Abstract In this paper, the identification problem of recovering the spatial source F ∈ L 2 ⁢ ( 0 , l ) {F\in L^{2}(0,l)} in the wave equation u t ⁢ t = u x ⁢ x + F ⁢ ( x ) ⁢ cos ⁡ ( ω ⁢ t ) {u_{tt}=u_{xx}+F(x)\cos(\omega t)} , with harmonically varying external source F ⁢ ( x ) ⁢ cos ⁡ ( ω ⁢ t ) {F(x)\cos(\omega t)} and with the homogeneous boundary u ⁢ ( 0 , t ) = u ⁢ ( l , t ) = 0 {u(0,t)=u(l,t)=0} , t ∈ ( 0 , T ) {t\in(0,T)} , and initial u ⁢ ( x , 0 ) = u t ⁢ ( x , 0 ) = 0 {u(x,0)=u_{t}(x,0)=0} , x ∈ ( 0 , l ) {x\in(0,l)} , conditions, is studied. As a measurement output g ⁢ ( t ) {g(t)} , the Neumann-type boundary measurement g ⁢ ( t ) := u x ⁢ ( 0 , t ) {g(t):=u_{x}(0,t)} , t ∈ ( 0 , T ) {t\in(0,T)} , at the left boundary x = 0 {x=0} is used. It is assumed that the observation g ∈ L 2 ⁢ ( 0 , T ) {g\in L^{2}(0,T)} may has a random noise. We propose combination of the boundary control for PDEs, adjoint method and Tikhonov regularization, for identification of the unknown source F ∈ L 2 ⁢ ( 0 , l ) {F\in L^{2}(0,l)} . Our approach based on weak solution theory of PDEs and, as a result, allows use of nonsmooth input/output data. Introducing the input-output operator Φ ⁢ F := u x ⁢ ( 0 , t ; F ) {\Phi F:=u_{x}(0,t;F)} , Φ : L 2 ⁢ ( 0 , l ) ↦ L 2 ⁢ ( 0 , T ) {\Phi:L^{2}(0,l)\mapsto L^{2}(0,T)} , where u ⁢ ( x , t ; F ) {u(x,t;F)} is the solution of the wave equation with above homogeneous boundary and initial conditions, we first prove the compactness of this operator. This allows to obtain the uniqueness of regularized solution of the identification problem, i.e. the minimum of the regularized cost functional J α ⁢ ( F ) := J ⁢ ( F ) + 1 2 ⁢ α ⁢ ∥ F ∥ L 2 ⁢ ( 0 , l ) 2 {J_{\alpha}(F):=J(F)+\frac{1}{2}\alpha\|F\|_{L^{2}(0,l)}^{2}} , where J ⁢ ( F ) = 1 2 ⁢ ∥ u x ⁢ ( 0 , ⋅ ; F ) - g ∥ L 2 ⁢ ( 0 , T ) 2 {J(F)=\frac{1}{2}\|u_{x}(0,\cdot\,;F)-g\|_{L^{2}(0,T)}^{2}} . Then the adjoint problem approach is used to derive a formula for the Fréchet gradient of the cost functional J ⁢ ( F ) {J(F)} . Use of the gradient formula in the conjugate gradient algorithm (CGA) allows to construct a fast algorithm for recovering the unknown source F ⁢ ( x ) {F(x)} . A comprehensive set of benchmark numerical examples, with up to 10 noise level random noisy data, illustrate the usefulness and effectiveness of the proposed approach.

中文翻译：

---

从诺依曼型边界测量识别波动方程中谐波变化的外部源

摘要 本文研究了波动方程ut ⁢ t = ux ⁢ x + 中空间源F ∈ L 2 ⁢ ( 0 , l ) {F\in L^{2}(0,l)} 的辨识问题F ⁢ ( x ) ⁢ cos ⁡ ( ω ⁢ t ) {u_{tt}=u_{xx}+F(x)\cos(\omega t)} ，具有谐波变化的外部源 F ⁢ ( x ) ⁢ cos ⁡ ( ω ⁢ t ) {F(x)\cos(\omega t)} 并且具有齐次边界 u ⁢ ( 0 , t ) = u ⁢ ( l , t ) = 0 {u(0,t)=u( l,t)=0} , t ∈ ( 0 , T ) {t\in(0,T)} ，并且初始 u ⁢ ( x , 0 ) = ut ⁢ ( x , 0 ) = 0 {u(x, 0)=u_{t}(x,0)=0} , x ∈ ( 0 , l ) {x\in(0,l)} ，条件，被研究。作为测量输出 g ⁢ ( t ) {g(t)} ，Neumann 型边界测量 g ⁢ ( t ) := ux ⁢ ( 0 , t ) {g(t):=u_{x}(0, t)} , t ∈ ( 0 , T ) {t\in(0,T)} ，在左边界使用 x = 0 {x=0} 。假设观察 g ∈ L 2 ⁢ ( 0 , T ) {g\in L^{2}(0,T)} 可能具有随机噪声。我们建议将偏微分方程的边界控制、伴随方法和 Tikhonov 正则化相结合，用于识别未知源 F ∈ L 2 ⁢ ( 0 , l ) {F\in L^{2}(0,l)} 。我们的方法基于 PDE 的弱解理论，因此允许使用非平滑输入/输出数据。介绍输入输出算子 Φ ⁢ F := ux ⁢ ( 0 , t ; F ) {\Phi F:=u_{x}(0,t;F)} , Φ : L 2 ⁢ ( 0 , l ) ↦ L 2 ⁢ ( 0 , T ) {\Phi:L^{2}(0,l)\mapsto L^{2}(0,T)} ，其中 u ⁢ ( x , t ; F ) {u(x ,t;F)} 是具有以上齐次边界和初始条件的波动方程的解，我们首先证明该算子的紧致性。这允许获得识别问题的正则化解的唯一性，即正则化成本函数的最小值 J α ⁢ ( F ) := J ⁢ ( F ) + 1 2 ⁢ α ⁢ ∥ F ∥ L 2 ⁢ ( 0 , l ) 2 {J_{\alpha}(F)：=J(F)+\frac{1}{2}\alpha\|F\|_{L^{2}(0,l)}^{2}} ，其中 J ⁢ ( F ) = 1 2 ⁢ ∥ ux ⁢ ( 0 , ⋅ ; F ) - g ∥ L 2 ⁢ ( 0 , T ) 2 {J(F)=\frac{1}{2}\|u_{x}(0,\cdot\,; F)-g\|_{L^{2}(0,T)}^{2}} . 然后使用伴随问题方法推导出成本函数 J ⁢ ( F ) {J(F)} 的 Fréchet 梯度的公式。在共轭梯度算法 (CGA) 中使用梯度公式可以构建一个快速算法来恢复未知源 F ⁢ ( x ) {F(x)} 。一组全面的基准数值示例，包含多达 10 个噪声级别的随机噪声数据，说明了所提出方法的实用性和有效性。然后使用伴随问题方法推导出成本函数 J ⁢ ( F ) {J(F)} 的 Fréchet 梯度的公式。在共轭梯度算法 (CGA) 中使用梯度公式可以构建一个快速算法来恢复未知源 F ⁢ ( x ) {F(x)} 。一组全面的基准数值示例，包含多达 10 个噪声级随机噪声数据，说明了所提出方法的实用性和有效性。然后使用伴随问题方法推导出成本函数 J ⁢ ( F ) {J(F)} 的 Fréchet 梯度的公式。在共轭梯度算法 (CGA) 中使用梯度公式可以构建一个快速算法来恢复未知源 F ⁢ ( x ) {F(x)} 。一组全面的基准数值示例，包含多达 10 个噪声级别的随机噪声数据，说明了所提出方法的实用性和有效性。