Compressed sensing and dynamic mode decomposition

Brunton, Steven L.; Proctor, Joshua L.; Tu, Jonathan H.; Kutz, J. Nathan

doi:10.3934/jcd.2015002

Cited by 146 publications

(110 citation statements)

References 58 publications

Supporting

Mentioning

109

Contrasting

Order By: Relevance

“…The magic of Compressive Sensing (CS) [8][9][10][11] is in its ability to overcome this constraint by seeking a solution that can be less sparse than the dimension of the chosen feature space using l 1 -norm regularized reconstruction. Such methods have been successfully applied in image processing using Fourier or wavelet basis and also to fluid flows [6,7,[28][29][30][31]. Compressive sensing essentially looks for a regularized sparse solution using l 1 norm minimization of the sparse coefficients by solving a computationally manageable convex optimization problem.…”

Section: Introductionmentioning

confidence: 99%

Interplay of Sensor Quantity, Placement and System Dimension in POD-Based Sparse Reconstruction of Fluid Flows

Jayaraman

Mamun

Chen

2019

Fluids

View full text Add to dashboard Cite

Sparse linear estimation of fluid flows using data-driven proper orthogonal decomposition (POD) basis is systematically explored in this work. Fluid flows are manifestations of nonlinear multiscale partial differential equations (PDE) dynamical systems with inherent scale separation that impact the system dimensionality. Given that sparse reconstruction is inherently an ill-posed problem, the most successful approaches require the knowledge of the underlying low-dimensional space spanning the manifold in which the system resides. In this paper, we adopt an approach that learns basis from singular value decomposition (SVD) of training data to recover sparse information. This results in a set of four design parameters for sparse recovery, namely, the choice of basis, system dimension required for sufficiently accurate reconstruction, sensor budget and their placement. The choice of design parameters implicitly determines the choice of algorithm as either l 2 minimization reconstruction or sparsity promoting l 1 minimization reconstruction. In this work, we systematically explore the implications of these design parameters on reconstruction accuracy so that practical recommendations can be identified. We observe that greedy-smart sensor placement, particularly interpolation points from the discrete empirical interpolation method (DEIM), provide the best balance of computational complexity and accurate reconstruction.

show abstract

Section: Introductionmentioning

confidence: 99%

Interplay of Sensor Quantity, Placement and System Dimension in POD-Based Sparse Reconstruction of Fluid Flows

Jayaraman

Mamun

Chen

2019

Fluids

View full text Add to dashboard Cite

show abstract

“…A lower RMSE value represents a better reconstruction. For the noisy cases, the peak value to noise ratio (PVNR), inspired by PVNR defined in [19] and as defined below was used as the comparison metric PVNR 20 log 10 max Z re f…”

Section: Resultsmentioning

confidence: 99%

“…As the first test, the vorticity of the double-gyre flow (as represented in [10]) was taken and used. The vorticity w is given as…”

Section: Dmd Mode-shapes Reconstructionmentioning

confidence: 99%

“…This makes the proposed algorithm (known as NU-DMD) impractical in the case the fully-sampled dataset is not available. Another approach for incorporating compressed sensing in DMD (known as csDMD) was developed by Kutz et al [10]. In csDMD, the DMD eigenvalues are obtained from a sub-sampled dataset (similar to NU-DMD), which has the advantage of reducing computation time, and then the full DMD mode shapes are reconstructed through using an l 1 -minimization scheme based on a chosen set of basis vectors.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Dynamic Denoising and Gappy Data Reconstruction Based on Dynamic Mode Decomposition and Discrete Cosine Transform

et al. 2018

View full text Add to dashboard Cite

Dynamic Mode Decomposition (DMD) is a data-driven method to analyze the dynamics, first applied to fluid dynamics. It extracts modes and their corresponding eigenvalues, where the modes are spatial fields that identify coherent structures in the flow and the eigenvalues describe the temporal growth/decay rates and oscillation frequencies for each mode. The recently introduced compressed sensing DMD (csDMD) reduces computation times and also has the ability to deal with sub-sampled datasets. In this paper, we present a similar technique based on discrete cosine transform to reconstruct the fully-sampled dataset (as opposed to DMD modes as in csDMD) from sub-sampled noisy and gappy data using l 1 minimization. The proposed method was benchmarked against csDMD in terms of denoising and gap-filling using three datasets. The first was the 2-D time-resolved plot of a double gyre oscillator which has about nine oscillatory modes. The second dataset was derived from a Duffing oscillator. This dataset has several modes associated with complex eigenvalues which makes them oscillatory. The third dataset was taken from the 2-D simulation of a wake behind a cylinder at Re = 100 and was used for investigating the effect of changing various parameters on reconstruction error. The Duffing and 2-D wake datasets were tested in presence of noise and rectangular gaps. While the performance for the double-gyre dataset is comparable to csDMD, the proposed method performs substantially better (lower reconstruction error) for the dataset derived from the Duffing equation and also, the 2-D wake dataset according to the defined reconstruction error metrics.

show abstract

“…Compressed DMD provides a computationally efficient framework to compute the dynamic mode decomposition on massively under-sampled or compressed data [7]. The method was originally devised to reconstruct highdimensional, full-resolution DMD modes from sparse, spatially under-resolved measurements by leveraging compressed sensing.…”

Section: Dmd For Real-time Background Modelingmentioning

confidence: 99%

Compressed dynamic mode decomposition for background modeling

Erichson

Brunton

Kutz

2016

J Real-Time Image Proc

121

View full text Add to dashboard Cite

We introduce the method of compressed dynamic mode decomposition (cDMD) for background modeling. The dynamic mode decomposition (DMD) is a regression technique that integrates two of the leading data analysis methods in use today: Fourier transforms and singular value decomposition. Borrowing ideas from compressed sensing and matrix sketching, cDMD eases the computational workload of high resolution video processing. The key principal of cDMD is to obtain the decomposition on a (small) compressed matrix representation of the video feed. Hence, the cDMD algorithm scales with the intrinsic rank of the matrix, rather then the size of the actual video (data) matrix. Selection of the optimal modes characterizing the background is formulated as a sparsity-constrained sparse coding problem. Our results show, that the quality of the resulting background model is competitive, quantified by the F-measure, Recall and Precision. A GPU (graphics processing unit) accelerated implementation is also presented which further boosts the computational efficiency of the algorithm.

show abstract

Compressed sensing and dynamic mode decomposition

Cited by 146 publications

References 58 publications

Interplay of Sensor Quantity, Placement and System Dimension in POD-Based Sparse Reconstruction of Fluid Flows

Interplay of Sensor Quantity, Placement and System Dimension in POD-Based Sparse Reconstruction of Fluid Flows

Dynamic Denoising and Gappy Data Reconstruction Based on Dynamic Mode Decomposition and Discrete Cosine Transform

Compressed dynamic mode decomposition for background modeling

Contact Info

Product

Resources

About