Speaker
Description
Dynamic Mode Decomposition (DMD) is a data-driven tool for capturing complex nonlinear dynamics. It can be used to identify, analyze and forecast dynamical systems $x_{k+1}=F(x_k)$ governed by an unknown or complex mapping $F$ using only observed snapshots $s_1,s_2,...,s_{n+1}$. If we denote $X := (s_1 \ s_2 \ \cdots \ s_n)$, $Y := (s_2 \ s_3 \ \cdots \ s_{n+1})$, finite-dimensional approximation of the governing function $F$ is a matrix $A$ such that $ A X \approx Y$ in the least-squares sense.
In a streaming application DMD has to be recomputed over a widening data window. If we denote a rank-revealing orthonormal decomposition of $X = Q_x\tilde X$ and $Y = Q_y \tilde Y$, Hemati et al. (2014) propose updating Rayleigh-Ritz matrix $$ Q_x^TAQ_x = (Q_x^TQ_y)(\tilde Y \tilde X^T)(\tilde X \tilde X^T)^{-1} =: (Q_x^TQ_y) G_{y,x} G_x^{-1} $$ via updates of $Q_x, Q_y, G_{y,x}$ and $G_x$. Smaller size and special structure of $G_{y,x}$, $G_x$ make them easy to update. However, since $$\tilde Y \tilde X^T (\tilde X \tilde X^T)^{-1} = \tilde Y \tilde X^\dagger$$ this formulation increases the condition number $\kappa(\tilde X\tilde X^T) = \kappa(\tilde X)^2$ making further computation potentially unstable. Additionally, since $X$ and $Y$ only differ up to one column, maintaining and updating two separate orthogonal matrices seems redundant. Our approach is based on low-rank orthogonal decomposition of $S = Q\tilde S$ where $S = (s_1 \ s_2 \ \cdots \ s_{n+1} )$. This way we only have to store and update one orthogonal basis $Q$. We propose an alternative formulation $$\tilde Y \tilde X^{-1} = (\tilde Y\tilde Q)T^{-1} =: G_{y,q}T^{-1}$$ using LQ or RQ decomposition of $\tilde X = T\tilde Q^T$ that avoids the squaring of the condition number while maintaining the same matrix sizes and simple updates. Our numerical analysis and numerical experiments demonstrate better numerical properties that result in better forecasting skill.