research

2024

1. Well-posedness of some Congested Models of Chemotaxis and their Singular Limits Toward Free Boundary Problems

   Michael Rozowski

   2024

   In Preparation
2. Hele-Shaw Flow with Surface Tension and Kinetic Undercooling as a Sharp Interface Limit of the Fully Parabolic Keller-Segel System with Nonlinear Diffusion

   Michael Rozowski

   2024

   In Preparation
3. Volume-Preserving Mean-Curvature Flow as a Singular Limit of a Diffusion-Aggregation Equation

   Antoine Mellet, and Michael Rozowski

   2024

   Abs arXiv Submitted

   The Patlak-Keller-Segel system of equations (PKS) is a classical example of aggregation-diffusion equation in which the repulsive effect of diffusion is in competition with the attractive chemotaxis term. Recent work on the Parabolic-Elliptic PKS model have shown that when the repulsion is modeled by a nonlinear diffusion term \(\rho \nabla \rho^{m-1}\) with \(m>2\), this competition leads to phase separation phenomena. Furthermore, in some asymptotic regime corresponding to a large population observed over a long enough time, the interface separating regions of high and low density evolves according to the Hele-Shaw free boundary problem with surface tension. In the present paper, we consider the counterpart of that model, namely the Elliptic-Parabolic PKS model and we prove that the same phase separation phenomena occurs, but the motion of the interface is now described (asymptotically) by a volume-preserving mean-curvature flow.

2022

1. Input layer regularization for magnetic resonance relaxometry biexponential parameter estimation

   Michael Rozowski, Jonathan Palumbo, Jay Bisen, and 4 more authors

   Magnetic Resonance in Chemistry, 2022

   Abs Published

   Many methods have been developed for estimating the parameters of biexponential decay signals, which arise throughout magnetic resonance relaxometry (MRR) and the physical sciences. This is an intrinsically ill-posed problem so that estimates can depend strongly on noise and underlying parameter values. Regularization has proven to be a remarkably efficient procedure for providing more reliable solutions to ill-posed problems, while, more recently, neural networks have been used for parameter estimation. We re-address the problem of parameter estimation in biexponential models by introducing a novel form of neural network regularization which we call input layer regularization (ILR). Here, inputs to the neural network are composed of a biexponential decay signal augmented by signals constructed from parameters obtained from a regularized nonlinear least-squares estimate of the two decay time constants. We find that ILR results in a reduction in the error of time constant estimates on the order of 15%–50% or more, depending on the metric used and signal-to-noise level, with greater improvement seen for the time constant of the more rapidly decaying component. ILR is compatible with existing regularization techniques and should be applicable to a wide range of parameter estimation problems.