dmd prediction of initial condition
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Assessment
- Difficulty
- 5/5
- Estimated time
- Over a week
- Newbie friendliness
- 20/100
Research direction
Start by reviewing the DMD documentation and equations referenced in the issue, then locate the existing DMD implementation and reproduce the reported initial-condition error. Done would require a clearly defined resolution explaining whether the observed behavior is expected or identifying the implementation change needed.
Written by the indexing model from the issue text.
Description
My DMD code seems to be working, but it seems to make a large error at the starting time. Shouldn't DMD be able to reproduce the initial condition exactly? Looking at
https://www.librom.net/features.html
I can see the last equation in the DMD section
$$
\mathbf{u}(t) = \mathbf{\Phi\Lambda}^{{t}/{\Delta t}}\mathbf{b}_0
$$
using $\mathbf{b}_0=\mathbf{\Phi}^\dagger\mathbf{u}_0$ we get
$$
\mathbf{u}(t) = \mathbf{\Phi\Lambda}^{{t}/{\Delta t}}\mathbf{\Phi}^\dagger\mathbf{u}_0
$$
At $t=0$ we have $\mathbf{\Lambda}^{{t}/{\Delta t}}=\mathbf{1}$ and hence,
$$
\mathbf{u}(0) = \mathbf{\Phi}\mathbf{\Phi}^\dagger\mathbf{u}_0
$$
Since $\mathbf{\Phi}\mathbf{\Phi}^\dagger=\mathbf{1}$ we get
$$
\mathbf{u}(0)=\mathbf{u}_0
$$
Hence, the initial condition is faithfully reproduced. Is my understanding correct?
- Dominant language
- C++
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- 47
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