Fast Fourier optimization

Vanderbei, Robert J.

doi:10.1007/s12532-011-0034-8

Cited by 16 publications

(12 citation statements)

References 24 publications

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“…This is left for future investigation. The Kronecker idea is inspired by the fast-Fourier transform where dense matrices are split into products of sparse matrices ( [39], [44]). Hence, any optimization method that accommodates sparse matrix multiplication operations can potentially be altered to benefit further from a Kronecker compressed sensing scheme.…”

Section: Discussionmentioning

confidence: 99%

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Revisiting compressed sensing: exploiting the efficiency of simplex and sparsification methods

et al. 2016

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We propose two approaches to solve large-scale compressed sensing problems. The first approach uses the parametric simplex method to recover very sparse signals by taking a small number of simplex pivots, while the second approach reformulates the problem using Kronecker products to achieve faster computation via a sparser problem formulation. In particular, we focus on the computational aspects of these methods in compressed sensing. For the first approach, if the true signal is very sparse and we initialize our solution to be the zero vector, then a customized parametric simplex method usually takes a small number of iterations to converge. Our numerical studies show that this approach is 10 times faster than state-of-the-art methods for recovering very sparse signals. The second approach can be used when the sensing matrix is the Kronecker product of two smaller matrices. We show that the best-known sufficient condition for the Kronecker compressed sensing (KCS) strategy to obtain a perfect recovery is more restrictive than the corresponding condition if using the first approach. However, KCS can be formulated as a linear program with a very sparse constraint matrix, whereas the first approach involves a completely dense constraint matrix. Hence, algorithms that benefit from sparse problem representation, such as interior point methods (IPMs), are expected to have computational advantages for the KCS problem. We numerically demonstrate that KCS combined with IPMs is up to 10The first author's research is supported by ONR Award N00014-13-1-0093, the third author's by NSF Grant III-1116730, and the fourth author's by NSF Grant DMS-1005539.

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Section: Discussionmentioning

confidence: 99%

“…Specifically, for linear programming, the sparsity of the constraint matrix is a significant contributor towards computational efficiency [25,39]. In fact, we can view the decomposition in (P 7 ) as a sparsification technique analogous to one step of the fast-Fourier optimization idea described in [44].…”

Section: Ipm Kcs: Minmentioning

confidence: 99%

Revisiting compressed sensing: exploiting the efficiency of simplex and sparsification methods

et al. 2016

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“…For our linear programming computations, we used GLPK within SageMath. 24 We believe that our code could be sped up significantly by incorporating the fast Fourier transform, 25 possibly giving new bounds on ω(G p ) for significantly larger primes p.…”

Section: Future Workmentioning

confidence: 99%

Linear programming bounds for cliques in Paley graphs

Magsino¹,

Mixon²,

Parshall³

2019

Wavelets and Sparsity XVIII

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The Lovász theta number is a semidefinite programming bound on the clique number of (the complement of) a given graph. Given a vertex-transitive graph, every vertex belongs to a maximal clique, and so one can instead apply this semidefinite programming bound to the local graph. In the case of the Paley graph, the local graph is circulant, and so this bound reduces to a linear programming bound, allowing for fast computations. Impressively, the value of this program with Schrijver's nonnegativity constraint rivals the state-of-the-art closed-form bound recently proved by Hanson and Petridis. We conjecture that this linear programming bound improves on the Hanson-Petridis bound infinitely often, and we derive the dual program to facilitate proving this conjecture.

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“…Shaped pupils, which were initially optimized for highcontrast imaging in one dimension (Spergel & Kasdin 2001;Vanderbei et al 2003;Kasdin et al 2007), can be numerically optimized in two dimensions (2D) for any telescope aperture Vanderbei 2012). Their versatility, robustness, and achromaticity make them good candidates for compact coronagraphic instruments: unlike APLCs, shaped pupils do not rely on a Lyot mask or a Lyot stop to create a high contrast, although a field stop is probably mandatory, given the limited A&A 566, A31 (2014) dynamic range of detectors.…”

Section: Introductionmentioning

confidence: 99%

Apodized phase mask coronagraphs for arbitrary apertures

Carlotti

Pueyo

Mawet

2014

A&A

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Context. With a clear circular aperture, the vortex coronagraph perfectly cancels an on-axis point source and offers a 0.9 or 1.75λ/D inner working angle for topological charge 2 or 4, respectively. Current and near-future large telescopes are on-axis, however, and the diffraction effects of the central obscuration, and the secondary supports are strong enough to prevent the detection of companions 10 −3 -10 −5 as bright as, or fainter than, their host star. Aims. Recent advances show that a ring apodizer can restore the performance of this coronagraph by compensating for the diffraction effects of a circular central obscuration in a 1D modeling of the pupil. Our aim is to extend this work and design optimal apodizers for arbitrary apertures in 2D in order to tackle the diffraction effects of the spiders and other noncircular artifacts in the pupil. Methods. We fold this analytical result into a numerical optimization scheme that yields hybrid coronagraph designs that combine the advantages of the vortex coronagraph (small in IWA) and of shaped pupils coronagraphs (robustness to central obscuration and pupil asymmetric structures). The transmission of the apodizer is maximized, while constraints are set on the extremum values of the electric field that is computed in chosen regions of the Lyot plane through closed form expressions derived even for topological charges. Optimal apodizers are computed for topological charges 2 and 4 vortex coronagraphs and for telescope apertures with 10-30% central obscurations and 0%, 0.5%, and 1% thick spiders. Results. We put the results of our numerical optimizations in perspective with the analytical solutions and show that our apodizations converge to the ring apodizations. We then characterize the impacts of the obscuration ratio and the thickness of the spiders on the throughput and the IWA. For the apodized charge-2 vortex coronagraph the throughputs are slightly below those of the ring apodized vortex coronagraph, and the inner working angle is mostly unaffected by the apodization. The throughputs of the apodizers for the charge-4 vortex coronagraph are higher than those of the ring apodized vortex coronagraph. This effect increases with the obscuration ratio, though the inner working angle does, too, and it ranges between 2 and 3λ/D. Conclusions. The results presented in this paper show that high contrast at small inner working angles can be obtained with a vortex coronagraph for on-axis telescopes, in spite of the presence of a secondary mirror and its secondary support structures.

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Fast Fourier optimization

Cited by 16 publications

References 24 publications

Revisiting compressed sensing: exploiting the efficiency of simplex and sparsification methods

Revisiting compressed sensing: exploiting the efficiency of simplex and sparsification methods

Linear programming bounds for cliques in Paley graphs

Apodized phase mask coronagraphs for arbitrary apertures

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