Kamran Mohseni

Liqian Peng, Kamran Mohseni

In this paper, a symplectic model reduction technique, proper symplectic decomposition (PSD) with symplectic Galerkin projection, is proposed to save the computational cost for the simplification of large-scale Hamiltonian systems while preserving the symplectic structure. As an analogy to the classical proper orthogonal decomposition (POD)-Galerkin approach, PSD is designed to build a symplectic subspace to fit empirical data, while the symplectic Galerkin projection constructs a reduced Hamiltonian system on the symplectic subspace. For practical use, we introduce three algorithms for PSD, which are based upon: the cotangent lift, complex singular value decomposition, and nonlinear programming. The proposed technique has been proven to preserve system energy and stability. Moreover, PSD can be combined with the discrete empirical interpolation method to reduce the computational cost for nonlinear Hamiltonian systems. Owing to these properties, the proposed technique is better suited than the classical POD-Galerkin approach for model reduction of Hamiltonian systems, especially when long-time integration is required. The stability, accuracy, and efficiency of the proposed technique are illustrated through numerical simulations of linear and nonlinear wave equations.

Liqian Peng, Kamran Mohseni

This paper reports a development in the proper symplectic decomposition (PSD) for model reduction of forced Hamiltonian systems. As an analogy to the proper orthogonal decomposition (POD), PSD is designed to build a symplectic subspace to fit empirical data. Our aim is two-fold. First, to achieve computational savings for large-scale Hamiltonian systems with external forces. Second, to simultaneously preserve the symplectic structure and the forced structure of the original system. We first reformulate d'Alembert's principle in the Hamiltonian form. Corresponding to the integral and local forms of d'Alembert's principle, we propose two different structure-preserving model reduction approaches to reconstruct low-dimensional systems, based on the variational principle and on the structure-preserving projection, respectively. These two approaches are proven to yield the same reduced system. Moreover, by incorporating the vector field into the data ensemble, we provided several algorithms for energy preservation. In a special case when the external force is described by the Rayleigh dissipative function, the proposed method automatically preserves the dissipativity, boundedness, and stability of the original system. The stability, accuracy, and efficiency of the proposed method are illustrated through numerical simulations of a dissipative wave equation.

Liqian Peng, Kamran Mohseni

In this paper, we propose a new approach to model reduction of parameterized partial differential equations (PDEs) based on the concept of adaptive reduced bases. The presented approach is particularly suited for large-scale nonlinear systems characterized by parameter variations. Instead of using a global basis to construct a global reduced model, the proposed method approximates the original system by multiple lower-dimensional subspaces. Each localized reduced basis is generated by the SVD of a weighted snapshot ensemble; here, each weighting coefficient is a function of the input parameter. Compared with a global model reduction method, such as the classical POD, the adaptive model reduction method could yield a more accurate solution with a fixed subspace dimension. Moreover, we combine the adaptive reduced model with the chord iteration to solve elliptic PDEs in a computationally efficient fashion. The potential of the method for achieving large speedups, while maintaining good accuracy, is demonstrated for both elliptic and parabolic PDEs in a few numerical examples.

Zhuoyuan Song, Kamran Mohseni

We present the concept of concurrent flow-based localization and mapping (FLAM) for autonomous field robots navigating within background flows. Different from the classical simultaneous localization and mapping (SLAM) problem, where the robot interacts with discrete features, FLAM utilizes the continuous flow fields as navigation references for mobile robots and provides flow field mapping capability with in-situ flow velocity observations. This approach is of importance to underwater vehicles in mid-depth oceans or aerial vehicles in GPS-denied atmospheric circulations. This article introduces the formulation of FLAM as a full SLAM solution motivated by the feature-based GraphSLAM framework. The performance of FLAM was demonstrated through simulation within artificial flow fields that represent typical geophysical circulation phenomena: a steady single-gyre flow field and a double-gyre flow field with unsteady turbulent perturbations. The results indicate that FLAM provides significant improvements in the robots' localization accuracy and a consistent approximation of the background flow field. It is also shown that FLAM leads to smooth robot trajectory estimates.

Zhuoyuan Song, Kamran Mohseni

A current-aided inertial navigation framework is proposed for small autonomous underwater vehicles in long-duration operations (> 1 hour), where neither frequent surfacing nor consistent bottom-tracking are available. We instantiate this concept through mid-depth, underwater navigation. This strategy mitigates dead-reckoning uncertainty of a traditional inertial navigation system by comparing the estimate of local, ambient flow velocity with preloaded ocean current maps. The proposed navigation system is implemented through a marginalized particle filter where the vehicle's states are sequentially tracked along with sensor bias and local turbulence that is not resolved by general flow prediction. The performance of the proposed approach is first analyzed through Monte Carlo simulations in two artificial background flow fields, resembling real-world ocean circulation patterns, superposed with smaller-scale, turbulent components with Kolmogorov energy spectrum. The current-aided navigation scheme significantly improves the dead-reckoning performance of the vehicle even when unresolved, small-scale flow perturbations are present. For a 6-hour navigation with an automotive-grade inertial navigation system, the current-aided navigation scheme results in positioning estimates with under 3% uncertainty per distance traveled (UDT) in a turbulent, double-gyre flow field, and under 7.3% UDT in a turbulent, meandering jet flow field. Further evaluation with field test data and actual ocean simulation analysis demonstrates consistent performance for a 6-hour mission, positioning result with under 25% UDT for a 24-hour navigation when provided direct heading measurements, and terminal positioning estimate with 16% UDT at the cost of increased uncertainty at an early stage of the navigation.