---
title: Krylov Dynamic Mode Decomposition (DMD)
url: https://www.emergentmind.com/topics/krylov-dynamic-mode-decomposition-dmd
type: topic
---

# Krylov Dynamic Mode Decomposition (DMD)

Krylov Dynamic Mode Decomposition (DMD) is an optimized variant of the Dynamic Mode Decomposition framework designed for efficient analysis of high-dimensional, highly oscillatory spatiotemporal datasets. By incorporating both time-delay coordinates (TDC) and Krylov-subspace-inspired projections, this methodology achieves significant computational and memory reductions with negligible loss in modal accuracy. Krylov DMD addresses bottlenecks present in standard DMD by projecting high-dimensional data onto a carefully constructed low-dimensional subspace before matrix factorization and spectral decomposition, enabling robust identification of oscillatory modes in large-scale systems [2001.03332].

## 1. Standard DMD and Computational Challenges

Standard DMD operates on snapshot matrices $X \in \mathbb{R}^{M \times N}$, where $M$ is the spatial dimension and $N$ the number of temporal samples. The snapshot pair is split as $X_1 = [x_1, ..., x_{N-1}]$ and $X_2 = [x_2, ..., x_N]$, seeking the best-fit linear operator $A$ such that $X_2 \approx A X_1$. The core algorithm involves the following steps:
1. Economic-size singular value decomposition (SVD): $X_1 = U \Sigma V^*$, truncated to rank $r$.
2. Low-dimensional operator construction: $\tilde{S} = U^* X_2 V \Sigma^{-1}$.
3. Spectral decomposition: $\tilde{S} y_k = \mu_k y_k$, DMD modes $\phi_k = U y_k$, eigenvalues $\omega_k = \ln(\mu_k)/\Delta t$.
4. State reconstruction: $x_{\text{DMD}}(t) = \Phi\,\mathrm{diag}(e^{\omega t})\,b$, with $b = \Phi^\dagger x_1$.

However, when $M$ and/or $N$ are large, the SVD step induces computational complexity $O(\min(M,N)^2 \max(M,N))$ and memory usage $O(MN)$. Data from many relevant phenomena are “big” ($M \gg 10^3$) and “highly oscillatory,” requiring dense sampling ($N$ large), which amplifies these bottlenecks. Standard DMD may also fail to resolve oscillatory modes in non-Markovian or coarsely-sampled systems [2001.03332].

## 2. Time-Delay Coordinates and the Need for Projection

Time-Delay Coordinates augment each snapshot $x_k$ by stacking $p$ past states, forming highly informative, tall Hankel-like matrices:

\[
X_{1,\text{aug}} = \begin{bmatrix}
x_1 & \hdots & x_{q-1} \\
x_2 & \hdots & x_q \\
\vdots \\
x_p & \hdots & x_{N-1}
\end{bmatrix}
\in \mathbb{R}^{(p+1)M \times (N-p)}
\]

This procedure effectively embeds hidden oscillatory dynamics into a higher-dimensional, approximately linear, manifold, enhancing the extractability of underlying coherent structures. However, it increases the row dimension to $qM$, drastically elevating the cost of subsequent SVD. Mitigation involves the application of projection operators $R \in \mathbb{R}^{a \times (qM)}$, with $a \ll qM$, resulting in compressed matrices $Z_{1,\text{aug}} = R X_{1,\text{aug}}$, $Z_{2,\text{aug}} = R X_{2,\text{aug}}$. The projection must preserve dominant DMD eigenvalues and modes while reducing both floating-point operations and memory requirements [2001.03332].

## 3. Krylov-Subspace Projections

The Krylov subspace, $\mathcal{K}_m(A, b) = \mathrm{span}\{b, Ab, A^2 b, ..., A^{m-1} b\}$ for $A \in \mathbb{R}^{n \times n}$ and seed vector $b \in \mathbb{R}^n$, is a classical construct from numerical linear algebra exploited here to define low-dimensional projectors. Its construction uses the Arnoldi process (a modified Gram–Schmidt):

- Inputs: random matrix $A$, seed vector $b$.
- Iteratively generate orthonormal basis $V_m = [v_1, ..., v_m]$ for the subspace.
- The projection operator is then $R_K = V^* \in \mathbb{R}^{(a+1) \times M}$.

For data $X \in \mathbb{R}^{M \times N}$, $Z = R_K X = V^* X \in \mathbb{R}^{(a+1) \times N}$ yields a compressed representation. The orthogonal projector is $P = VV^*$, but $R_K$ suffices for right-multiplicative projection. This basis, once constructed, is data-agnostic and can be reused if $A$ and $b$ are fixed.

## 4. Krylov-Projected TDC-DMD: Algorithmic Workflow

The Krylov-projected time-delay DMD framework proceeds as follows:
1. **TDC Embedding:** Form augmented snapshot matrices $X_{1,\text{aug}}$, $X_{2,\text{aug}}$ of size $qM \times (N-q)$.
2. **Projector Construction:** Use Arnoldi on $A$, $b$ to obtain $V \in \mathbb{R}^{M \times (a+1)}$; set $R_K = V^*$.
3. **Data Projection:** $Z_1 = R_K X_{1,\text{aug}}$, $Z_2 = R_K X_{2,\text{aug}}$.
4. **SVD:** $Z_1 = U_Z \Sigma_Z V_Z^*$, truncated to $r \ll a+1$.
5. **Koopman Matrix:** $\tilde{S} = U_Z^* Z_2 V_Z \Sigma_Z^{-1}$.
6. **Spectral Decomposition:** $\tilde{S} y_k = \mu_k y_k$, $\phi_k = U_Z y_k$.
7. **Amplitudes and Frequencies:** $b = \Phi^\dagger x_1$, $\omega_k = \ln(\mu_k)/\Delta t$.
8. **Reconstruction:** $x_{\rm DMD}(t) = \Phi \,\mathrm{diag}(e^{\omega t}) b$.

This workflow compresses the effective row-dimension from $qM$ to $a+1$, where $a+1$ is the Krylov subspace dimension, delivering a dramatic reduction in resource consumption [2001.03332].

## 5. Computational Complexity and Memory Analysis

A direct comparison of various DMD strategies for data of dimension $M$ and projection size $a$ is as follows:

| Method                 | SVD Cost                               | Projection/Arnoldi Cost     | Memory Usage         |
|------------------------|----------------------------------------|-----------------------------|----------------------|
| Standard TDC‐DMD       | $O(\min(qM, N)^2 \max(qM, N))$         | —                           | $O(qM \cdot N)$      |
| Sampling/Gaussian Proj | $O(a^2 N)$ (after proj)                | $O(a q M N)$                | $O(a N)$             |
| Krylov‐DMD             | $O(a^2 N)$ (after proj)                | $O(a M^2)$ (Arnoldi, once); $O(a M q N)$ (proj) | $O(M a)$ (basis) + $O(a N)$ (proj)|

For $M = 10^4$–$10^6$ and $a = 10^2$–$10^3$, Krylov DMD achieves a reduction of one to two orders of magnitude in both FLOPs and memory. If the Krylov basis $V$ is constructed once and reused, the amortized cost is much lower than standard DMD. This enables practical application to very large datasets that would otherwise be infeasible with direct SVD [2001.03332].

## 6. Experimental Benchmarks and Modal Accuracy

Empirical evaluation is performed on two canonical data sets:
- Double Gyre vorticity ($[0,2] \times [0,1]$, $M=10^4$, $N=200$).
- 2D compressible signal, two-frequency ($M=10^4$, $N=200$).

Time-delay embedding uses $q=2$. Projection dimensions for Krylov DMD and comparators are selected as $a=100$ (Double Gyre) and $a=50$ (Signal). The singular value truncation rank is $r=20$.

**Results summary:**
- Eigenvalue spectra (Im$(\lambda)$ vs Re$(\lambda)$) are nearly identical across projection variants, forming symmetric clusters near the unit circle, indicative of stable dynamics.
- Long-run reconstruction errors $\|x_{\text{true}}(t) - x_{\text{DMD}}(t)\|_2$ for Krylov DMD and sparse projection DMD are nearly indistinguishable from classic DMD when $a \cdot q \geq r$ (e.g., $100 \cdot 2 \geq 20$).
- Sampling-based methods can omit weak modes, resulting in occasional long-term drift.
- Runtime benchmarks (per trial): classic TDC-DMD $\sim3.2$s, sampling DMD $\sim0.12$s, Gaussian projection $\sim0.18$s, sparse projection $\sim0.15$s, Krylov DMD (amortized) $\sim0.20$s (Arnoldi step $\sim0.05$s).

## 7. Theoretical Considerations, Limitations, and Extensions

Krylov DMD offers substantive advantages:
- High data reduction: compresses $M q$ to $a$, with $a \ll M q$, without significant loss in modal accuracy.
- Robust mode identification in highly oscillatory regimes through joint TDC and Krylov projection.
- Compatibility with sparse storage where appropriate.

Potential limitations include:
- Initial Arnoldi cost $O(a M^2)$, which can be burdensome for very large $M$ unless $V$ is reused.
- The accuracy and stability depend on the choice of $a$, $A$, and $b$ (projection dimension and seeding), necessitating problem-dependent cross-validation.
- Error bounds for combined TDC/Krylov projection are not fully characterized.

Possible directions for further development include streaming/online Arnoldi for time-varying data, adaptive selection of $a$ for energy target capture, application to $\ell_1$-based compressive DMD, and extension to PDE-constrained inverse problems or control-oriented settings [2001.03332].

Source: https://www.emergentmind.com/topics/krylov-dynamic-mode-decomposition-dmd