Distance in the space of elasticity tensors: optimal choices of Hookean models

在我的研究中，我努力提高地震现象和地球的物质属性的定量描述。我的工作属于地球科学背景下的应用数学领域。为了定量地描述构成地球的物质，我使用了由胡克固体概念提供的数学模型。这种假想固体的性质，即描述它的数学表达式的性质，可以作为实际材料物理性质的类比。正如我通过我的研究所证明的那样，胡克固体具有八种对称性。这在我的工作中至关重要，因为真实的材料的行为可以通过考虑具有特定对称性的胡克固体来近似。然而，由于真实的材料不是胡克固体，我们不能说给定的材料属于特定的对称类。我们只能考虑这种材料与特定类的接近程度。两个概念必须援引来检查这种接近，这是我目前的研究重点。首先，我们必须在胡克立体的抽象空间中引入“距离”的概念。第二，为了给现场数据提供一种实用的方法，我们必须在实验误差的背景下考虑距离;我们可能会问，数据分辨率是否允许我们区分两种不同对称性的胡克固体的距离。为了评估这些问题答案的可靠性，我将使用不确定性分析，这需要先进的分析方法和有效的计算技术。这项研究的实用重要性的动机是最近增加的准确性，在实验装置，激发了进一步的进步的定量配方。换句话说，在所获取的数据中存在比通过标准技术可以提取的更多的信息。在其直接应用中，我的研究对关注全球和勘探问题的地球科学家很重要。它提供了有关材料特性的信息及其准确性的估计，以增加基于这些信息的科学和工业决策的信心。

In my research, I strive to enhance the quantitative description of seismic phenomena and material properties of the Earth. My work belongs to the realm of applied mathematics in the context of geosciences. To describe quantitatively materials that constitute the Earth, I use a mathematical model provided by the concept of a Hookean solid. Properties of such a hypothetical solid, which are properties of mathematical expressions that describe it, serve as analogies for physical properties of actual materials. As I have demonstrated through my research, Hookean solids exhibit eight symmetry classes. This is of paramount importance in my work, since the behaviour of a real material can be approximated by considering a Hookean solid of a particular symmetry. However, since real materials are not Hookean solids, we cannot say that a given material belongs to a particular symmetry class. We can consider only the proximity of this material to a particular class. Two concepts must be invoked to examine such a proximity, which is the focus of my present research. First, we must invoke a concept of "distance" within the abstract space of Hookean solids. Second, to provide a pragmatic methodology for field data, we have to consider distance in the context of experimental errors; we might ask whether or not the data resolution allows us to distinguish between distances to Hookean solids of two distinct symmetries. To evaluate the reliability of answers to such questions, I will use uncertainty analysis, which requires both advanced analytical methods and efficient computational techniques. The pragmatic importance of this research is motivated by recent increase of accuracy in experimental apparatus that inspires further advancements of quantitative formulations. In other words, there is more information in acquired data than can be extracted by standard techniques. In its immediate application, my research is important to geoscientists concerned with both global and exploration issues. It provides information on material properties with estimates of their accuracy to increase confidence in both scientific and industrial decisions based on such information.