Related papers: A homogeneous Rayleigh quotient with applications …
Optimization of quadratic functions and the quotient of those are relevant in subspace and iterative optimization methods. In this paper, the calculation of the generalized operator norm and extremal generalized Rayleigh quotient is…
Three fundamental variational principles used for solving elastodynamic eigenvalue problems are studied within the context of elastic wave propagation in periodic composites (phononics). We study the convergence of the eigenvalue problems…
We present an iterative method to diagonalise large matrices. The basic idea is the same as the conjugated gradient (CG) method, i.e, minimizing the Rayleigh quotient via its gradient and avoiding reintroduce errors to the directions of…
The main topic of this note is a discussion of applicability conditions of the Nehari manifold method depending on the value of parameters of equations. As the main tool, we apply the nonlinear generalized Rayleigh quotient method.
We present two novel methods for approximating minimizers of the abstract Rayleigh quotient $\Phi(u)/ \|u\|^p$. Here $\Phi$ is a strictly convex functional on a Banach space with norm $\|\cdot\|$, and $\Phi$ is assumed to be positively…
We propose a new stepsize for the gradient method. It is shown that this new stepsize will converge to the reciprocal of the largest eigenvalue of the Hessian, when Dai-Yang's asymptotic optimal gradient method (Computational Optimization…
Finding the eigenvalues connected to the covariance operator of a centred Hilbert-space valued Gaussian process is genuinely considered a hard problem in several mathematical disciplines. In statistics this problem arises for instance in…
We propose a theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach much like the Rayleigh quotient for symmetric matrix eigenvalues. These notions are particularly…
We introduce variational methods for finding approximate eigenfunctions and eigenvalues of quantum Hamiltonians by constructing a set of orthogonal wave functions which approximately solve the eigenvalue equation.
Extracting approximate eigenpairs from a prescribed subspace is of fundamental importance in eigenvalue computation. While projecting the target eigenvector onto the subspace yields satisfactory accuracy, extracting an approximate eigenpair…
The convergence of the Rayleigh-Ritz method with nonlinear parameters optimized through minimization of the trace of the truncated matrix is demonstrated by a comparison with analytically known eigenstates of various quasi-solvable systems.…
We are interested in computing a simple eigenpair $(\lambda,{\bf x})$ of a large non-Hermitian matrix $A$, by a general harmonic Rayleigh-Ritz projection method. Given a search subspace $\mathcal{K}$ and a target point $\tau$, we focus on…
It is well known that a family of $n\times n$ commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the $n$ joint eigenvalues of the family. In…
The generalized eigenvalue (GE) problems are of particular importance in various areas of science engineering and machine learning. We present a variational quantum algorithm for finding the desired generalized eigenvalue of the GE problem,…
We develop a quantitative theory of stochastic homogenization for linear, uniformly parabolic equations with coefficients depending on space and time. Inspired by recent works in the elliptic setting, our analysis is focused on certain…
We study a non-linear eigenvalue problem for vector-valued eigenfunctions and give a succinct uniqueness proof for minimizers of the associated Rayleigh quotient.
We study the rate of convergence for (variational) eigenvalues of several non-linear problems involving oscillating weights and subject to different kinds of boundary conditions in bounded domains.
We consider a convex minimization problem for which the objective is the sum of a homogeneous polynomial of degree four and a linear term. Such task arises as a subproblem in algorithms for quadratic inverse problems with a…
Thermoacoustic instabilities are one of the most challenging problems faced by gas turbine and rocket motor manufacturers. The key instability mechanism is described by the {\it Rayleigh criterion}. The Rayleigh criterion does not directly…
In this article we further develop a perturbation approach to the Rayleigh--Ritz approximations from our earlier work. We both sharpen the estimates and extend the applicability of the theory to nonnegative definite operators . The…