Inverse problems for linear hyperbolic equations using mixed formulations
Inverse problems for linear hyperbolic equations using mixed formulations
复制标题
使用混合公式的线性双曲方程的反问题
DOI：
10.1088/0266-5611/31/7/075001
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发表时间：
2015
期刊：
影响因子：
2.1
通讯作者：
A. Münch
中科院分区：
文献类型：
--
作者：
N. Cîndea;A. Münch
我们介绍了一种直接的方法，它允许解决线性双曲型方程的数值反问题。我们首先考虑在×（0，T）？>中提出的方程的完全解的重构问题。-Ω是R N的有界子集？> - 从局部分布的观察。我们采用最小二乘技术，并尽量减少从观察到任何解决方案的距离的L2范数。以双曲型方程作为问题的主要约束条件，最优性条件被简化为一个包含重构状态和一个拉格朗日乘子的混合形式。在通常的几何光学条件下，我们证明了这种混合制剂的适定性（特别是inf-sup条件），然后引入一个基于时空有限元离散的数值近似。我们证明了逼近的强收敛性，然后讨论了N = 1和N = 2的几个例子。本文还讨论了状态项和源项的重构问题。
We introduce a direct method which allows the solving of numerically inverse problems for linear hyperbolic equations. We first consider the reconstruction of the full solution of the equation posed in &OHgr; × ( 0 , T ) ?> —Ω being a bounded subset of R N ?> —from a partial distributed observation. We employ a least-squares technique and minimize the L2-norm of the distance from the observation to any solution. Taking the hyperbolic equation as the main constraint of the problem, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. Under usual geometric optic conditions, we show the well-posedness of this mixed formulation (in particular the inf–sup condition) and then introduce a numerical approximation based on space-time finite element discretization. We prove the strong convergence of the approximation and then discuss several examples for N = 1 and N = 2. The problem of the reconstruction of both the state and the source terms is also addressed.