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.
In recent years, many efforts have been addressed on collision avoidance of collectively moving agents. In this paper, we propose a modified version of the Vicsek model with adaptive speed, which can guarantee the absence of collisions. However, this strategy leads to an aggregated state with slowly moving agents. We therefore further add a certain repulsion, which results in both faster consensus and longer safe distance among agents, and thus provides a powerful mechanism for collective motions in biological and technological multiagent systems.
Data I/O poses a significant bottleneck in large-scale CFD simulations; thus, practitioners would like to significantly reduce the number of times the solution is saved to disk, yet retain the ability to recover any field quantity (at any time instance) a posteriori. The objective of this work is therefore to accurately recover missing CFD data a posteriori at any time instance, given that the solution has been written to disk at only a relatively small number of time instances. We consider in particular high-order discretizations (e.g., discontinuous Galerkin), as such techniques are becoming increasingly popular for the simulation of highly separated flows. To satisfy this objective, this work proposes a methodology consisting of two stages: 1) dimensionality reduction and 2) dynamics learning. For dimensionality reduction, we propose a novel hierarchical approach. First, the method reduces the number of degrees of freedom within each element of the high-order discretization by applying autoencoders from deep learning. Second, the methodology applies principal component analysis to compress the global vector of encodings. This leads to a low-dimensional state, which associates with a nonlinear embedding of the original CFD data. For dynamics learning, we propose to apply regression techniques (e.g., kernel methods) to learn the discrete-time velocity characterizing the time evolution of this low-dimensional state. A numerical example on a large-scale CFD example characterized by nearly 13 million degrees of freedom illustrates the suitability of the proposed method in an industrial setting.
Abstract. This article discusses a newly developed online manifold learning method, subspace iteration using reduced models (SIRM), for the dimensionality reduction of dynamical systems. This method may be viewed as subspace iteration combined with a model reduction procedure. Specifically, starting with a test solution, the method solves a reduced model to obtain a more precise solution, and it repeats this process until sufficient accuracy is achieved. The reduced model is obtained by projecting the full model onto a subspace that is spanned by the dominant modes of an extended data ensemble. The extended data ensemble in this article contains not only the state vectors of some snapshots of the approximate solution from the previous iteration but also the associated tangent vectors. Therefore, the proposed manifold learning method takes advantage of the information of the original dynamical system to reduce the dynamics. Moreover, the learning procedure is computed in the online stage, as opposed to being computed offline, which is used in many projection-based model reduction techniques that require prior calculations or experiments. After providing an error bound of the classical POD-Galerkin method in terms of the projection error and the initial condition error, we prove that the sequence of approximate solutions converge to the actual solution of the original system as long as the vector field of the full model is locally Lipschitz on an open set that contains the solution trajectory. Good accuracy of the proposed method has been demonstrated in two numerical examples, from a linear advection-diffusion equation to a nonlinear Burgers equation. In order to save computational cost, the SIRM method is extended to a local model reduction approach by partitioning the entire time domain into several subintervals and obtaining a series of local reduced models of much lower dimensionality. The accuracy and efficiency of the local SIRM are shown through the numerical simulation of the Navier-Stokes equation in a lid-driven cavity flow problem.
SUMMARYIn this article, we propose a new approach for model reduction of parameterized partial differential equations (PDEs) by a locally weighted proper orthogonal decomposition (LWPOD) method. 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, LWPOD approximates the original system by multiple local reduced bases. Each local reduced basis is generated by the singular value decomposition of a weighted snapshot matrix. Compared with global model reduction methods, such as the classical proper orthogonal decomposition, LWPOD can yield more accurate solutions with a fixed subspace dimension. As another contribution, we combine LWPOD 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.
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