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1. Sparsity-promoting algorithms for the discovery of informative Koopman-invariant subspaces

   https://doi.org/10.1017/jfm.2021.271  

   Pan, Shaowu; Arnold-Medabalimi, Nicholas; Duraisamy, Karthik (June 2021, Journal of Fluid Mechanics)

   null (Ed.)

   Koopman decomposition is a nonlinear generalization of eigen-decomposition, and is being increasingly utilized in the analysis of spatio-temporal dynamics. Well-known techniques such as the dynamic mode decomposition (DMD) and its linear variants provide approximations to the Koopman operator, and have been applied extensively in many fluid dynamic problems. Despite being endowed with a richer dictionary of nonlinear observables, nonlinear variants of the DMD, such as extended/kernel dynamic mode decomposition (EDMD/KDMD) are seldom applied to large-scale problems primarily due to the difficulty of discerning the Koopman-invariant subspace from thousands of resulting Koopman eigenmodes. To address this issue, we propose a framework based on a multi-task feature learning to extract the most informative Koopman-invariant subspace by removing redundant and spurious Koopman triplets. In particular, we develop a pruning procedure that penalizes departure from linear evolution. These algorithms can be viewed as sparsity-promoting extensions of EDMD/KDMD. Furthermore, we extend KDMD to a continuous-time setting and show a relationship between the present algorithm, sparsity-promoting DMD and an empirical criterion from the viewpoint of non-convex optimization. The effectiveness of our algorithm is demonstrated on examples ranging from simple dynamical systems to two-dimensional cylinder wake flows at different Reynolds numbers and a three-dimensional turbulent ship-airwake flow. The latter two problems are designed such that very strong nonlinear transients are present, thus requiring an accurate approximation of the Koopman operator. Underlying physical mechanisms are analysed, with an emphasis on characterizing transient dynamics. The results are compared with existing theoretical expositions and numerical approximations. 

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2. Modal analysis of blood flows in saccular aneurysms

   https://doi.org/10.1063/5.0243383  

   Nguyen, Thien-Tam; Kasperski, Davina; Huynh, Phat Kim; Le, Trung Quoc; Le, Trung Bao (January 2025, Physics of Fluids)

   Currently, it is challenging to investigate aneurismal hemodynamics based on current in vivo data such as Magnetic Resonance Imaging or Computed Tomography due to the limitations in both spatial and temporal resolutions. In this work, we investigate the use of modal analysis at various resolutions to examine its usefulness for analyzing blood flows in brain aneurysms. Two variants of Dynamic Mode Decomposition (DMD): (i) Hankel-DMD; and (ii) Optimized-DMD, are used to extract the time-dependent dynamics of blood flows during one cardiac cycle. First, high-resolution hemodynamic data in patient-specific aneurysms are obtained using Computational Fluid Dynamics. Second, the dynamics modes, along with their spatial amplitudes and temporal magnitudes are calculated using the DMD analysis. Third, an examination of DMD analyses using a range of spatial and temporal resolutions of hemodynamic data to validate the applicability of DMD for low-resolution data, similar to ones in clinical practices. Our results show that DMD is able to characterize the inflow jet dynamics by separating large-scale structures and flow instabilities even at low spatial and temporal resolutions. Its robustness in quantifying the flow dynamics using the energy spectrum is demonstrated across different resolutions in all aneurysms in our study population. Our work indicates that DMD can be used for analyzing blood flow patterns of brain aneurysms and is a promising tool to be explored in in vivo. 

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3. Femtosecond 3D photolithography through a digital micromirror device and a microlens array

   https://doi.org/10.1364/AO.457847  

   Jakkinapalli, Aravind; Baskar, Balaji; Wen, Sy-Bor (May 2022, Applied Optics)

   Based on the microscale 3D point cloud projection with a digital micromirror device (DMD) and a microlens array (MLA) developed recently, we explore the capabilities of this specific type of 3D projection in 3D lithography with femtosecond light in this study. Unlike 3D point cloud projection with UV continuous light demonstrated before, high accuracy positioning between the DMD and the MLA is required to have rays simultaneously arrive at the designed voxel positions to induce two-photon absorption with femtosecond light. Because of this additional requirement, a new positioning method through direct microscope inspection of the relative positions of the DMD and the MLA is developed in this study. Because of the usage of a rectangular MLA, around four rays can arrive at each projecting voxel at the same time. Thus, to the best of our knowledge, a new algorithm for determining the pixel map on the DMD to the 3D point cloud projection with a femtosecond laser is also developed. It is observed that a very long exposure time is required to generate 3D patterns with the new 3D projection scheme because of the very limited number of rays used for projecting each voxel with the new algorithm. It is also found that 3D structures with desired shapes should be projected far away from the MLA ( $\sim <\#comment/>15f$to $30f$, with $f$being the focal distance of the MLA) in the 3D lithography with this femtosecond 3D point cloud projection. For patterns projected closer than $10f$, shapes are distorted because of unwanted voxels cured with the 3D projection technique using a DMD and MLA. 

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4. Data-Driven Model Reduction for Coupled Flow and Geomechanics Based on DMD Methods

   https://doi.org/10.3390/fluids4030138  

   Bao, Anqi; Gildin, Eduardo; Narasingam, Abhinav; Kwon, Joseph S. (September 2019, Fluids)

   Learning reservoir flow dynamics is of primary importance in creating robust predictive models for reservoir management including hydraulic fracturing processes. Physics-based models are to a certain extent exact, but they entail heavy computational infrastructure for simulating a wide variety of parameters and production scenarios. Reduced-order models offer computational advantages without compromising solution accuracy, especially if they can assimilate large volumes of production data without having to reconstruct the original model (data-driven models). Dynamic mode decomposition (DMD) entails the extraction of relevant spatial structure (modes) based on data (snapshots) that can be used to predict the behavior of reservoir fluid flow in porous media. In this paper, we will further enhance the application of the DMD, by introducing sparse DMD and local DMD. The former is particularly useful when there is a limited number of sparse measurements as in the case of reservoir simulation, and the latter can improve the accuracy of developed DMD models when the process dynamics show a moving boundary behavior like hydraulic fracturing. For demonstration purposes, we first show the methodology applied to (flow only) single- and two-phase reservoir models using the SPE10 benchmark. Both online and offline processes will be used for evaluation. We observe that we only require a few DMD modes, which are determined by the sparse DMD structure, to capture the behavior of the reservoir models. Then, we applied the local DMDc for creating a proxy for application in a hydraulic fracturing process. We also assessed the trade-offs between problem size and computational time for each reservoir model. The novelty of our method is the application of sparse DMD and local DMDc, which is a data-driven technique for fast and accurate simulations. 

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5. Beyond expectations: residual dynamic mode decomposition and variance for stochastic dynamical systems

   https://doi.org/10.1007/s11071-023-09135-w  

   Colbrook, Matthew_J; Li, Qin; Raut, Ryan_V; Townsend, Alex (December 2023, Nonlinear Dynamics)

   Abstract Koopman operators linearize nonlinear dynamical systems, making their spectral information of crucial interest. Numerous algorithms have been developed to approximate these spectral properties, and dynamic mode decomposition (DMD) stands out as the poster child of projection-based methods. Although the Koopman operator itself is linear, the fact that it acts in an infinite-dimensional space of observables poses challenges. These include spurious modes, essential spectra, and the verification of Koopman mode decompositions. While recent work has addressed these challenges for deterministic systems, there remains a notable gap in verified DMD methods for stochastic systems, where the Koopman operator measures the expectation of observables. We show that it is necessary to go beyond expectations to address these issues. By incorporating variance into the Koopman framework, we address these challenges. Through an additional DMD-type matrix, we approximate the sum of a squared residual and a variance term, each of which can be approximated individually using batched snapshot data. This allows verified computation of the spectral properties of stochastic Koopman operators, controlling the projection error. We also introduce the concept of variance-pseudospectra to gauge statistical coherency. Finally, we present a suite of convergence results for the spectral information of stochastic Koopman operators. Our study concludes with practical applications using both simulated and experimental data. In neural recordings from awake mice, we demonstrate how variance-pseudospectra can reveal physiologically significant information unavailable to standard expectation-based dynamical models. 

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