Free Boundary Problems and Level-Set Methods

三个主要的研究方向是：（i）自动正则化参数的选择;（ii）低秩和稀疏张量分解的黎曼优化;（iii）心脏中的接触问题。数学图像处理中正则化参数自动选择规则的研究主要从两个方面展开：（1）将图像恢复问题包含在约束集中的双层优化问题进行了理论分析和数值求解。上层目标编码统计相关的信息，它涉及的正则化参数和未知的图像作为变量。这样的问题是臭名昭著的退化，需要先进的工具，从集值分析得出尖锐的平稳性条件，和复杂的求解器。(2)更一般的准平衡配方被认为是一个上层目标的制定是不可能的。然后通过参数更新步骤来增强恢复问题，该步骤涉及应用于统计量（诸如局部图像残差）的置信度技术。在与INVERSE 2.0和IMAGE的联合研究中，局部正则化将应用于扩散张量成像。高阶低秩和稀疏张量分解的黎曼优化研究需要适当的秩概念和多线性秩张量流形的描述。后者是一个重要的积木在交替最小化算法求解变分问题建模的低秩/稀疏分解没有任何凸松弛。为了对非刚性运动进行建模，模型包含一个参数，该参数可能与光流问题中的逐像素位移场有关。张量分解将导致SFB内的几个合作：（i）使用INVERSE 2.0，计划在数值和分析上与PCA/ICA进行比较;（ii）使用IMAGE，将考虑重建或运动校正任务的时空正则化;（iii）利用MRI，将研究形态学和时间特征的分离以及来自生物标志物的信息的分解。为了准确地捕捉房室平面位移，该项目将开发一个解决方案，用于解决涉及外膜和周围膜的接触问题，这对HEART的机电模型很重要。该研究涉及引入顺应性无摩擦接触模型，其中顺应性来自心外膜上的位移依赖性边界条件（通过心包隐含），而摩擦的缺乏是由于心包液。由此产生的准平衡问题是离散的Galerkin有限元几乎不可压缩的材料。我们的算法解决方案的过程将是基于一个顺序近似方案，每个子问题类似于一个Signorini型接触问题的刚性障碍。

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.