Wiener’s lemma: localization and various approaches

Shin, Chang Eon; Sun, Qiyu

doi:10.1007/s11766-013-3215-6

Cited by 22 publications

(10 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…First we follow the argument used in the proof of Theorem IV.4 in [19] to show that the inverse of a positive definite matrix with limited geodesic-width has exponential off-diagonal decay. The well-localization for the inverse of matrices is of great importance in applied harmonic analysis, numerical analysis, distributed optimization and many mathematical and engineering fields, see [13,16,21,37,38] for historical remarks and recent advances. where ω is a positive integer and c, L, M are positive constants with 0 < c < L. Then A −1 = (G 1 (i, j)) i,j∈V and A −1 B = (G 2 (i, j)) i,j∈V have exponential off-diagonal decay,…”

Section: Proofsmentioning

confidence: 99%

A Divide-and-Conquer Algorithm for Distributed Optimization on Networks

Emirov¹,

Song²,

Sun³

2021

Preprint

View full text Add to dashboard Cite

In this paper, we consider networks with topologies described by some connected undirected graph G = (V, E) and with some agents (fusion centers) equipped with processing power and local peer-to-peer communication, and optimization problem minx F (x) = i∈V fi(x) with local objective functions fi depending only on neighboring variables of the vertex i ∈ V . We introduce a divide-and-conquer algorithm to solve the above optimization problem in a distributed and decentralized manner. The proposed divide-and-conquer algorithm has exponential convergence, its computational cost is almost linear with respect to the size of the network, and it can be fully implemented at fusion centers of the network. Our numerical demonstrations also indicate that the proposed divide-and-conquer algorithm has superior performance than popular decentralized optimization methods do for the least squares problem with/without 1 penalty.

show abstract

Section: Proofsmentioning

confidence: 99%

A Divide-and-Conquer Algorithm for Distributed Optimization on Networks

Emirov¹,

Song²,

Sun³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Такая алгебра операторов называлась в [15] алгеброй Гохберга-Баскакова-Сёструнда. В тривиальном случае, если u(i, j) = u 0 (i−j), где u…”

Section: классы операторов и основные определенияunclassified

“…1 определения 7. Напомним, что End u H -полное пространство [15]. Оператор J m определяется формулой (11).…”

Section: нб усковаunclassified

“…Оператор X ∈ M = S 2,α (H) есть решение нелинейного операторного уравнения (9), рассматриваемого в пространстве M = S 2,α (H) с трансформаторами J m и Γ m , определяемыми формулами (18). Имеет место равенство (14), (15).…”

Section: нб усковаunclassified

See 1 more Smart Citation

The matrix analysis of spectral projections for the perturbed self-adjoint operators

Ускова¹

2019

Sib. Elektron. Mat. Izv.

View full text Add to dashboard Cite

We study bounded perturbations of an unbounded positive definite self-adjoint operator with discrete spectrum. The spectrum has semi-simple eigenvalues with finite geometric multiplicity and the perturbation belongs to operator space defined by rate of the off-diagonal decay of the operator matrix. We show that the spectral projections and the resolvent of the perturbed operator belong to the same space as the perturbation. These results are applied to the Hill operator and the operator with matrix potential. We also consider the inverse problem and the modified Galerkin method.

show abstract

“…The reader may refer to [2,3,4,6,10,11,12,13] for historical remarks and more properties of the above three classes of matrices. For 1 ≤ p ≤ ∞, a weight u is called a p-submultiplicative weight if there exists another weight v satisfying…”

Section: Differentiable Matrix Algebrasmentioning

confidence: 99%