Probing the Pareto Frontier for Basis Pursuit Solutions

Berg, E. van den; Friedlander, Michael P.

doi:10.1137/080714488

Cited by 1,650 publications

(1,057 citation statements)

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“…The SPGL1 method used for CS was implemented into OCTOPUS based on a Fortran translation of the original MatLab code of van den Berg and Friedlander (26). We plan to release this implementation as a stand-alone tool in the near future (for the moment the code can be obtained from the OCTOPUS repository).…”

Section: Methodsmentioning

confidence: 99%

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Application of compressed sensing to the simulation of atomic systems

Andrade

Sanders

Aspuru‐Guzik

2012

Proc. Natl. Acad. Sci. U.S.A.

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Compressed sensing is a method that allows a significant reduction in the number of samples required for accurate measurements in many applications in experimental sciences and engineering. In this work, we show that compressed sensing can also be used to speed up numerical simulations. We apply compressed sensing to extract information from the real-time simulation of atomic and molecular systems, including electronic and nuclear dynamics. We find that, compared to the standard discrete Fourier transform approach, for the calculation of vibrational and optical spectra the total propagation time, and hence the computational cost, can be reduced by approximately a factor of five.sparse signal reconstruction | molecular dynamics | electron dynamics A recent development in the field of data analysis is the compressed sensing (CS) (or compressive sampling) method (1, 2). The foundation of the method is the concept of sparsity: A signal expanded in a certain basis is said to be sparse when most of the expansion coefficients are zero. This extra information can be used by the CS method to significantly reduce the number of measurements needed to reconstruct a signal. CS has been successfully applied to data acquisition in many different areas (3), including the improvement of the resolution of medical magnetic resonance imaging (4) and the experimental study of atomic and quantum systems (5-7).In this article we show that CS can also be an invaluable tool for some numerical simulations with a considerable reduction of the computational cost. We focus on atomistic simulations of nanoscopic systems by using CS to extract frequency-resolved information from real-time methods such as molecular dynamics (MD) and real-time electron dynamics.MD (8, 9) is one of the most widely used methods to study atomistic systems computationally as it can be used to compute many static and dynamical properties. In MD the trajectory of the atomic nuclei is obtained by integrating their equations of motion either with parametrized force fields or else by explicitly modeling the electrons (10). Given the importance of MD, developing methods that can improve the precision and reduce the computational cost of this method, especially for ab initio MD, can have a large impact in the field of atomistic simulation.Real-time electron dynamics, in particular real-time timedependent density functional theory (TDDFT) (11), plays a similarly important role in the study of linear and nonlinear electronic properties (12-15). Because of its scalability and parallelizability, real-time TDDFT is particularly efficient for large electronic systems (16), so an additional reduction in the computational cost can extend the boundaries of the system sizes that can be studied.Many physical properties are represented by frequency-dependent quantities. To obtain these from real-time information, usually a discrete Fourier transform (FT) is used. Our approach is to replace this FT by a calculation of the Fourier coefficients based on the CS method. To obtain a given ...

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Section: Methodsmentioning

confidence: 99%

“…6. This problem is not trivial, so we rely on the spectral projected gradient for 1-norm minimization (SPGL1) algorithm developed by van den Berg and Friedlander (26). To avoid numerical stability issues we work with a normalized BPDN problem, where the prefactor 2Δω∕π of the F matrix, Eq.…”

Section: Methodsmentioning

confidence: 99%

Application of compressed sensing to the simulation of atomic systems

Andrade

Sanders

Aspuru‐Guzik

2012

Proc. Natl. Acad. Sci. U.S.A.

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“…The particular BP implementation is publicly available [4] from the authors of [3]. The result appears in Figure 4, where the blue line reproduces the noise-free CBF solution from Figure 2, and the red line shows the sparse BP solution.…”

Section: B Basis Pursuit Approach To Beamformingmentioning

confidence: 93%

Basis Pursuit for robust passive acoustic beamforming

Shapo

Kreucher

2009

2009 Conference Record of the Forty-Third Asilomar Conference on Signals, Systems and Computers

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1 -Beamforming is a process that supplies directional gain to sensor array processing. One modality where beamforming adds great value is passive sonar, where real-aperture arrays receive signals emitted by acoustic sources. In passive sonar systems, the beamformer is the backbone of a processing structure that detects, localizes, and classifies external targets. Conventional beamformers use deterministic time-delays (often implemented as phase shifts) to arrange coherent addition of plane-wave signals at each sensor. Recently, adaptive beamformers take advantage of signal time history by imposing a model on the environment. Basis Pursuit is another reconstruction approach used in Compressed Sensing that also enforces a physics-based model -in this case a model of scene sparsity. This paper describes an application of this technique to the beamforming problem. The main benefit of the Basis Pursuit beamforming approach is that it is robust to missing array elements, providing nearly full-aperture performance in a reduced sensor environment. This result is advantageous in the case of processing with inoperative hydrophones. It may also provide cost savings by allowing array design with fewer hydrophones. I . INTRODUCTIONRecently, the signal processing community has shown great interest in Compressed Sensing ("CS") [1]. In CS, fewer (sometimes significantly fewer) samples of a physical phenomenon than traditional Nyquist bounds would mandate reconstruct the original signal via a technique such as Basis Pursuit (BP). Good reconstruction is possible under the assumption that the sampled signals are actually sparse. Passive sonar often obeys this assumption, with 180-(or 360)-degree field of view but few targets. This paper describes a beamforming method that exploits the sparsity constraints used in BP. Like the Conventional Beamformer ("CBF"), we assume each signal impinging on a passive array of sensors arrives as a planewave [2]. In addition, we also assume the number of planewave signals is small compared to the number of arrival angles considered ("beams").Our method produces results comparable to CBF in the case of a full-aperture array. However, when the number of sensors decreases (either by hyrdrophones becoming unexpectedly inoperable or by design), our method produces higher-fidelity results than the CBF approach operating on the same reduced input data. This paper proceeds as follows. Section II reviews a standard model of plane-wave physics and derives the CBF. A simulated example demonstrates the power of this classic approach, and highlights the benefits of using data in the wide frequency ranges available with broadband sensors.Section III builds on the model presented in Section II, and gives a brief description of our BP approach to beamforming. This section shows results from a BPbeamformer operating on the same simulated data input from Section II.Section IV demonstrates the benefits of the methodology on Navy collected at-sea data by comparing CBF and BP beamformer outputs in the case wh...

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“…which can be solved using off-the-shelf algorithms tuned to solve 1 -recovery problems [19]. The parameters 1 ≥ 0 and 2 ≥ 0 can be used to "tweak" the recovery performance.…”

Section: Recovery Of Full-resolution Framesmentioning

confidence: 99%

CS-MUVI: Video compressive sensing for spatial-multiplexing cameras

Sankaranarayanan

Studer

Baraniuk

2012

2012 IEEE International Conference on Computational Photography (ICCP)

132

108

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Compressive sensing (CS)-based spatial-multiplexing cameras (SMCs) sample a scene through a series of coded projections using a spatial light modulator and a few optical sensor elements. SMC architectures are particularly useful when imaging at wavelengths for which full-frame sensors are too cumbersome or expensive. While existing recovery algorithms for SMCs perform well for static images, they typically fail for time-varying scenes (videos). In this paper, we propose a novel CS multi-scale video (CS-MUVI) sensing and recovery framework for SMCs. Our framework features a co-designed video CS sensing matrix and recovery algorithm that provide an efficiently computable low-resolution video preview. We estimate the scene's optical flow from the video preview and feed it into a convexoptimization algorithm to recover the high-resolution video. We demonstrate the performance and capabilities of the CS-MUVI framework for different scenes.

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Probing the Pareto Frontier for Basis Pursuit Solutions

Cited by 1,650 publications

References 35 publications

Application of compressed sensing to the simulation of atomic systems

Application of compressed sensing to the simulation of atomic systems

Basis Pursuit for robust passive acoustic beamforming

CS-MUVI: Video compressive sensing for spatial-multiplexing cameras

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