基于单一偏振附近局部数据的各向异性刚度张量重构
Reconstruction of anisotropic stiffness tensors from partial data around one polarization
AI总结：
本文利用代数几何工具研究各向异性弹性反问题，证明了在二维及最各向异性的三维情形下，单一偏振附近的局部数据可唯一重构刚度张量，并指出弹性反问题会随不对称性增加而更易处理。
摘要：
我们利用代数几何工具研究各向异性弹性中的反问题。在 $n$ 维空间中，具有各向异性刚度张量的弹性波方程解的奇异性具有传播动力学，这种动力学由所谓的慢度面捕捉。慢度面是 $\mathbb{R}^n$ 余切丛中的超曲面，最终证明它们是代数簇。利用慢度面族的代数几何性质，我们证明了：对于二维各向异性刚度张量空间中稠密开子集内的张量，单一慢度面上某一偏振附近的少量数据能唯一确定整个慢度面及其刚度张量。在三维情形下，对于一般的正交各向异性和单斜刚度张量，少数异常伴随张量会产生相同的慢度面；然而，我们推测在最各向异性的设定（三斜）下，张量是唯一的，正如二维情况一样。确定张量所需的局部数据自然源自地震学测量或地震反问题的几何化版本。此外，我们解释了如何使用 Gröbner 基有效地执行刚度张量的重构。对于对称材料（例如完全各向同性），我们的唯一性或有限性结果失效，这印证了弹性反问题会随着不对称性的增加而变得更易处理这一反直觉的主张。
英文摘要：
We study inverse problems in anisotropic elasticity using tools from algebraic geometry. The singularities of solutions to the elastic wave equation in dimension $n$ with an anisotropic stiffness tensor have propagation kinematics captured by so-called slowness surfaces, which are hypersurfaces in the cotangent bundle of $\mathbb{R}^n$ that turn out to be algebraic varieties. Leveraging the algebraic geometry of families of slowness surfaces we show that, for tensors in a dense open subset in a space of anisotropic two-dimensional stiffness tensors, a small amount of data around one polarization in an individual slowness surface uniquely determines the entire slowness surface and its stiffness tensor. In three dimensions, for generic orthorhombic and monoclinic stiffness tensors, a small number of anomalous companions give rise to the same slowness surface; nevertheless, we conjecture that in the most anisotropic setting (triclinic) the tensor is unique, as in two dimensions. The partial data needed to determine a tensor arises naturally from seismological measurements or geometrized versions of seismic inverse problems. Additionally, we explain how the reconstruction of the stiffness tensor can be carried out effectively, using Gröbner bases. Our uniqueness or finiteness results fail for symmetric materials (e.g., fully isotropic), evidencing the counterintuitive claim that inverse problems in elasticity can become more tractable with increasing asymmetry.
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