Reduced order models for imaging and inversion with waves and diffusive fields

该项目的重点是开发新的技术，用于计算反演和在医学成像、地球物理勘探、无损评估和测试以及其他应用中出现的成像问题，在这些应用中，必须在不直接接触物体内部的情况下确定物体的内部结构和特性。物体受到探测场的作用，并对其响应进行测量。电磁场和声波最常用于探测。传统上，根据测量确定对象属性的逆问题被表示为测量数据和正向模型的预测之间的失配的最小化。众所周知，这些问题很难解决，因为测量对模型属性的高度非线性依赖。提出了一种基于降阶模型(ROMs)理论的反演成像方法。这种方法旨在缓解逆问题的非线性，从而使其更容易求解。它允许人们获得没有传统方法通常难以去除的各种类型的伪像的图像。所提出的框架具有足够的通用性，既可用于波的反演和成像，也可用于扩散区的反演和成像。首先，将偏微分方程(PDE)算子在时间域或频域中的偏微分方程解快照子空间上的投影构造为只读存储器。这确保了只读存储器响应内插测量数据。即使在反演中PDE算子和解的快照都不能直接访问，但是可以使用线性代数工具从测量数据计算投影。在构建了只读存储器之后，它可以至少以两种方式使用。首先，它可以用于成像算法。由于只读存储器是PDE运算符的投影，因此可以从只读存储器的反投影来构造图像。模型降阶考虑了反射器之间的非线性相互作用，从而允许消除由多次反射引起的伪影。这是对传统成像方法的巨大改进，传统的成像方法通常基于线性化(Born，Kirchhoff)，而线性化错过或误解了非线性效果。其次，只读存储器可以用来重新表述传统的优化问题，以最大限度地减少只读存储器失配而不是数据失配。这样的优化目标应该是更凸的，这使得求逆不太容易陷入局部极小。另一个后果是加速了趋同。提出了用于反演和成像的ROMS的以下几个具体方面：(1)新的成像函数；(2)后向散射和非配置源/接收器数据测量设置；(3)非线性数据预处理；(4)传统优化方法的重构；(5)非迭代反演方法。

The project's focus is on development of novel techniques for computational inversion and imaging problems arising in medical imaging, geophysical exploration, nondestructive evaluation and testing and other applications where the internal structure and properties of objects must be determined without direct access to the object's interior. The object is subjected to probing fields and the measurements of its response are made. Electromagnetic fields and acoustic waves are most often used for probing. Conventionally an inverse problem of determining an object's properties from the measurements is formulated as a minimization of a misfit between the measured data and the prediction of a forward model. Such problems are notoriously difficult to solve due to a highly nonlinear dependence of measurements on model properties. We propose an approach to inversion and imaging based on the theory of reduced order models (ROMs). This approach aims to alleviate the nonlinearity of the inverse problem thus making it much easier to solve. It allows one to obtain images free from various types of artifacts that conventional methods often struggle to remove. The proposed framework is general enough and can be applied to inversion and imaging both with waves and in diffusive regimes. First, a ROM is constructed as a projection of the partial differential equation (PDE) operator on subspaces of PDE solution snapshots either in the time or the frequency domain. This ensures that the ROM response interpolates the measured data. Even though neither the PDE operator nor the solution snapshots are directly accessible in inversion, projections can be computed from the measured data using the tools of linear algebra. After the ROM is constructed it may be used in at least two ways. First, it can be used in an imaging algorithm. Since the ROM is a projection of the PDE operator, an image can be constructed from the back-projection of a ROM. Model reduction takes into account nonlinear interactions between the reflectors and thus allows one to eliminate artifacts caused by multiple reflections. This is a vast improvement over conventional imaging approaches that are often based on linearizations (Born, Kirchhoff) which miss or misinterpret the nonlinear effects. Second, the ROM can be used to reformulate conventional optimization problems to minimize ROM misfit instead of data misfit. Such optimization objective is expected to be more convex which makes inversion less prone to local minima. Another consequence is accelerated convergence. The following specific aspects of ROMs for inversion and imaging are proposed: (1) new imaging functionals;(2) backscattering and non-collocated source/receiver data measurement settings; (3) nonlinear data preprocessing; (4) reformulations of conventional optimization approaches; (5) non-iterative inversion methods.