Marc D. Ryser

Marc D. Ryser, Nilima Nigam, Paul F. Tupper

White noise-driven nonlinear stochastic partial differential equations (SPDEs) of parabolic type are frequently used to model physical and biological systems in space dimensions d = 1,2,3. Whereas existence and uniqueness of weak solutions to these equations are well established in one dimension, the situation is different for d \geq 2. Despite their popularity in the applied sciences, higher dimensional versions of these SPDE models are generally assumed to be ill-posed by the mathematics community. We study this discrepancy on the specific example of the two dimensional Allen-Cahn equation driven by additive white noise. Since it is unclear how to define the notion of a weak solution to this equation, we regularize the noise and introduce a family of approximations. Based on heuristic arguments and numerical experiments, we conjecture that these approximations exhibit divergent behavior in the continuum limit. The results strongly suggest that a series of published numerical studies are problematic: shrinking the mesh size in these simulations does not lead to the recovery of a physically meaningful limit.

Xuan Liu, Yinhao Ren, Marc D. Ryser et al.

Accurate lesion tracking in temporal mammograms is essential for monitoring breast cancer progression and facilitating early diagnosis. However, automated lesion correspondence across exams remains a challenges in computer-aided diagnosis (CAD) systems, limiting their effectiveness. We propose MammoTracker, a mask-guided lesion tracking framework that automates lesion localization across consecutively exams. Our approach follows a coarse-to-fine strategy incorporating three key modules: global search, local search, and score refinement. To support large-scale training and evaluation, we introduce a new dataset with curated prior-exam annotations for 730 mass and calcification cases from the public EMBED mammogram dataset, yielding over 20000 lesion pairs, making it the largest known resource for temporal lesion tracking in mammograms. Experimental results demonstrate that MammoTracker achieves 0.455 average overlap and 0.509 accuracy, surpassing baseline models by 8%, highlighting its potential to enhance CAD-based lesion progression analysis. Our dataset will be available at https://gitlab.oit.duke.edu/railabs/LoGroup/mammotracker.