Dynamic Mode Decomposition (DMD) is a model order reduction technique that helps reduce the complexity and high dimensionality of computational models. Since each decomposed mode has only a single frequency, the result of DMD is frequently easier to interpret physically than the conventional Proper Orthogonal Decomposition (POD). The DMD can also produce the eigenvalues of each mode to show the trend of the mode, establishing the rate of growth or decay, but the original DMD cannot produce the contributing weights of the modes. As a result, a challenge with the technique is selecting the important modes to build a reduced order model. DMD variants have been developed to estimate the weights of each mode in the DMD. One of the popular methods is called Optimal Mode Decomposition (OMD). This method decomposes the data matrix into a product of the DMD modes, the diagonal weight matrix, and the Vandermonde matrix by minimizing the error between this product and the original data. The weight matrix can be used to rank the importance of the mode contributions and ultimately leads to the reduced order model for prediction and controlling purpose. We are currently applying DMD and OMD to a large-scale numerical simulation of the San Francisco Bay, which features complicated coastal geometry, multiple frequency components, and high periodicity. Since DMD defines modes with specific frequencies, we expect DMD would produce a good approximation of the original model, but the preliminary results show that the predictability of the DMD is poor if unimportant modes are dropped according to the OMD. We are currently testing other DMD variants and will report our findings in the presentation.

1. Predictability of the Dynamic Mode
Decomposition in Coastal Processes
Ruo-Qian Wang, Mark Stacey
Civil and Environmental Engineering @ UC Berkeley
Liv Herdman, Patrick Barnard
USGS Pacific Coastal and Marine Science Center

2. Outline
• Introduction
• First time to introduce DMD to analyze coastal processes
• Application of DMD
• DMD can capture the spatial-temporal characteristics of
a dynamic system of tidal basin
• Predictability of DMD

3. DMD Algorithm
Xt =USVT
Xt+1 = AXt
S =UT
AU =UT
Xt+1VS-1
SW =WL
F =UW
Linear Operator
Singular Value Decomposition (SVD)
Similarity Transform
Eigen Mode Decomposition
DMD modes
A = Xt+1VS-1
UT
Pseudo-inversion

4. Comparison to POD (PCA or EMD)
• DMD
Spatial-temporal
Clear physical meaning
Growth/Decay rates
• POD
Spatial
Difficult to interpret

5. Computational Domain

6. DMD Stability

7. DMD Modes (Semidiurnal and Diurnal)

8. DMD Modes (Overtide and Long-term)

9. To predict
• Optimal DMD
• Prediction

10. Prediction with 2 Modes
Jan 1 Jan 10 Jan 20 Jan 30

11. Prediction with All Modes
Jan 1 Jan 10 Jan 20 Jan 30

12. Prediction Error
err =
ˆX - X
2
X 2

13. Summary
• First time to apply DMD in coastal processes
• DMD reveal spatial-temporal patterns
• Semidiurnal / Diurnal
• Overtide / Long-term
• Prediction with 2 modes is the best

14. Thank you for your attention!
Special thank to Dr. Joel Tchoufag