基于全局严格凸代价泛函的炸药介电常数恢复
Recovering of dielectric constants of explosives via a globally strictly convex cost functional
摘要：
本文讨论了用散射电场的一个分量的边界测量值估算炸药介电常数的反问题。它被公式化为一个双曲型微分方程的系数反问题。在应用拉普拉斯变换后，构造了一个新的成本泛函，并制定了变分问题。该泛函的关键特征是拉普拉斯算子的Carleman权函数的存在。证明了该泛函在任意大小的Hilbert空间的有界集上的严格凸性。这允许建立梯度下降法的全局收敛性。给出了数值试验的一些结果。
英文摘要：
The inverse problem of estimating dielectric constants of explosives using boundary measurements of one component of the scattered electric field is addressed. It is formulated as a coefficient inverse problem for a hyperbolic differential equation. After applying the Laplace transform, a new cost functional is constructed and a variational problem is formulated. The key feature of this functional is the presence of the Carleman Weight Function for the Laplacian. The strict convexity of this functional on a bounded set in a Hilbert space of an arbitrary size is proven. This allows for establishing the global convergence of the gradient descent method. Some results of numerical experiments are presented.
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