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A hybrid data assimilation algorithm is developed for complex dynamical systems with partial observations. The method starts with applying a spectral decomposition to the entire spatiotemporal fields, followed by creating a machine learning model that builds a nonlinear map between the coefficients of observed and unobserved state variables for each spectral mode. A cheap low‐order nonlinear stochastic parameterized extended Kalman filter (SPEKF) model is employed as the forecast model in the ensemble Kalman filter to deal with each mode associated with the observed variables. The resulting ensemble members are then fed into the machine learning model to create an ensemble of the corresponding unobserved variables. In addition to the ensemble spread, the training residual in the machine learning‐induced nonlinear map is further incorporated into the state estimation, advancing the diagnostic quantification of the posterior uncertainty. The hybrid data assimilation algorithm is applied to a precipitating quasi‐geostrophic (PQG) model, which includes the effects of water vapor, clouds, and rainfall beyond the classical two‐level QG model. The complicated nonlinearities in the PQG equations prevent traditional methods from building simple and accurate reduced‐order forecast models. In contrast, the SPEKF forecast model is skillful in recovering the intermittent observed states, and the machine learning model effectively estimates the chaotic unobserved signals. Utilizing the calibrated SPEKF and machine learning models under a moderate cloud fraction, the resulting hybrid data assimilation remains reasonably accurate when applied to other geophysical scenarios with nearly clear skies or relatively heavy rainfall, implying the robustness of the algorithm for extrapolation.

 

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 classical 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. 

There are two main strategies for improving the projection-based reduced order model (ROM) accuracy—(i) improving the ROM, that is, adding new terms to the standard ROM; and (ii) improving the ROM basis, that is, constructing ROM bases that yield more accurate ROMs. In this paper, we use the latter. We propose two new Lagrangian inner products that we use together with Eulerian and Lagrangian data to construct two new Lagrangian ROMs, which we denote α-ROM and λ-ROM. We show that both Lagrangian ROMs are more accurate than the standard Eulerian ROMs, that is, ROMs that use standard Eulerian inner product and data to construct the ROM basis. Specifically, for the quasi-geostrophic equations, we show that the new Lagrangian ROMs are more accurate than the standard Eulerian ROMs in approximating not only Lagrangian fields (e.g., the finite time Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction). In particular, the α-ROM can be orders of magnitude more accurate than the standard Eulerian ROMs. We emphasize that the new Lagrangian ROMs do not employ any closure modeling to model the effect of discarded modes (which is standard procedure for low-dimensional ROMs of complex nonlinear systems). Thus, the dramatic increase in the new Lagrangian ROMs’ accuracy is entirely due to the novel Lagrangian inner products used to build the Lagrangian ROM basis. 

The vagina undergoes large finite deformations and has complex geometry and microstructure, resulting in material and geometric nonlinearities, complicated boundary conditions, and nonhomogeneities within finite element (FE) simulations. These nonlinearities pose a significant challenge for numerical solvers, increasing the computational time by several orders of magnitude. Simplifying assumptions can reduce the computational time significantly, but this usually comes at the expense of simulation accuracy. This study proposed the use of reduced order modeling (ROM) techniques to capture experimentally measured displacement fields of rat vaginal tissue during inflation testing in order to attain both the accuracy of higher‐fidelity models and the speed of simpler simulations. The proper orthogonal decomposition (POD) method was used to extract the significant information from FE simulations generated by varying the luminal pressure and the parameters that introduce the anisotropy in the selected constitutive model. A new data‐driven (DD) variational multiscale (VMS) ROM framework was extended to obtain the displacement fields of rat vaginal tissue under pressure. For comparison purposes, we also investigated the classical Galerkin ROM (G‐ROM). In our numerical study, both the G‐ROM and the DD‐VMS‐ROM decreased the FE computational cost by orders of magnitude without a significant decrease in numerical accuracy. Furthermore, the DD‐VMS‐ROM improved the G‐ROM accuracy at a modest computational overhead. Our numerical investigation showed that ROM has the potential to provide efficient and accurate computational tools to describe vaginal deformations, with the ultimate goal of improving maternal health.