Rigged Dynamic Mode Decomposition: Data-Driven Generalized Eigenfunction Decompositions for Koopman Operators

• URL: http://arxiv.org/abs/2405.00782v1
• Date: Wed, 1 May 2024 18:00:18 GMT
• Title: Rigged Dynamic Mode Decomposition: Data-Driven Generalized Eigenfunction Decompositions for Koopman Operators
• Authors: Matthew J. Colbrook, Catherine Drysdale, Andrew Horning,
• Abstract summary: We introduce the Rigged Dynamic Mode Decomposition (Rigged DMD) algorithm, which computes generalized eigenfunction decompositions of Koopman operators. Rigged DMD addresses challenges with a data-driven methodology that approximates the Koopman operator's resolvent and its generalized eigenfunctions. We provide examples, including systems with a Lebesgue spectrum, integrable Hamiltonian systems, the Lorenz system, and a high-Reynolds number lid-driven flow in a two-dimensional square cavity.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We introduce the Rigged Dynamic Mode Decomposition (Rigged DMD) algorithm, which computes generalized eigenfunction decompositions of Koopman operators. By considering the evolution of observables, Koopman operators transform complex nonlinear dynamics into a linear framework suitable for spectral analysis. While powerful, traditional Dynamic Mode Decomposition (DMD) techniques often struggle with continuous spectra. Rigged DMD addresses these challenges with a data-driven methodology that approximates the Koopman operator's resolvent and its generalized eigenfunctions using snapshot data from the system's evolution. At its core, Rigged DMD builds wave-packet approximations for generalized Koopman eigenfunctions and modes by integrating Measure-Preserving Extended Dynamic Mode Decomposition with high-order kernels for smoothing. This provides a robust decomposition encompassing both discrete and continuous spectral elements. We derive explicit high-order convergence theorems for generalized eigenfunctions and spectral measures. Additionally, we propose a novel framework for constructing rigged Hilbert spaces using time-delay embedding, significantly extending the algorithm's applicability. We provide examples, including systems with a Lebesgue spectrum, integrable Hamiltonian systems, the Lorenz system, and a high-Reynolds number lid-driven flow in a two-dimensional square cavity, demonstrating Rigged DMD's convergence, efficiency, and versatility. This work paves the way for future research and applications of decompositions with continuous spectra.

Related papers

• Multiplicative Dynamic Mode Decomposition [4.028503203417233]
  We introduce Multiplicative Dynamic Mode Decomposition (MultDMD), which enforces the multiplicative structure inherent in the Koopman operator within its finite-dimensional approximation. MultDMD presents a structured approach to finite-dimensional approximations and can accurately reflect the spectral properties of the Koopman operator. We elaborate on the theoretical framework of MultDMD, detailing its formulation, optimization strategy, and convergence properties.
  arXiv  Detail & Related papers  (2024-05-08T18:09:16Z)
• On the Convergence of Hermitian Dynamic Mode Decomposition [4.028503203417233]
  We study the convergence of Hermitian Dynamic Mode Decomposition to the spectral properties of self-adjoint Koopman operators. We numerically demonstrate our results by applying them to two-dimensional Schr"odinger equations.
  arXiv  Detail & Related papers  (2024-01-06T11:13:16Z)
• Multi-Grid Tensorized Fourier Neural Operator for High-Resolution PDEs [93.82811501035569]
  We introduce a new data efficient and highly parallelizable operator learning approach with reduced memory requirement and better generalization. MG-TFNO scales to large resolutions by leveraging local and global structures of full-scale, real-world phenomena. We demonstrate superior performance on the turbulent Navier-Stokes equations where we achieve less than half the error with over 150x compression.
  arXiv  Detail & Related papers  (2023-09-29T20:18:52Z)
• Hierarchical Integration Diffusion Model for Realistic Image Deblurring [71.76410266003917]
  Diffusion models (DMs) have been introduced in image deblurring and exhibited promising performance. We propose the Hierarchical Integration Diffusion Model (HI-Diff), for realistic image deblurring. Experiments on synthetic and real-world blur datasets demonstrate that our HI-Diff outperforms state-of-the-art methods.
  arXiv  Detail & Related papers  (2023-05-22T12:18:20Z)
• Score-based Diffusion Models in Function Space [140.792362459734]
  Diffusion models have recently emerged as a powerful framework for generative modeling. We introduce a mathematically rigorous framework called Denoising Diffusion Operators (DDOs) for training diffusion models in function space. We show that the corresponding discretized algorithm generates accurate samples at a fixed cost independent of the data resolution.
  arXiv  Detail & Related papers  (2023-02-14T23:50:53Z)
• The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems [0.0]
  We introduce measure-preserving extended dynamic mode decomposition ($textttmpEDMD$), the first truncation method whose eigendecomposition converges to the spectral quantities of Koopman operators. $textttmpEDMD$ is flexible and easy to use with any pre-existing DMD-type method, and with different types of data.
  arXiv  Detail & Related papers  (2022-09-06T06:37:54Z)
• Residual Dynamic Mode Decomposition: Robust and verified Koopmanism [0.0]
  Dynamic Mode Decomposition (DMD) describes complex dynamic processes through a hierarchy of simpler coherent features. We present Residual Dynamic Mode Decomposition (ResDMD), which overcomes challenges through the data-driven computation of residuals associated with the full infinite-dimensional Koopman operator. ResDMD computes spectra and pseudospectra of general Koopman operators with error control, and computes smoothed approximations of spectral measures (including continuous spectra) with explicit high-order convergence theorems.
  arXiv  Detail & Related papers  (2022-05-19T18:02:44Z)
• Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems [2.0305676256390934]
  This paper describes algorithms with rigorous convergence guarantees for computing spectral information of Koopman operators. We compute smoothed approximations of spectral measures associated with general measure-preserving dynamical systems. We demonstrate our algorithms on the tent map, circle rotations, Gauss iterated map, nonlinear pendulum, double pendulum, and Lorenz system.
  arXiv  Detail & Related papers  (2021-11-29T19:01:26Z)
• Dynamic Mode Decomposition in Adaptive Mesh Refinement and Coarsening Simulations [58.720142291102135]
  Dynamic Mode Decomposition (DMD) is a powerful data-driven method used to extract coherent schemes. This paper proposes a strategy to enable DMD to extract from observations with different mesh topologies and dimensions.
  arXiv  Detail & Related papers  (2021-04-28T22:14:25Z)
• Estimating Koopman operators for nonlinear dynamical systems: a nonparametric approach [77.77696851397539]
  The Koopman operator is a mathematical tool that allows for a linear description of non-linear systems. In this paper we capture their core essence as a dual version of the same framework, incorporating them into the Kernel framework. We establish a strong link between kernel methods and Koopman operators, leading to the estimation of the latter through Kernel functions.
  arXiv  Detail & Related papers  (2021-03-25T11:08:26Z)
• Graph Gamma Process Generalized Linear Dynamical Systems [60.467040479276704]
  We introduce graph gamma process (GGP) linear dynamical systems to model real multivariate time series. For temporal pattern discovery, the latent representation under the model is used to decompose the time series into a parsimonious set of multivariate sub-sequences. We use the generated random graph, whose number of nonzero-degree nodes is finite, to define both the sparsity pattern and dimension of the latent state transition matrix.
  arXiv  Detail & Related papers  (2020-07-25T04:16:34Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.