Related papers: Adaptive Eigenvalue Computation - Complexity Estim…
We present a new algorithm for solving an eigenvalue problem for a real symmetric matrix which is a rank-one modification of a diagonal matrix. The algorithm computes each eigenvalue and all components of the corresponding eigenvector with…
In this article a unified approach to iterative soft-thresholding algorithms for the solution of linear operator equations in infinite dimensional Hilbert spaces is presented. We formulate the algorithm in the framework of generalized…
We propose a geometric numerical analysis of SDEs admitting Lie symmetries which allows us to individuate a symmetry adapted coordinates system where the given SDE has notable invariant properties. An approximation scheme preserving the…
We describe a novel algorithm for solving general parametric (nonlinear) eigenvalue problems. Our method has two steps: first, high-accuracy solutions of non-parametric versions of the problem are gathered at some values of the parameters;…
In this paper, it is shown that any well-posed 2nd order PDE can be reformulated as a well-posed first order least squares system. This system will be solved by an adaptive wavelet solver in optimal computational complexity. The…
We consider the eigenvalue problem for the case where the input matrix is symmetric and its entries perturb in some given intervals. We present a characterization of some of the exact boundary points, which allows us to introduce an inner…
The goal of this paper is to survey the properties of the eigenvalue relaxation for least squares binary problems. This relaxation is a convex program which is obtained as the Lagrangian dual of the original problem with an implicit compact…
In this paper we discuss spectral properties of operators associated with the least-squares finite element approximation of elliptic partial differential equations. The convergence of the discrete eigenvalues and eigenfunctions towards the…
Consider a symmetric matrix $A(v)\in\RR^{n\times n}$ depending on a vector $v\in\RR^n$ and satisfying the property $A(\alpha v)=A(v)$ for any $\alpha\in\RR\backslash{0}$. We will here study the problem of finding $(\lambda,v)\in\RR\times…
The solution of (generalized) eigenvalue problems for symmetric or Hermitian matrices is a common subtask of many numerical calculations in electronic structure theory or materials science. Solving the eigenvalue problem can easily amount…
Many eigenvalue problems arising in practice are often of the generalized form $A\x=\lambda B\x$. One particularly important case is symmetric, namely $A, B$ are Hermitian and $B$ is positive definite. The standard algorithm for solving…
In this paper, we discuss approximating the eigenvalue problem of biharmonic equation. We first present an equivalent mixed formulation which admits amiable nested discretization. Then, we construct multi-level finite element schemes by…
We consider scalar semilinear elliptic PDEs, where the nonlinearity is strongly monotone, but only locally Lipschitz continuous. To linearize the arising discrete nonlinear problem, we employ a damped Zarantonello iteration, which leads to…
Sequential data collection has emerged as a widely adopted technique for enhancing the efficiency of data gathering processes. Despite its advantages, such data collection mechanism often introduces complexities to the statistical inference…
In this paper, we propose a decomposition approach for eigenvalue problems with spatial symmetries, including the formulation, discretization as well as implementation. This approach can handle eigenvalue problems with either Abelian or…
We consider the minimization or maximization of the $J$th largest eigenvalue of an analytic and Hermitian matrix-valued function, and build on Mengi et al. (2014, SIAM J. Matrix Anal. Appl., 35, 699-724). This work addresses the setting…
Eigenvalue problems serve as fundamental substrates for applications in large-scale scientific simulations and machine learning, often requiring computation on massively parallel platforms. As these platforms scale to hundreds of thousands…
We analyze adaptive mesh-refining algorithms for conforming finite element discretizations of certain non-linear second-order partial differential equations. We allow continuous polynomials of arbitrary, but fixed polynomial order. The…
We derive a formula that simplifies the original asymptotic iteration method formulation to find the energy eigenvalues for the analytically solvable cases. We then show that there is a connection between the asymptotic iteration and the…
We tackle the problem of building adaptive estimation procedures for ill-posed inverse problems. For general regularization methods depending on tuning parameters, we construct a penalized method that selects the optimal smoothing sequence…