arXiv subjects

Peter Frame

Publications and source records attributed to Peter Frame.

5 recordsLinked to original sources

arXiv2025

Data-Driven Transient Growth Analysis

The transient growth of disturbances made possible by the non-normality of the linearized Navier-Stokes equations plays an important role in bypass transition for many shear flows. Transient growth is typically quantified by the maximum energy growth among all possible initial disturbances, which is given by the largest squared singular value of the matrix exponential of the linearized Navier-Stokes operator. In this paper, we propose a data-driven approach to studying transient growth wherein we calculate optimal initial conditions, the resulting responses, and the corresponding energy growth directly from flow data. Mathematically, this is accomplished by optimizing the growth over linear combinations of input and output data pairs. We also introduce a regularization to mitigate the sensitivity to noisy measurements and unwanted nonlinearity. The data-driven method simplifies and broadens the application of transient growth analysis -- it removes the burden of writing a new code or linearizing an existing one, alleviates the computational expense for large problems, eliminates the challenge of obtaining a well-posed spatial propagator for spatial growth analyses, and enables the direct application of transient growth analysis to experimental data. We validate the data-driven method using a linearized Ginzburg-Landau model problem corrupted by process and measurement noise and obtain good agreement between the data-driven and the standard operator-based results. We then apply the method to study the spatial transient growth of disturbances in a transitional boundary layer using data from the Johns Hopkins Turbulence Database. Our method successfully identifies the optimal output response and provides plausible estimates of the transient spatial energy growth at various spanwise wavenumbers.

physics.flu-dyn↗

arXiv2024

Nonlinear space-time model reduction in the frequency domain

We propose a space-time reduced-order model (ROM) for nonlinear dynamical systems, building upon previous work on linear systems. Whereas most ROMs are space-only in that they reduce only the spatial dimension of the state, the proposed method leverages an efficient encoding of the entire trajectory of the state on the time interval $[0,T]$, enabling significant additional reduction. Trajectories are encoded using SPOD modes, a spatial basis at each temporal frequency tailored to the structures that appear at that frequency. These modes have a number of properties that make them an ideal choice for space-time model reduction, including separability and near-optimality for long trajectories. We derive a system of algebraic equations involving the SPOD coefficients, forcing, and initial condition by projecting an implicit solution of the governing equations onto the set of SPOD modes in a space-time inner product. We therefore refer to the method as spectral solution operator projection (SSOP). The online phase of SSOP comprises solving this system for the SPOD coefficients, given the initial condition and forcing. We find that SSOP gives two orders of magnitude lower error than POD-Galerkin projection at the same number of modes and CPU time across a suite of tests, including ones that use out-of-sample forcings and affine parameter variation. In fact, the method is substantially more accurate even than the projection of the solution onto the POD modes, which is a lower bound for the error of any method based on a linear space-only encoding of the state.

math.NA↗

arXiv2024

Linear model reduction using spectral proper orthogonal decomposition

Most model reduction methods reduce the state dimension and then temporally evolve a set of coefficients that encode the state in the reduced representation. In this paper, we instead employ an efficient representation of the entire trajectory of the state over some time interval of interest and then solve for the static coefficients that encode the trajectory on the interval. We use spectral proper orthogonal decomposition (SPOD) modes, which are provably optimal for representing long trajectories and substantially outperform any representation of the trajectory in a purely spatial basis (e.g., POD). We develop a method to solve for the SPOD coefficients that encode the trajectories for forced linear dynamical systems given the forcing and initial condition, thereby obtaining the accurate prediction of the dynamics afforded by the SPOD representation of the trajectory. The method, which we refer to as spectral solution operator projection (SSOP), is derived by projecting the general time-domain solution for a linear time-invariant system onto the SPOD modes. We demonstrate the new method using two examples: a linearized Ginzburg-Landau equation and an advection-diffusion problem. In both cases, the error of the proposed method is orders of magnitude lower than that of POD-Galerkin projection and balanced truncation. The method is also fast, with CPU time comparable to or lower than both benchmarks in our examples. Finally, we describe a data-free space-time method that is a derivative of the proposed method and show that it is also more accurate than balanced truncation in most cases.

math.NA↗

arXiv2023

Beyond optimal disturbances: a statistical framework for transient growth

The theory of transient growth describes how linear mechanisms can cause temporary amplification of disturbances even when the linearized system is asymptotically stable as defined by its eigenvalues. This growth is traditionally quantified by finding the initial disturbance that generates the maximum response, in terms of energy gain, at the peak time of its evolution. While this bounds the growth, it can vastly overstate the growth of a real disturbance. In this paper, we introduce a statistical perspective on transient growth that models statistics of the energy amplification of the disturbances. We derive a formula for the mean energy amplification in terms of the two-point spatial correlation of the initial disturbance. We also derive an accurate approximation of the probability density function of the energy of the growing disturbance, from which confidence bounds on the growth can be obtained. Applying our analysis to Poisseuille flow yields a number of observations. First, the mean gain can be drastically smaller than the maximum, especially when the disturbances are broadband in wavenumber content. In these cases, it is exceedingly unlikely to achieve near-optimal growth due to the exponential behavior which we observe in the probability density function. Second, the characteristic length scale of the initial disturbances has a significant impact on the expected growth; specifically, large-scale initial disturbances produce orders-of-magnitude-larger expected growth than smaller scales, indicating that the length scale of incoming disturbances may be key in determining whether transient growth leads to transition for a particular flow. Finally, while the optimal growth scales quadratically with Reynolds number, we observe that the mean energy amplification scales only linearly for certain reasonable choices of the initial correlations.

physics.flu-dyn↗

arXiv2022

Space-time POD and the Hankel matrix

Time-delay embedding is an increasingly popular starting point for data-driven reduced-order modeling efforts. In particular, the singular value decomposition (SVD) of a block Hankel matrix formed from successive delay embeddings of the state of a dynamical system lies at the heart of several popular reduced-order modeling methods. In this paper, we show that the left singular vectors of this Hankel matrix are a discrete approximation of classical space-time proper orthogonal decomposition (POD) modes, and the singular values are square roots of the POD energies. This connection establishes a clear interpretation of the Hankel modes grounded in classical theory, and we gain insights into the Hankel modes by instead analyzing the equivalent discrete space-time POD modes in terms of the correlation matrix formed by multiplying the Hankel matrix by its conjugate transpose. These insights include the distinct meaning of rows and columns, the implied norm in which the modes are optimal, the impact of the time step between snapshots on the modes, and an interpretation of the embedding dimension/height of the Hankel matrix in terms of the time window on which the modes are optimal. Moreover, the connections we establish offer opportunities to improve the convergence and computation time in certain practical cases, and to improve the accuracy of the modes with the same data. Finally, popular variants of POD, namely the standard space-only POD and spectral POD, are recovered in the limits that snapshots used to form each column of the Hankel matrix represent flow evolution over short and long times, respectively.

math.DS↗