An adaptive algorithm based on the shifted inverse iteration for the Steklov eigenvalue problem

This paper proposes and analyzes an a posteriori error estimator for the finite element multi-scale discretization approximation of the Steklov eigenvalue problem. Based on the a posteriori error estimates, an adaptive algorithm of shifted inverse iteration type is designed. Finally, numerical experiments comparing the performances of three kinds of different adaptive algorithms are provided, which illustrate the efficiency of the adaptive algorithm proposed here

Hessian-based adaptive sparse quadrature for infinite-dimensional Bayesian inverse problems

In this work we propose and analyze a Hessian-based adaptive sparse quadrature to compute infinite-dimensional integrals with respect to the posterior distribution in the context of Bayesian inverse problems with Gaussian prior. Due to the concentration of the posterior distribution in the domain of the prior distribution, a prior-based parametrization and sparse quadrature may fail to capture the posterior distribution and lead to erroneous evaluation results. By using a parametrization based on the Hessian of the negative log-posterior, the adaptive sparse quadrature can effectively allocate the quadrature points according to the posterior distribution. A dimension-independent convergence rate of the proposed method is established under certain assumptions on the Gaussian prior and the integrands. Dimension-independent and faster convergence than $O(N^{-1/2})$ is demonstrated for a linear as well as a nonlinear inverse problem whose posterior distribution can be effectively approximated by a Gaussian distribution at the MAP point

Superconvergent Two-grid Methods For Elliptic Eigenvalue Problems

Some numerical algorithms for elliptic eigenvalue problems are proposed, analyzed, and numerically tested. The methods combine advantages of the two-grid algorithm, two-space method, the shifted inverse power method, and the polynomial preserving recovery technique . Our new algorithms compare favorably with some existing methods and enjoy superconvergence property.Comment: 18 pages, 9 figure

Convergence Rate and Quasi-Optimal Complexity of Adaptive Finite Element Computations for Multiple Eigenvalues

In this paper, we study an adaptive finite element method for multiple eigenvalue problems of a class of second order elliptic equations. By using some eigenspace approximation technology and its crucial property which is also presented in this paper, we extend the results in \cite{dai-xu-zhou08} to multiple eigenvalue problems, we obtain both convergence rate and quasi-optimal complexity of the adaptive finite element eigenvalue approximation.Comment: 38 pages, 9 figures, 34 references. arXiv admin note: text overlap with arXiv:1201.2308 by other authors. text overlap with arXiv:1201.2308 by other authors without attributio

Perturbed preconditioned inverse iteration for operator eigenvalue problems with applications to adaptive wavelet discretization

In this paper we discuss an abstract iteration scheme for the calculation of the smallest eigenvalue of an elliptic operator eigenvalue problem. A short and geometric proof based on the preconditioned inverse iteration (PINVIT) for matrices [Knyazev and Neymeyr, (2009)] is extended to the case of operators. We show that convergence is retained up to any tolerance if one only uses approximate applications of operators which leads to the perturbed preconditioned inverse iteration (PPINVIT). We then analyze the Besov regularity of the eigenfunctions of the Poisson eigenvalue problem on a polygonal domain, showing the advantage of an adaptive solver to uniform refinement when using a stable wavelet base. A numerical example for PPINVIT, applied to the model problem on the L-shaped domain, is shown to reproduce the predicted behaviour.Comment: submitted to Adv. Comp. Mat

Low-Rank Solution Methods for Stochastic Eigenvalue Problems

We study efficient solution methods for stochastic eigenvalue problems arising from discretization of self-adjoint partial differential equations with random data. With the stochastic Galerkin approach, the solutions are represented as generalized polynomial chaos expansions. A low-rank variant of the inverse subspace iteration algorithm is presented for computing one or several minimal eigenvalues and corresponding eigenvectors of parameter-dependent matrices. In the algorithm, the iterates are approximated by low-rank matrices, which leads to significant cost savings. The algorithm is tested on two benchmark problems, a stochastic diffusion problem with some poorly separated eigenvalues, and an operator derived from a discrete stochastic Stokes problem whose minimal eigenvalue is related to the inf-sup stability constant. Numerical experiments show that the low-rank algorithm produces accurate solutions compared to the Monte Carlo method, and it uses much less computational time than the original algorithm without low-rank approximation

Accurate Inverses for Computing Eigenvalues of Extremely Ill-conditioned Matrices and Differential Operators

This paper is concerned with computations of a few smaller eigenvalues (in absolute value) of a large extremely ill-conditioned matrix. It is shown that smaller eigenvalues can be accurately computed for a diagonally dominant matrix or a product of diagonally dominant matrices by combining a standard iterative method with the accurate inversion algorithms that have been developed for such matrices. Applications to the finite difference discretization of differential operators are discussed. In particular, a new discretization is derived for the 1-dimensional biharmonic operator that can be written as a product of diagonally dominant matrices. Numerical examples are presented to demonstrate the accuracy achieved by the new algorithms

Globally Constructed Adaptive Local Basis Set for Spectral Projectors of Second Order Differential Operators

Spectral projectors of second order differential operators play an important role in quantum physics and other scientific and engineering applications. In order to resolve local features and to obtain converged results, typically the number of degrees of freedom needed is much larger than the rank of the spectral projector. This leads to significant cost in terms of both computation and storage. In this paper, we develop a method to construct a basis set that is adaptive to the given differential operator. The basis set is systematically improvable, and the local features of the projector is built into the basis set. As a result the required number of degrees of freedom is only a small constant times the rank of the projector. The construction of the basis set uses a randomized procedure, and only requires applying the differential operator to a small number of vectors on the global domain, while each basis function itself is supported on strictly local domains and is discontinuous across the global domain. The spectral projector on the global domain is systematically approximated from such a basis set using the discontinuous Galerkin (DG) method. The global construction procedure is very flexible, and allows a local basis set to be consistently constructed even if the operator contains a nonlocal potential term. We verify the effectiveness of the globally constructed adaptive local basis set using one-, two- and three-dimensional linear problems with local potentials, as well as a one dimensional nonlinear problem with nonlocal potentials resembling the Hartree-Fock problem in quantum physics

A study on anisotropic mesh adaptation for finite element approximation of eigenvalue problems with anisotropic diffusion operators

Anisotropic mesh adaptation is studied for the linear finite element solution of eigenvalue problems with anisotropic diffusion operators. The M-uniform mesh approach is employed with which any nonuniform mesh is characterized mathematically as a uniform one in the metric specified by a metric tensor. Bounds on the error in the computed eigenvalues are established for quasi-M-uniform meshes. Numerical examples arising from the Laplace-Beltrami operator on parameterized surfaces, nonlinear diffusion, thermal diffusion in a magnetic field in plasma physics, and the Laplacian operator on an L-shape domain are presented. Numerical results show that anisotropic adaptive meshes can lead to more accurate computed eigenvalues than uniform or isotropic adaptive meshes. They also confirm the second order convergence of the error that is predicted by the theoretical analysis. The effects of approximation of curved boundaries on the computation of eigenvalue problems is also studied in two dimensions. It is shown that the initial mesh used to define the geometry of the physical domain should contain at least \sqrt{N} boundary points to keep the effects of boundary approximation at the level of the error of the finite element approximation, where N is the number of the elements in the final adaptive mesh. Only about N^{1/3} boundary points in the initial mesh are needed for boundary value problems. This implies that the computation of eigenvalue problems is more sensitive to the boundary approximation than that of boundary value problems.Comment: 29 page

A geometric method for eigenvalue problems with low rank perturbations

We consider the problem of finding the spectrum of an operator taking the form of a low-rank (rank one or two) non-normal perturbation of a well-understood operator, motivated by a number of problems of applied interest which take this form. We use the fact that the system is a low rank perturbation of a solved problem, together with a simple idea of classical differential geometry (the envelope of a family of curves) to completely analyze the spectrum. We use these techniques to analyze three problems of this form: a model of the oculomotor integrator due to Anastasio and Gad (2007), a continuum integrator model, and a nonlocal model of phase separation due to Rubinstein and Sternberg (1992)