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Register a new userBounds on the convergence of Ritz values from Krylov subspaces to interior eigenvalues of Hermitean matrices
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We consider bounds on the convergence of Ritz values from a sequence of Krylov subspaces to interior eigenvalues of Hermitean matrices. These bounds are useful in regions of low spectral density, for example near voids in the spectrum, as is required in many applications. Our bounds are obtained by considering the usual Kaniel-Paige-Saad formalism applied to the shifted and squared matrix.
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A priori, a posteriori, and mixed type upper bounds for the absolute change in Ritz values of self-adjoint matrices in terms of submajorization relations are obtained. Some of our results prove recent conjectures by Knyazev, Argentati, and Zhu, which extend several known results for one dimensional subspaces to arbitrary subspaces. In addition, we improve Nakatsukasas version of the $tan Theta$ theorem of Davis and Kahan. As a consequence, we obtain new quadratic a posteriori bounds for the absolute change in Ritz values.
Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep Ritz method (DRM) cite{wan11} for second order elliptic equations with Neumann boundary conditions. We establish the first nonasymptotic convergence rate in $H^1$ norm for DRM using deep networks with $mathrm{ReLU}^2$ activation functions. In addition to providing a theoretical justification of DRM, our study also shed light on how to set the hyper-parameter of depth and width to achieve the desired convergence rate in terms of number of training samples. Technically, we derive bounds on the approximation error of deep $mathrm{ReLU}^2$ network in $H^1$ norm and on the Rademacher complexity of the non-Lipschitz composition of gradient norm and $mathrm{ReLU}^2$ network, both of which are of independent interest.
We give upper and lower bounds on the determinant of a perturbation of the identity matrix or, more generally, a perturbation of a nonsingular diagonal matrix. The matrices considered are, in general, diagonally dominant. The lower bounds are best possible, and in several cases they are stronger than well-known bounds due to Ostrowski and other authors. If $A = I-E$ is an $n times n$ matrix and the elements of $E$ are bounded in absolute value by $varepsilon le 1/n$, then a lower bound of Ostrowski (1938) is $det(A) ge 1-nvarepsilon$. We show that if, in addition, the diagonal elements of $E$ are zero, then a best-possible lower bound is [det(A) ge (1-(n-1)varepsilon),(1+varepsilon)^{n-1}.] Corresponding upper bounds are respectively [det(A) le (1 + 2varepsilon + nvarepsilon^2)^{n/2}] and [det(A) le (1 + (n-1)varepsilon^2)^{n/2}.] The first upper bound is stronger than Ostrowskis bound (for $varepsilon < 1/n$) $det(A) le (1 - nvarepsilon)^{-1}$. The second upper bound generalises Hadamards inequality, which is the case $varepsilon = 1$. A necessary and sufficient condition for our upper bounds to be best possible for matrices of order $n$ and all positive $varepsilon$ is the existence of a skew-Hadamard matrix of order $n$.
The main of this work is to use the unit lower triangular matrices for solving inverse eigenvalue problem of nonnegative matrices and present the easier method to solve this problem.
We provide upper bounds on the perturbation of invariant subspaces of normal matrices measured using a metric on the space of vector subspaces of $mathbb{C}^n$ in terms of the spectrum of both the unperturbed & perturbed matrices, as well as, spectrum of the unperturbed matrix only. The results presented give tighter bounds than the Davis-Khan $sinTheta$ theorem. We apply the result to a graph perturbation problem.
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