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1. Reduced Order Models for the Quasi-Geostrophic Equations: A Brief Survey

   https://doi.org/10.3390/fluids6010016  

   Mou, Changhong ; Wang, Zhu ; Wells, David R. ; Xie, Xuping ; Iliescu, Traian ( January 2021 , Fluids)

   Reduced order models (ROMs) are computational models whose dimension is significantly lower than those obtained through classical numerical discretizations (e.g., finite element, finite difference, finite volume, or spectral methods). Thus, ROMs have been used to accelerate numerical simulations of many query problems, e.g., uncertainty quantification, control, and shape optimization. Projection-based ROMs have been particularly successful in the numerical simulation of fluid flows. In this brief survey, we summarize some recent ROM developments for the quasi-geostrophic equations (QGE) (also known as the barotropic vorticity equations), which are a simplified model for geophysical flows in which rotation plays a central role, such as wind-driven ocean circulation in mid-latitude ocean basins. Since the QGE represent a practical compromise between efficient numerical simulations of ocean flows and accurate representations of large scale ocean dynamics, these equations have often been used in the testing of new numerical methods for ocean flows. ROMs have also been tested on the QGE for various settings in order to understand their potential in efficient numerical simulations of ocean flows. In this paper, we survey the ROMs developed for the QGE in order to understand their potential in efficient numerical simulations of more complex ocean flows: We explain how classicalmore »numerical methods for the QGE are used to generate the ROM basis functions, we outline the main steps in the construction of projection-based ROMs (with a particular focus on the under-resolved regime, when the closure problem needs to be addressed), we illustrate the ROMs in the numerical simulation of the QGE for various settings, and we present several potential future research avenues in the ROM exploration of the QGE and more complex models of geophysical flows.« less
2. Leveraging reduced-order models for state estimation using deep learning

   https://doi.org/10.1017/jfm.2020.409  

   Nair, Nirmal J. ; Goza, Andres ( August 2020 , Journal of Fluid Mechanics)

   State estimation is key to both analysing physical mechanisms and enabling real-time control of fluid flows. A common estimation approach is to relate sensor measurements to a reduced state governed by a reduced-order model (ROM). (When desired, the full state can be recovered via the ROM.) Current methods in this category nearly always use a linear model to relate the sensor data to the reduced state, which often leads to restrictions on sensor locations and has inherent limitations in representing the generally nonlinear relationship between the measurements and reduced state. We propose an alternative methodology whereby a neural network architecture is used to learn this nonlinear relationship. A neural network is a natural choice for this estimation problem, as a physical interpretation of the reduced state–sensor measurement relationship is rarely obvious. The proposed estimation framework is agnostic to the ROM employed, and can be incorporated into any choice of ROMs derived on a linear subspace (e.g. proper orthogonal decomposition) or a nonlinear manifold. The proposed approach is demonstrated on a two-dimensional model problem of separated flow around a flat plate, and is found to outperform common linear estimation alternatives.
3. Reduced Order Model Hessian Approximations in Newton Methods for Optimal Control

   https://doi.org/10.1007/978-3-030-95157-3_18  

   Heinkenschloss, Matthias ; Magruder, Caleb ( July 2022 , Realization and Model Reduction of Dynamical Systems - A Festschrift in Honor of the 70th Birthday of Thanos Antoulas)

   Beattie, C.A. ; Benner, P. ; Embree, M. ; Gugercin, S. ; Lefteriu, S. (Ed.)

   This paper introduces reduced order model (ROM) based Hessian approximations for use in inexact Newton methods for the solution of optimization problems implicitly constrained by a large-scale system, typically a discretization of a partial differential equation (PDE). The direct application of an inexact Newton method to this problem requires the solution of many PDEs per optimization iteration. To reduce the computational complexity, a ROM Hessian approximation is proposed. Since only the Hessian is approximated, but the original objective function and its gradient is used, the resulting inexact Newton method maintains the first-order global convergence property, under suitable assumptions. Thus even computationally inexpensive lower fidelity ROMs can be used, which is different from ROM approaches that replace the original optimization problem by a sequence of ROM optimization problem and typically need to accurately approximate function and gradient information of the original problem. In the proposed approach, the quality of the ROM Hessian approximation determines the rate of convergence, but not whether the method converges. The projection based ROM is constructed from state and adjoint snapshots, and is relatively inexpensive to compute. Numerical examples on semilinear parabolic optimal control problems demonstrate that the proposed approach can lead to substantial savings in termsmore »of overall PDE solves required.« less
4. Multiscale Modeling of Composite Materials under Volumetric and Interfacial Damage: Achieving Adaptive Model Order Reduction

   https://doi.org/10.2514/6.2023-0138  

   Lin, Min ; Brandyberry, David ; Zhang, Xiang ( January 2023 , AIAA SCITECH 2023 Forum)

   In this manuscript, we present a multiscale Adaptive Reduced-Order Modeling (AROM) framework to efficiently simulate the response of heterogeneous composite microstructures under interfacial and volumetric damage. This framework builds on the eigendeformation-based reduced-order homogenization model (EHM), which is based on the transformation field analysis (TFA) and operates in the context of computational homogenization with a focus on model order reduction of the microscale problem. EHM pre-computes certain microstructure information by solving a series of linear elastic problems defined over the fully resolved microstructure (i.e., concentration tensors, interaction tensors) and approximates the microscale problem using a much smaller basis spanned over subdomains (also called parts) of the microstructure. Using this reduced basis, and prescribed spatial variation of inelastic response fields over the parts, the microscale problem leads to a set of algebraic equations with part-wise responses as unknowns, instead of node-wise displacements as in finite element analysis. The volumetric and interfacial influence functions are calculated by using the Interface enriched Generalized Finite Element Method (IGFEM) to compute the coefficient tensors, in which the finite element discretization does not need to conform to the material interfaces. AROM takes advantage of pre-computed coefficient tensors associated with the finest ROM and efficiently computes themore »coefficient tensors of a series of gradually coarsening ROMs. During the multiscale analysis stage, the simulation starts with a coarse ROM which can capture the initial elastic response well. As the loading continues and response in certain parts of the microstructure starts to localize, the analysis adaptively switches to the next level of refined ROM to better capture those local responses. The performance of AROM is evaluated by comparing the results with regular EHM (no adaptive refinement) and IGFEM under different loading conditions and failure modes for various 2D and 3D microstructures. The proposed AROM provides an efficient way to model history-dependent nonlinear responses for composite materials under localized interface failure and phase damage.« less
5. Modeling of Collective Cell Behaviors Interacting With Extracellular Matrix Using Dual Faceted Linearization

   https://doi.org/10.1115/DSCC2018-9164  

   Mayalu, Michaëlle N. ; Kim, Min-Cheol ; Asada, H. Harry ( September 2018 , 2018 ASME Dynamic Systems and Control Conference)

   Cells interacting over an extracellular matrix (ECM) exhibit emergent behaviors, which are often observably different from single-cell dynamics. Fibroblasts embedded in a 3-D ECM, for example, compact the surrounding gel and generate an anisotropic strain field, which cannot be observed in single cellinduced gel compaction. This emergent matrix behavior results from collective intracellular mechanical interaction and is crucial to explain the large deformations and mechanical tensions that occur during embryogenesis, tissue development and wound healing. Prediction of multi-cellular interactions entails nonlinear dynamic simulation, which is prohibitively complex to compute using first principles especially as the number of cells increase. Here, we introduce a new methodology for predicting nonlinear behaviors of multiple cells interacting mechanically through a 3D ECM. In the proposed method, we first apply Dual- Faceted Linearization to nonlinear dynamic systems describing cell/matrix behavior. Using this unique linearization method, the original nonlinear state equations can be expressed with a pair of linear dynamic equations by augmenting the independent state variables with auxiliary variables which are nonlinearly dependent on the original states. Furthermore, we can find a reduced order latent space representation of the dynamic equations by orthogonal projection onto the basis of a lower dimensional linear manifold within themore »augmented variable space. Once converted to latent variable equations, we superpose multiple dynamic systems to predict their collective behaviors. The method is computationally efficient and accurate as demonstrated through its application for prediction of emergent cell induced ECM compaction.« less