Modeling and Computation in Elastography

弹性成像是一种新兴的成像手段，它寻求从弹性/粘弹性介质对外力的响应来非破坏性地确定其力学性质。它在医学诊断和无损检测中有着广泛的应用。在这种情况下，物理实验的相关性和高昂的成本推动了对新的和有效的数值算法的需求，以加速相应的设计和识别过程。然而，到目前为止，现有的模型和数值解决方案已被证明不能满足这一目的。这项研究项目启动了一个综合方案，通过开发基于严格数学分析的广泛适用的模型的快速方法来补救这种情况。这项工作的结果应该会对其中的一些情况产生直接影响，因为PI可以从领先的从业者那里获得真实数据和实验。改进的重建方案应该导致更准确的识别；它们的快速实施将允许在改进的成像系统的设计中真正地指导。本项目的目标是研究基本的数学问题，并开发有效的计算方法来解决弹性成像中的正问题和反问题。对于正演模拟，应针对固体和固体/流体相互作用问题开发快速而准确的基于积分式的求解器，目的是为弹性成像中的底层物理模型提供完全高效和可靠的模拟基础设施。所提出的方法将解决与设计超奇异积分的高阶求积相关的一些重大挑战，加速评估长程势，以及开发合适的预条件。为了解决反问题，应发展结合多频延拓和长波长快速线性逼近的新方法，以解决与反问题的非线性和不适定特性有关的经典困难。这反过来又使真正的虚拟设计成为可能，从而促进了PDE约束优化领域的进步。

Elastography is an emerging imaging modality that seeks to non-destructively determine the mechanical properties of elastic/viscoleastic media from their response to external forces. It has a wide range of applications in medical diagnostics and non-destructive testing. The relevance and high cost of physical experimentation in such instances has driven the need for novel and efficient numerical algorithms to accelerate the corresponding design and identification process. To date, however, existing models and numerical schemes for their resolution have proven inadequate for this purpose. This research project initiates an integrated program for the remediation of this situation through the development of fast methods of broadly applicable models that are based on rigorous mathematical analysis. The results of the work should have a direct impact in a number of these instances, enhanced by the PI's access to real data and experiments from leading practitioners. Improved reconstruction schemes should lead to more accurate identification; their fast implementation, in turn, will allow for true guidance in the design of improved imaging systems.The objective of this project is to examine fundamental mathematical issues and develop efficient computational methods for solving the direct and inverse problems in elastography. For the forward modeling, fast and accurate integral equation based solvers shall be developed for solid and solid/fluid interaction problems, aimed at attaining fully efficient and reliable simulation infrastructures for the underlying physical models in elastography. The approach proposed will address a number of significant challenges associated with the design of high-order quadratures for hypersingular integrals, with the accelerated evaluation of long-range potentials, and with the development of suitable preconditioners. To tackle the inverse problems, novel procedures with a combination of the multi-frequency continuation with fast linear approximations at long wavelengths shall be developed to address classical difficulties related to the nonlinear and ill-posed character of the inverse problems. This, in turn, should enable the possibility of true virtual design leading to advances in the area of PDE-constrained optimization.