34,703 research outputs found
On the Compressive Spectral Method
The authors of [Proc. Natl. Acad. Sci. USA, 110 (2013), pp. 6634--6639] proposed sparse Fourier domain approximation of solutions to multiscale PDE problems by soft thresholding. We show here that the method enjoys a number of desirable numerical and analytic properties, including convergence for linear PDEs and a modified equation resulting from the sparse approximation. We also extend the method to solve elliptic equations and introduce sparse approximation of differential operators in the Fourier domain. The effectiveness of the method is demonstrated on homogenization examples, where its complexity is dependent only on the sparsity of the problem and constant in many cases
Compressive Split-Step Fourier Method
In this paper an approach for decreasing the computational effort required
for the split-step Fourier method (SSFM) is introduced. It is shown that using
the sparsity property of the simulated signals, the compressive sampling
algorithm can be used as a very efficient tool for the split-step spectral
simulations of various phenomena which can be modeled by using differential
equations. The proposed method depends on the idea of using a smaller number of
spectral components compared to the classical split-step Fourier method with a
high number of components. After performing the time integration with a smaller
number of spectral components and using the compressive sampling technique with
l1 minimization, it is shown that the sparse signal can be reconstructed with a
significantly better efficiency compared to the classical split-step Fourier
method. Proposed method can be named as compressive split-step Fourier method
(CSSFM). For testing of the proposed method the Nonlinear Schrodinger Equation
and its one-soliton and two-soliton solutions are considered
Spectral Compressive Sensing with Model Selection
The performance of existing approaches to the recovery of frequency-sparse
signals from compressed measurements is limited by the coherence of required
sparsity dictionaries and the discretization of frequency parameter space. In
this paper, we adopt a parametric joint recovery-estimation method based on
model selection in spectral compressive sensing. Numerical experiments show
that our approach outperforms most state-of-the-art spectral CS recovery
approaches in fidelity, tolerance to noise and computation efficiency.Comment: 5 pages, 2 figures, 1 table, published in ICASSP 201
Compressive split-step Fourier method
In this paper an approach for decreasing the computational effort required for the split-step Fourier method (SSFM) is introduced. It is shown that using the sparsity property of the simulated signals, the compressive sampling algorithm can be used as a very efficient tool for the split-step spectral simulations of various phenomena which can be modeled by using differential equations. The proposed method depends on the idea of using a smaller number of spectral components compared to the classical split-step Fourier method with a high number of components. After performing the time integration with a smaller number of spectral components and using the compressive sampling technique with l(1) minimization, it is shown that the sparse signal can be reconstructed with a significantly better efficiency compared to the classical split-step Fourier method. Proposed method can be named as compressive split-step Fourier method (CSSFM). For testing of the proposed method the Nonlinear Schrodinger Equation and its one-soliton and two-soliton solutions are considered.Publisher's Versio
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