Modeling and Computation in Elastography

弹性成像是一种新兴的成像方式，其寻求从弹性/粘弹性介质对外力的响应中非破坏性地确定其机械特性。它在医疗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成本已经驱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，现有的模型和数值方案，其决议已被证明不足以达到这一目的。该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模型的快速方法来补救这种情况。研究结果应在许多此类情况下产生直接影响，并通过PI获得来自主要从业者的真实的数据和实验而得到加强。改进的重建方案应导致更准确的识别，他们的快速实施，反过来，将允许真正的指导，在设计改进的成像systems.The项目的目标是研究基本的数学问题，并制定有效的计算方法，解决弹性成像的正问题和逆问题。对于正演建模，应针对固体和固体/流体相互作用问题开发快速准确的基于积分方程的求解器，旨在为弹性成像中的底层物理模型实现充分有效和可靠的模拟基础设施。所提出的方法将解决一些重大的挑战与设计的高阶求积超奇异积分，与加速评估的长程潜力，并与适当的预条件的发展。为了解决反问题，新的程序相结合的多频连续与快速线性近似在长波长应开发解决经典的困难与反问题的非线性和不适定性。反过来，这应该使真正的虚拟设计的可能性，导致在该地区的偏微分方程约束优化的进步。

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.