scholarly journals Finite-Rank ADI Iteration for Operator Lyapunov Equations

2013 ◽  
Vol 51 (5) ◽  
pp. 4084-4117 ◽  
Author(s):  
Mark R. Opmeer ◽  
Timo Reis ◽  
Winnifried Wollner
Keyword(s):  
Finite Rank ◽  
Lyapunov Equations ◽  
Adi Iteration

scholarly journals ADI iteration for Lyapunov equations: A tangential approach and adaptive shift selection

2016 ◽  
Vol 109 ◽  
pp. 85-95
Author(s):  
Thomas Wolf ◽  
Heiko K.F. Panzer ◽  
Boris Lohmann
Keyword(s):  
Lyapunov Equations ◽  
Adaptive Shift ◽  
Adi Iteration

scholarly journals Low-Rank Alternating Direction Implicit Iteration in pyMOR

2020 ◽  
Vol 2 (1) ◽  
pp. 1-13
Author(s):  
Linus Balicki
Keyword(s):  
Large Scale ◽  
Low Rank ◽  
Balanced Truncation ◽  
Lyapunov Equations ◽  
Software Library ◽  
Alternating Direction Implicit ◽  
Truncation Method ◽  
Alternating Direction ◽  
Adi Iteration ◽  
Implicit Iteration

The low-rank alternating direction implicit (LR-ADI) iteration is an effective method for solving large-scale Lyapunov equations. In the software library pyMOR, solutions to Lyapunov equations play an important role when reducing a model using the balanced truncation method. In this article we introduce the LR-ADI iteration as well as pyMOR, while focusing on its features which are relevant for integrating the iteration into the library. We compare the run time of the iteration's pure pyMOR implementation with those achieved by external libraries available within the pyMOR framework.

scholarly journals Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

scholarly journals Coarse median algebras: the intrinsic geometry of coarse median spaces and their intervals

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Graham A. Niblo ◽  
Nick Wright ◽  
Jiawen Zhang
Keyword(s):  
Finite Rank ◽  
Intrinsic Geometry ◽  
Bounded Geometry ◽  
Higher Dimensional ◽  
Median Algebra ◽  
Asymptotic Geometry ◽  
Median Operator

AbstractThis paper establishes a new combinatorial framework for the study of coarse median spaces, bridging the worlds of asymptotic geometry, algebra and combinatorics. We introduce a simple and entirely algebraic notion of coarse median algebra which simultaneously generalises the concepts of bounded geometry coarse median spaces and classical discrete median algebras. We study the coarse median universe from the perspective of intervals, with a particular focus on cardinality as a proxy for distance. In particular we prove that the metric on a quasi-geodesic coarse median space of bounded geometry can be constructed up to quasi-isometry using only the coarse median operator. Finally we develop a concept of rank for coarse median algebras in terms of the geometry of intervals and show that the notion of finite rank coarse median algebra provides a natural higher dimensional analogue of Gromov’s concept of $$\delta $$ δ -hyperbolicity.

scholarly journals Mutual information for low-rank even-order symmetric tensor estimation

2020 ◽  
Author(s):  
Clément Luneau ◽  
Jean Barbier ◽  
Nicolas Macris
Keyword(s):  
Mutual Information ◽  
Finite Rank ◽  
Interpolation Method ◽  
Symmetric Tensor ◽  
Low Rank ◽  
Tensor Factorization ◽  
Adaptive Interpolation ◽  
Odd Order ◽  
Rank One ◽  
Even Order

Abstract We consider a statistical model for finite-rank symmetric tensor factorization and prove a single-letter variational expression for its asymptotic mutual information when the tensor is of even order. The proof applies the adaptive interpolation method originally invented for rank-one factorization. Here we show how to extend the adaptive interpolation to finite-rank and even-order tensors. This requires new non-trivial ideas with respect to the current analysis in the literature. We also underline where the proof falls short when dealing with odd-order tensors.

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