Product Eigenvalue Problems

Watkins, David S.

doi:10.1137/s0036144504443110

Cited by 38 publications

(33 citation statements)

References 88 publications

(141 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This eigenvalue relation of block cyclic matrices has also been observed in earlier studies [34], [40]- [42]. One immediate consequence of this eigenfamily structure of the M -block cyclic graph is that eigenvalues can be real only for M 2.…”

Section: -Isupporting

confidence: 81%

Extending Classical Multirate Signal Processing Theory to Graphs—Part I: Fundamentals

Teke

Vaidyanathan

2017

IEEE Trans. Signal Process.

View full text Add to dashboard Cite

Abstract-Signal processing on graphs finds applications in many areas. In recent years renewed interest on this topic was kindled by two groups of researchers. Narang and Ortega constructed two-channel filter banks on bipartitie graphs described by Laplacians. Sandryhaila and Moura developed the theory of linear systems, filtering, and frequency responses for the case of graphs with arbitrary adjacency matrices, and showed applications in signal compression, prediction, etc. Inspired by these contributions, this paper extends classical multirate signal processing ideas to graphs. The graphs are assumed to be general with a possibly non-symmetric and complex adjacency matrix. The paper revisits ideas such as noble identities, aliasing, and polyphase decompositions in graph multirate systems. Drawing such a parallel to classical systems allows one to design filter banks with polynomial filters, with lower complexity than arbitrary graph filters. It is shown that the extension of classical multirate theory to graphs is nontrivial, and requires certain mathematical restrictions on the graph. Thus, classical noble identities cannot be taken for granted. Similarly, one cannot claim that the so-called delay chain system is a perfect reconstruction system (as in classical filter banks). It will also be shown that M -partite extensions of the bipartite filter bank results will not work for M -channel filter banks, but a more restrictive condition called M -block cyclic property should be imposed. Such graphs are studied in detail. A detailed theory for M -channel filter banks is developed in a companion paper.

show abstract

Section: -Isupporting

confidence: 81%

Extending Classical Multirate Signal Processing Theory to Graphs—Part I: Fundamentals

Teke

Vaidyanathan

2017

IEEE Trans. Signal Process.

View full text Add to dashboard Cite

show abstract

“…17,18 This eigenvalue relation of block cyclic matrices has also been observed in earlier studies. [19][20][21][22] This property is as follows: Theorem 2 (Eigen-families of M -Block cyclic graphs). Eigenvalues and eigenvectors of the adjacency matrix of an M -Block cyclic graph come as families of size M .…”

Section: Eigen-properties Of M -Block Cyclic Graphsmentioning

confidence: 99%

Extending classical multirate signal processing theory to graphs

Teke

Vaidyanathan

2017

Wavelets and Sparsity XVII

View full text Add to dashboard Cite

A variety of different areas consider signals that are defined over graphs. Motivated by the advancements in graph signal processing, this study first reviews some of the recent results on the extension of classical multirate signal processing to graphs. In these results, graphs are allowed to have directed edges. The possibly non-symmetric adjacency matrix A is treated as the graph operator. These results investigate the fundamental concepts for multirate processing of graph signals such as noble identities, aliasing, and perfect reconstruction (PR). It is shown that unless the graph satisfies some conditions, these concepts cannot be extended to graph signals in a simple manner. A structure called M -Block cyclic structure is shown to be sufficient to generalize the results for bipartite graphs on two-channels to M -channel filter banks. Many classical multirate ideas can be extended to graphs due to the unique eigenstructure of M -Block cyclic graphs. For example, the PR condition for filter banks on these graphs is identical to PR in classical theory, which allows the use of well-known filter bank design techniques. In order to utilize these results, the adjacency matrix of an M -Block cyclic graph should be given in the correct permutation. In the final part, this study proposes a spectral technique to identify the hidden M -Block cyclic structure from a graph with noisy edges whose adjacency matrix is given under a random permutation. Numerical simulation results show that the technique can recover the underlying M -Block structure in the presence of random addition and deletion of the edges.

show abstract

“…The periodic Schur decomposition, and more accurate algorithms to compute it are described in [Bojanczyk et al, 1992], [Varga & Dooren, 2001], and [Kressner, 2005]. See also the review [Watkins, 2005] on product eigenvalue problems. In the context of the multiple shooting computation of periodic orbits it has been used in [Lust, 1997] in the framework of the Newton-Picard method, and in [Lust, 2001] to improve the computation of Floquet multipliers.…”

Section: Proposition 2 Consider Now a Partial Periodic Schur Decomposmentioning

confidence: 99%

On the Multiple Shooting Continuation of Periodic Orbits by Newton–krylov Methods

Sánchez

Net

2010

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

The application of the multiple shooting method to the continuation of periodic orbits in large-scale dissipative systems is analyzed. A preconditioner for the linear systems which appear in the application of Newton's method is presented. It is based on the knowledge of invariant subspaces of the Jacobians at nearby solutions. The possibility of speeding up the process by using parallelism is studied for the thermal convection of a binary mixture of fluids in a rectangular domain, with positive results.

show abstract

Product Eigenvalue Problems

Cited by 38 publications

References 88 publications

Extending Classical Multirate Signal Processing Theory to Graphs—Part I: Fundamentals

Extending Classical Multirate Signal Processing Theory to Graphs—Part I: Fundamentals

Extending classical multirate signal processing theory to graphs

On the Multiple Shooting Continuation of Periodic Orbits by Newton–krylov Methods

Contact Info

Product

Resources

About