Scalable frames

Kutyniok, Gitta; Okoudjou, Kasso A.; Philipp, Friedrich; Tuley, Elizabeth K.

doi:10.1016/j.laa.2012.10.046

Cited by 53 publications

(55 citation statements)

References 15 publications

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“…For general background references on frames in Hilbert space, we refer to [8,12,16,24,25,30], and for electrical networks, see [3,4,9,15,29,33,35]. The facts on electrical networks we need are the laws of Kirchhoff and Ohm, and our computation of the frame coefficients as electrical currents is based on this, in part.…”

Section: Electrical Current As Frame Coefficientsmentioning

confidence: 99%

“…We proved in Lemma 4.2 (3) that u, ∆u l 2 ≥ 0 holds for all u ∈ l 2 where ·, · l 2 referes to the un-weighted l 2 -inner product. The connection between the two inner products is as follows: u, cu l 2 = u 2 l 2 ( c) which yields the following: Using (6.23) and (6.24), we get 25) so cP u = cu − ∆u, and as a point-wise identity on V . Hence…”

Section: Proofmentioning

confidence: 99%

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Frames and factorization of graph Laplacians

Jørgensen¹,

Tian²

2015

OpMath

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Abstract. Using functions from electrical networks (graphs with resistors assigned to edges), we prove existence (with explicit formulas) of a canonical Parseval frame in the energy Hilbert space HE of a prescribed infinite (or finite) network. Outside degenerate cases, our Parseval frame is not an orthonormal basis. We apply our frame to prove a number of explicit results: With our Parseval frame and related closable operators in HE we characterize the Friedrichs extension of the HE-graph Laplacian. We consider infinite connected network-graphs G = (V, E), V for vertices, and E for edges. To every conductance function c on the edges E of G, there is an associated pair (HE, ∆) where HE in an energy Hilbert space, and ∆ (= ∆c) is the c-graph Laplacian; both depending on the choice of conductance function c. When a conductance function is given, there is a current-induced orientation on the set of edges and an associated natural Parseval frame in HE consisting of dipoles. Now ∆ is a well-defined semibounded Hermitian operator in both of the Hilbert l 2 (V ) and HE. It is known to automatically be essentially selfadjoint as an l 2 (V )-operator, but generally not as an HE operator. Hence as an HE operator it has a Friedrichs extension. In this paper we offer two results for the Friedrichs extension: a characterization and a factorization. The latter is via l 2 (V ).

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Section: Electrical Current As Frame Coefficientsmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

Frames and factorization of graph Laplacians

Jørgensen¹,

Tian²

2015

OpMath

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“…The following theorem gives a formula for rewriting the subspace correction scheme based on a W -splitting (27) as a subspace correction method based on a V -splitting (29).…”

Section: Algebraic Transformationsmentioning

confidence: 99%

“…i offers additional improvements. Note that a similar question has been discussed for frames in [27]. We keep the basic setup of given auxiliary problems in ¹H i ; b i º and consider the family of iterations…”

Section: Introducing Multiple Scaling Parametersmentioning

confidence: 99%

Optimal scaling parameters for sparse grid discretizations

Griebel

Hullmann

Oswald

2014

Numer. Linear Algebra Appl.

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We apply iterative subspace correction methods to elliptic PDE problems discretized by generalized sparse grid systems. The involved subspace solvers are based on the combination of all anisotropic full grid spaces that are contained in the sparse grid space. Their relative scaling is at our disposal and has significant influence on the performance of the iterative solver. In this paper, we follow three approaches to obtain close-to-optimal or even optimal scaling parameters of the subspace solvers and thus of the overall subspace correction method. We employ a Linear Program that we derive from the theory of additive subspace splittings, an algebraic transformation that produces partially negative scaling parameters that result in improved asymptotic convergence properties, and finally, we use the OptiCom method as a variable nonlinear preconditioner

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“…A much more simple procedure, which can also be regarded as preconditioning by a diagonal matrix, is to scale each frame vector to generate a Parseval frame. Characterizing conditions -also of geometric type -for a frame to be scalable in this sense were obtained by Kutyniok et al (2013).…”

Section: Construction Of Tight Framesmentioning

confidence: 99%

Sparse matrices in frame theory

2013

Self Cite

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Frame theory is closely intertwined with signal processing through a canon of methodologies for the analysis of signals using (redundant) linear measurements. The canonical dual frame associated with a frame provides a means for reconstruction by a least squares approach, but other dual frames yield alternative reconstruction procedures. The novel paradigm of sparsity has recently entered the area of frame theory in various ways. Of those different sparsity perspectives, we will focus on the situations where frames and (not necessarily canonical) dual frames can be written as sparse matrices. The objective for this approach is to ensure not only low-complexity computations, but also high compressibility. We will discuss both existence results and explicit constructions.

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Scalable frames

Cited by 53 publications

References 15 publications

Frames and factorization of graph Laplacians

Frames and factorization of graph Laplacians

Optimal scaling parameters for sparse grid discretizations

Sparse matrices in frame theory

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