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Register a new userHigh order finite difference Hermite WENO fast sweeping methods for static Hamilton-Jacobi equations
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Added by
Yupeng Ren
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In this paper, we propose a novel Hermite weighted essentially non-oscillatory (HWENO) fast sweeping method to solve the static Hamilton-Jacobi equations efficiently. During the HWENO reconstruction procedure, the proposed method is built upon a new finite difference fifth order HWENO scheme involving one big stencil and two small stencils. However, one major novelty and difference from the traditional HWENO framework lies in the fact that, we do not need to introduce and solve any additional equations to update the derivatives of the unknown function $phi$. Instead, we use the current $phi$ and the old spatial derivative of $phi$ to update them. The traditional HWENO fast sweeping method is also introduced in this paper for comparison, where additional equations governing the spatial derivatives of $phi$ are introduced. The novel HWENO fast sweeping methods are shown to yield great savings in both computational time and storage, which improves the computational efficiency of the traditional HWENO scheme. In addition, a hybrid strategy is also introduced to further reduce computational costs. Extensive numerical experiments are provided to validate the accuracy and efficiency of the proposed approaches.
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In this paper, we combine the nonlinear HWENO reconstruction in cite{newhwenozq} and the fixed-point iteration with Gauss-Seidel fast sweeping strategy, to solve the static Hamilton-Jacobi equations in a novel HWENO framework recently developed in cite{mehweno1}. The proposed HWENO frameworks enjoys several advantages. First, compared with the traditional HWENO framework, the proposed methods do not need to introduce additional auxiliary equations to update the derivatives of the unknown function $phi$. They are now computed from the current value of $phi$ and the previous spatial derivatives of $phi$. This approach saves the computational storage and CPU time, which greatly improves the computational efficiency of the traditional HWENO scheme. In addition, compared with the traditional WENO method, reconstruction stencil of the HWENO methods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller on the same mesh. Second, the fixed-point fast sweeping method is used to update the numerical approximation. It is an explicit method and does not involve the inverse operation of nonlinear Hamiltonian, therefore any Hamilton-Jacobi equations with complex Hamiltonian can be solved easily. It also resolves some known issues, including that the iterative number is very sensitive to the parameter $varepsilon$ used in the nonlinear weights, as observed in previous studies. Finally, in order to further reduce the computational cost, a hybrid strategy is also presented. Extensive numerical experiments are performed on two-dimensional problems, which demonstrate the good performance of the proposed fixed-point fast sweeping HWENO methods.
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A tensor decomposition approach for the solution of high-dimensional, fully nonlinear Hamilton-Jacobi-Bellman equations arising in optimal feedback control of nonlinear dynamics is presented. The method combines a tensor train approximation for the value function together with a Newton-like iterative method for the solution of the resulting nonlinear system. The tensor approximation leads to a polynomial scaling with respect to the dimension, partially circumventing the curse of dimensionality. A convergence analysis for the linear-quadratic case is presented. For nonlinear dynamics, the effectiveness of the high-dimensional control synthesis method is assessed in the optimal feedback stabilization of the Allen-Cahn and Fokker-Planck equations with a hundred of variables.
In the past decade, there are many works on the finite element methods for the fully nonlinear Hamilton--Jacobi--Bellman (HJB) equations with Cordes condition. The linearised systems have large condition numbers, which depend not only on the mesh size, but also on the parameters in the Cordes condition. This paper is concerned with the design and analysis of auxiliary space preconditioners for the linearised systems of $C^0$ finite element discretization of HJB equations [Calcolo, 58, 2021]. Based on the stable decomposition on the auxiliary spaces, we propose both the additive and multiplicative preconditoners which converge uniformly in the sense that the resulting condition number is independent of both the number of degrees of freedom and the parameter $lambda$ in Cordes condition. Numerical experiments are carried out to illustrate the efficiency of the proposed preconditioners.
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