Dynamic mode decomposition as an analysis tool for time-dependent partial differential equations

Rot, Miha; Horvat, Martin; Kosec, Gregor

doi:10.23919/splitech55088.2022.9854243

Cited by 5 publications

(5 citation statements)

References 12 publications

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“…More specifically, it has been shown that the DMD algorithm is capable of approximating the modes of infinite‐dimensional linear Koopman operator, which can be defined to represent the dynamics of any nonlinear system 16,32 . Therefore, for a nonlinear dynamic system, the DMD algorithm provides a linear mapping that best approximates the nonlinear dynamics 17,24 . As such, DMD can be utilized to analyze a nonlinear system, 21 as it might be the case for in vivo ventilation and perfusion measurements 9 …”

Section: Discussionmentioning

confidence: 99%

“…16,32 Therefore, for a nonlinear dynamic system, the DMD algorithm provides a linear mapping that best approximates the nonlinear dynamics. 17,24 As such, DMD can be utilized to analyze a nonlinear system, 21 as it might be the case for in vivo ventilation and perfusion measurements. 9 In this work, the DMD was utilized with a fixed rank to identify dominant modes.…”

Section: Discussionmentioning

confidence: 99%

“…In data stacking, multiple time‐shifted copies of the data are stacked into a larger augmented matrix to increase the rank of the measurement matrix. As such, data stacking helps mitigate rank deficiencies 24 and can improve the accuracy of the reduced‐order system in extraction of information 15 . Afterward,

r=15

most dominant modes were identified with DMD, and the signal modulations caused by ventilation and perfusion were obtained as:

{S}_{vent}=\sum \limits_{k=1}^v{b}_k{\phi}_k,{S}_{perf}=\sum \limits_{k=1}^q{b}_k{\phi}_k,

where

v

and

q

indicate the number of identified respiratory and cardiac frequencies,

{b}_k

is the respective DMD mode amplitude, and

{\phi}_k

is the respective DMD mode.…”

Section: Methodsmentioning

confidence: 99%

“…In data stacking, multiple time-shifted copies of the data are stacked into a larger augmented matrix to increase the rank of the measurement matrix. As such, data stacking helps mitigate rank deficiencies 24 and can improve the accuracy of the reduced-order system in extraction of information. 15 Afterward, r = 15 most dominant modes were identified with DMD, and the signal modulations caused by ventilation and perfusion were obtained as:…”

Section: Dmd Of the Respiratory Signal And Image Analysesmentioning

confidence: 99%

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Dynamic mode decomposition of dynamic MRI for assessment of pulmonary ventilation and perfusion

Ilicak

Ozdemir

Zapp

et al. 2023

Magnetic Resonance in Med

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To introduce dynamic mode decomposition (DMD) as a robust alternative for the assessment of pulmonary functional information from dynamic non-contrast-enhanced acquisitions.Methods: Pulmonary fractional ventilation and normalized perfusion maps were obtained using DMD from simulated phantoms as well as in vivo dynamic acquisitions of healthy volunteers at 1.5T. The performance of DMD was compared with conventional Fourier decomposition (FD) and matrix pencil (MP) methods in estimating functional map values. The proposed method was evaluated based on estimated signal amplitude in functional maps across varying number of measurements. Results: Quantitative assessments performed on phantoms and in vivo measurements indicate that DMD is capable of successfully obtaining pulmonary functional maps. Specifically, compared to FD and MP methods, DMD is able to reduce variations in estimated amplitudes across different number of measurements. This improvement is evident in the fractional ventilation and normalized perfusion maps obtain from phantom simulations with frequency variations and noise, as well as in the maps obtained from in vivo measurements. Conclusions:A robust method for accurately estimating pulmonary ventilation and perfusion related signal changes in dynamic acquisitions is presented. The proposed method uses DMD to obtain functional maps reliably, while reducing amplitude variations caused by differences in number of measurements.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

r=15

most dominant modes were identified with DMD, and the signal modulations caused by ventilation and perfusion were obtained as:

{S}_{vent}=\sum \limits_{k=1}^v{b}_k{\phi}_k,{S}_{perf}=\sum \limits_{k=1}^q{b}_k{\phi}_k,

where

v

and

q

indicate the number of identified respiratory and cardiac frequencies,

{b}_k

is the respective DMD mode amplitude, and

{\phi}_k

is the respective DMD mode.…”

Section: Methodsmentioning

confidence: 99%

Section: Dmd Of the Respiratory Signal And Image Analysesmentioning

confidence: 99%

See 2 more Smart Citations

Dynamic mode decomposition of dynamic MRI for assessment of pulmonary ventilation and perfusion

Ilicak

Ozdemir

Zapp

et al. 2023

Magnetic Resonance in Med

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show abstract

“…and Z is the matrix of left eigenvectors of the matrix Ã. One uses mode power (P) to rank the calculated modes (Rot et al 2022).…”

Section: Brief Outline Of the Dmd Techniquementioning

confidence: 99%

Numerical analysis of tulip flame by dynamic mode decomposition

Joarder

2024

Fluid Dyn. Res.

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Inversion of a premixed flame front to the shape of a tulip flower during the last stage of its propagation in a channel or tube of aspect ratio greater than two has intrigued the scientific community for decades. In the present work, the tulip inversion of hydrogen-air premixed flame is simulated numerically in two dimensional domains of aspect ratio 7.32, and 3.66. Dynamic Mode Decomposition (DMD) is carried out on the two-dimensional pressure distribution obtained from the numerical simulation to understand the reasons behind the flame inversion. A plot of real and imaginary components of the eigenvalues over a unit circle clearly showed the presence of unstable acoustic modes during the tulip inversion for the domain of aspect ratio 7.32. The tulip inversion is not observed for the domain of aspect ratio 3.66. The distribution of the power and frequency of various modes are also plotted to identify the unstable modes. The present study shows generation of additional pressure waves during the tulip inversion process, which in term creates a velocity field that facilitates the flame inversion process.

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Dynamic mode decomposition with core sketch

et al. 2022

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With the increase in collected data volumes, either from experimental measurements or high fidelity simulations, there is an ever-growing need to develop computationally efficient tools to process, analyze, and interpret these data sets. Modal analysis techniques have gained great interest due to their ability to identify patterns in the data and extract valuable information about the system being considered. Dynamic mode decomposition (DMD) relies on elements of Koopman approximation theory to compute a set of modes, each associated with a fixed oscillation frequency and decay/growth rate. Extracting these details from large data sets can be computationally expensive due to the need to implement singular value decomposition of the input data matrix. Sketching algorithms have become popular in numerical linear algebra where statistical theoretic approaches are utilized to reduce the cost of major operations. A sketch of a matrix is another matrix which is significantly smaller, but still sufficiently approximates the original system. We put forth an efficient DMD framework, SketchyDMD, based on a core sketching algorithm that captures information about the range and co-range (an their mutual relationship) of input data. The proposed sketching-based framework can accelerate various portions of the DMD routines, compared to classical methods that operate directly on the raw input data. We conduct numerical experiments using the spherical shallow water equations as a prototypical model in the context of geophysical flows. We show that the proposed SketchyDMD is superior to existing randomized DMD methods that are based on capturing only the range of the input data.

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Dynamic mode decomposition as an analysis tool for time-dependent partial differential equations

Cited by 5 publications

References 12 publications

Dynamic mode decomposition of dynamic MRI for assessment of pulmonary ventilation and perfusion

Dynamic mode decomposition of dynamic MRI for assessment of pulmonary ventilation and perfusion

Numerical analysis of tulip flame by dynamic mode decomposition

Dynamic mode decomposition with core sketch

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