Recovering a potential in damped wave equation from Neumann-to-Dirichlet operator
Recovering a potential in damped wave equation from Neumann-to-Dirichlet operator
复制标题
从诺依曼到狄利克雷算子恢复阻尼波动方程中的势
DOI：
10.1088/1361-6420/abb8e8
复制
发表时间：
2020
期刊：
影响因子：
2.1
通讯作者：
A. Hasanov
中科院分区：
文献类型：
--
作者：
V. Romanov;A. Hasanov
阻尼波动方程 m(x)utt+μ(x)ut=r(x)uxx+q(x)u 中恢复势 q(x) 的反系数问题，(x, t) ∈ Ω T ≔ (0, ℓ) × (0, T) 受边界条件 r(0)u x (0, t) = f(t), u(ℓ, t) = 0 影响，由研究了狄利克雷边界测量输出ν(t) ≔ u(0, t), t ε (0, T]。对由特征定义的子域以及沿着这些特征定义的直接问题解的规律性进行了详细的微局部分析。在此分析的基础上，导出了必要的规律性结果和能量估计。证明了狄利克雷边界测量输出唯一地确定了区间[0, h(T/2)]内的势q(x)，并且该解属于到 C(0, h(T/2)) 且 T < T*，其中 h(z) 是方程 z=∫0h(z)m(x)/r(x)dx 的根，T*=2∫0ℓm(x)/r(x)dx 。此外，还证明了 Neumann-to-Dirichlet 算子的紧致性、可逆性和 Lipschitz 连续性。 Φf[⋅]:Q⊂C(0,ℓ)↦L2(0,T) , Φ f [q](t) ≔ u(0, t; q) 被证明，这也使我们能够证明定义为吉洪诺夫泛函 J(q)≔(1/2)‖Φf[⋅]−ν‖L2(0,T)2 最小值的反问题拟解的存在性。其 Fréchet 可微分性是通过利用相应伴随问题的唯一解推导得出的 Fréchet 梯度的显式公式，从而得出非常有效的基于梯度的计算识别算法。
The inverse coefficient problem of recovering the potential q(x) in the damped wave equation m(x)utt+μ(x)ut=r(x)uxx+q(x)u , (x, t) ∈ Ω T ≔ (0, ℓ) × (0, T) subject to the boundary conditions r(0)u x (0, t) = f(t), u(ℓ, t) = 0, from the Dirichlet boundary measured output ν(t) ≔ u(0, t), t ∈ (0, T] is studied. A detailed microlocal analysis of regularity of the direct problem solution in the subdomains defined by the characteristics as well as along these characteristics is provided. Based on this analysis, necessary regularity results and energy estimates are derived. It is proved that the Dirichlet boundary measured output uniquely determines the potential q(x) in the interval [0, h(T/2)] and this solution belongs to C(0, h(T/2)) with T < T*, where h(z) is the root of the equation z=∫0h(z)m(x)/r(x)dx , T*=2∫0ℓm(x)/r(x)dx . Moreover, the global uniqueness theorem is proved. Compactness, invertibility and Lipschitz continuity of the Neumann-to-Dirichlet operator Φf[⋅]:Q⊂C(0,ℓ)↦L2(0,T) , Φ f [q](t) ≔ u(0, t; q) is proved. This allows us to prove an existence of a quasi-solution of the inverse problem defined as a minimum of the Tikhonov functional J(q)≔(1/2)‖Φf[⋅]−ν‖L2(0,T)2 as well as its Fréchet differentiability. An explicit formula for the Fréchet gradient is derived by making use of the unique solution to corresponding adjoint problem. The proposed approach is leads to very effective gradient based computational identification algorithm.