Free Boundary Problems and Level-Set Methods

主要研究方向有：(1)自动正则化参数选择；(2)低阶稀疏张量分解的黎曼优化；(3)心脏接触问题。数学图像处理中综合自动正则化参数选择规则的研究主要有两个方面：(1)对约束集中包含图像恢复问题的双层优化问题进行了描述、分析和数值求解。上层目标编码统计相关信息，将正则化参数和未知图像作为变量。这类问题是出了名的退化，需要集值分析的高级工具来推导出尖锐的平稳性条件，以及复杂的求解器。(2)当不可能制定一个较高级别的目标时，考虑更一般的准平衡公式。然后，通过参数更新步骤来增加恢复问题，该步骤涉及应用于统计量(例如局部图像残差)的置信度技术。在与逆2.0和图像的联合研究中，局部正则化将被应用于扩散张量成像。高阶、低阶和稀疏张量分解的黎曼优化研究需要一个合适的秩概念和多线性秩张量流形的描述。后者是交替最小化算法中的一个重要组成部分，该算法用来模拟无任何凸松弛的低阶/稀疏分解的变分问题。为了模拟非/刚性运动，该模型包含一个参数，该参数可能与光流问题中的像素方向位移场有关。张量分解将导致SFB内的几个合作：(I)对于逆2.0，计划在数值和解析上与PCA/ICA进行比较；(Ii)将考虑重建或运动校正任务的图像时空正则化；(Iii)将研究形态和时间特征的分离以及来自生物标记物的信息分解。最后，为了准确捕捉房室平面位移，该项目将开发一个解决涉及心外膜和心周膜接触问题的解算器，这对于心脏的机电模型非常重要。这项研究包括引入柔顺无摩擦接触模型，其中柔顺是由心外膜上的位移相关边界条件(通过心包隐含)产生的，而无摩擦是由于心包液体引起的。由此产生的准平衡问题被伽辽金有限元离散为几乎不可压缩的材料。我们的算法求解过程将基于序列近似方案，其中每个子问题类似于带有刚性障碍物的西诺里尼型接触问题。

Three major research directions are pursued: (i) Automated regularization parameter choices; (ii) Riemannian optimization for low-rank and sparse tensor decomposition; (iii) contact problems in the heart. The research on integrated automated regularization parameter choice rules in mathematical image processing pursues two approaches: (1) Bilevel optimization problems, which contain the image restoration problem in the constraint set, are formulated, analyzed and numerically solved. The upper level objective encodes statistically relevant information and it involves the regularization parameter and the unknown image as variables. Such problems are notoriously degenerate and require advanced tools from set-valued analysis for deriving sharp stationarity conditions, and sophisticated solvers. (2) More general quasi-equilibrium formulations are considered whenever the formulation of an upper level objective is not possible. Then the restoration problem is augmented by a parameter update step, which involves a confidence technique applied to a statistical quantity (such as local image residuals). In a joint research effort with INVERSE 2.0 and IMAGE local regularization will be applied to diffusion tensor imaging.The research on Riemannian optimization for higher-order low-rank and sparse tensor decomposition requires an appropriate rank concept and a description of the manifold of tensors of multilinear rank. The latter is an important building block in an alternating minimization algorithm for solving a variational problem modeling the low-rank/sparse decomposition without any convex relaxation. To model non/rigid motions, the model contains a parameter, which may be related to the pixel-wise displacement field as in optical flow problems. The tensor decomposition will lead to several cooperations within the SFB: (i) With INVERSE 2.0 it is planned to compare numerically and analytically with PCA/ICA; (ii) with IMAGE temporal-spatial regularization of reconstruction or motion correction tasks will be considered; (iii) with MRI the separation of morphological and temporal features and the decomposition of information from biomarkers will be investigated.Finally, to accurately capture the atrio-ventricular plane displacement, the project will develop a solver for contact problems involving the epi- and peri-cardium which is important for the electromechanical model of HEART. The research involves the introduction of a compliant frictionless contact model, where the compliance results from the displacement dependent boundary condition (implied through the pericardium) on the epicardium and the absence of friction is due to the liquor pericardial. The resulting quasi-equilibrium problem is discretized by a Galerkin finite elements for nearly incompressible materials. Our algorithmic solution procedure will be based on a sequential approximation scheme where each subproblem resembles a Signorini type contact problem with a rigid obstacle.