example of a self-adjoint operator for which the second representation theorem does not hold is also provided.

On the eigenvalue decay of solutions to operator Lyapunov equations

This paper is concerned with the eigenvalue decay of the solution to operator Lyapunov equations with right-hand sides of finite rank. We show that the kth (generalized) eigenvalue decays exponentially in root k, provided that the involved operator A generates an exponentially stable analytic semigroup, and A is either self-adjoint or diagonalizable with its eigenvalues contained in a strip around the real axis. Numerical experiments with discretizations of 1D and 2D PDE control problems confirm this decay

The Tan 2

A version of the Davis-Kahan Tan

\Theta

2\Theta

Theorem for indefinite quadratic forms

theorem [SIAM J. Numer. Anal. \textbf{7} (1970), 1 -- 46] for not necessarily semibounded linear operators defined by quadratic forms is proven. This theorem generalizes a recent result by Motovilov and Selin [Integr. Equat. Oper. Theory \textbf{56} (2006), 511 -- 542].

Efficient and reliable hp-FEM estimates for quadratic eigenvalue problems and photonic crystal applications

We present a-posteriori analysis of higher order finite element approximations (hp-FEM) for quadratic Fredholm-valued operator functions. Residual estimates for approximations of the algebraic eigenspaces are derived and we reduce the analysis of the estimator to the analysis of an associated boundary value problem. For the reasons of robustness we also consider approximations of the associated invariant pairs. We show that our estimator inherits the efficiency and reliability properties of the underlying boundary value estimator. As a model problem we consider spectral problems arising in analysis of photonic crystals. In particular, we present an example where a targeted family of eigenvalues cannot be guaranteed to be semisimple. Numerical experiments with hp-FEM show the predicted convergence rates. The measured effectivities of the estimator compare favorably with the performance of the same estimator on the associated boundary value problem. We also present a benchmark estimator, based on the dual weighted residual (DWR) approach, which is more expensive to compute but whose measured effectivities are close to one.

The rotation of eigenspaces of perturbed matrix pairs II

This paper studies the perturbation theory for spectral projections of Hermitian matrix pairs , where is a non-singular Hermitian matrix which can be factorized as , and is positive definite. The class of allowed perturbations is so restricted that the corresponding perturbed pair must have the form , and is positive definite. The main contribution of the paper is a theorem which generalizes the main result from the first part of the paper to this more general setting. Our estimate, in its most general form, depends on a uniform norm bound on a set of all -unitary matrices which diagonalize . The second main contribution is a new sharp uniform estimate of a norm of all -unitary matrices which diagonalize such that is a quasi-definite matrix. The case of a quasi-definite pair is therefore the case where our bounds are most competitive. We present numerical experiments to corroborate the theory.

A Subspace Iteration Algorithm for Fredholm Valued Functions

We present an algorithm for approximating an eigensubspace of a spectral component of an analytic Fredholm valued function. Our approach is based on numerical contour integration and the analytic Fredholm theorem. The presented method can be seen as a variant of the FEAST algorithm for infinite dimensional nonlinear eigenvalue problems. Numerical experiments illustrate the performance of the algorithm for polynomial and rational eigenvalue problems.

Discrete perturbation estimates for eigenpairs of Fredholm operator-valued functions

We present perturbation estimates for eigenvalue and eigenvector approximations for a class of Fredholm operator-valued functions. Our approach is based on perturbation estimates for the generalized resolvents and the exponential convergence of the contour integration by the trapezoidal rule. We use discrete residual functions to estimate the resolvents a posteriori. Numerical experiments are also presented.