Numerical viscosity solutions to Hamilton-Jacobi equations via a Carleman estimate and the convexification method
Numerical viscosity solutions to Hamilton-Jacobi equations via a Carleman estimate and the convexification method
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DOI：
10.1016/j.jcp.2021.110828
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发表时间：
2021-04
期刊：
影响因子：
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通讯作者：
M. Klibanov;L. Nguyen;H. Tran
中科院分区：
文献类型：
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作者：
M. Klibanov;L. Nguyen;H. Tran
我们提出了一种全局收敛的数值方法，称为凸化，数值计算的粘性解的一阶Hamilton-Jacobi方程通过消失的粘性过程的粘性参数是一个固定的小数字。通过凸化，我们意味着我们采用合适的Carleman权函数来凸化直接从所考虑的Hamilton-Jacobi方程的形式定义的成本泛函。这个功能的严格凸性严格证明使用一个新的Carleman估计。我们还证明了这个严格凸泛函的唯一极小点可以通过梯度下降法达到。此外，我们表明，极小化很好地逼近的粘性解的Hamilton-Jacobi方程的边界数据中包含的噪声趋于零。一些有趣的数值例子。
We propose a globally convergent numerical method, called the convexification, to numerically compute the viscosity solution to first-order Hamilton-Jacobi equations through the vanishing viscosity process where the viscosity parameter is a fixed small number. By convexification, we mean that we employ a suitable Carleman weight function to convexify the cost functional defined directly from the form of the Hamilton-Jacobi equation under consideration. The strict convexity of this functional is rigorously proved using a new Carleman estimate. We also prove that the unique minimizer of this strictly convex functional can be reached by the gradient descent method. Moreover, we show that the minimizer well approximates the viscosity solution of the Hamilton-Jacobi equation as the noise contained in the boundary data tends to zero. Some interesting numerical illustrations are presented.