Adaptive Basis

When the adaptive ROM model is used (solver_mode = AROM), the reduced basis is updated dynamically. Several options for basis adaptation are available in CompFlowLab that can be selected via parameter `arom_method` in input file.

All of the methods described below are applicable only when the adaptive ROM framework is used, where the basis evolves dynamically as new snapshots become available.

direct: Direct Method

Solves a minimization problem to compute a new basis that best fits the training window of snapshots.

$$V^{n} = argmin\left\|V^{n} (V^{n-1})^{\dagger} {Q}^n - {Q}^n \right\|_F^2,$$

where $Q^n$ denotes a data matrix collecting estimated full-order states over a temporal window.

svd: Performing SVD from scratch

Updates the basis by performing a full SVD on the training snapshot matrix (F) and retaining the dominant left singular vectors. This method is computationally more expensive but provides an optimal low-rank representation of the accumulated snapshots.

ojas: Oja’s Rule

Employs Oja’s rule, a simple rank-one update of the form

$$(V^{n} = V^{n-1} + \eta \, q_{new} (q_{new}^T V^{n-1}))$$

where $\eta$ is the learning rate. The updated basis is orthonormalized via QR decomposition.

grouse: GROUSE Algorithm

Implements the GROUSE (Grassmannian Rank-One Update Subspace Estimation) algorithm. The basis is updated by moving along a geodesic on the Grassmann manifold:

$$V^{n} = V^{n-1} + \left( \frac{\cos(\alpha) - 1}{\|p\|} p + \frac{\sin(\alpha)}{\|r\|} r \right) \frac{q^T}{\|q\|}$$

where $p = V^{n-1} q$ is the projection of the new snapshot onto the current subspace,

$$r = q_{new} - p$$

is the residual, and $\alpha = \eta |r| |p|$ is the step size parameter.

isvd: Incremental SVD

Performs incremental SVD (iSVD) with a forgetting factor. The algorithm updates the basis by augmenting the current basis $V^{n-1}$ with the normalized residual of the new snapshot:

$$U_{\text{aug}} = [V^{n-1}, \; q_{\perp}]$$

where

$$q_{\perp} = frac{(q_{\text{new}} - V^{n-1} p)}{|q_{\text{new}} - V^{n-1} p|}$$

and

$$p = (V^{n-1})^T q_{\text{new}}$$

The SVD of a small $(K+1) \times (K+1)$ core matrix is computed, and the basis is truncated to the desired rank. A forgetting factor allows exponential decay of older information.

past: PAST Algorithm

Utilizes the PAST (Projection Approximation Subspace Tracking) algorithm, which approximates the subspace by minimizing a projection error criterion. The method uses a recursive least squares (RLS) update with forgetting factor $\lambda$:

$$P^{n} = \frac{1}{\lambda} \left( P^{n-1} - \frac{P^{n-1} w w^T P^{n-1}}{\lambda + w^T P^{n-1} w} \right)$$ $$V^{n} = V^{n-1} + (q_{\text{new}} - V^{n-1} w) (P^{n} w)^T$$

where

$$w = (V^{n-1})^T q_{\text{new}}$$

This efficiently tracks the dominant subspace without performing full SVD.