Volume 23, Issue 2

In a free boundary domain we consider inverse problems for determination a time-dependent coefficient at the first derivative of an unknown function in a generally degenerate parabolic equation with different overdetermination conditions. There are established conditions of existence and uniqueness of the classical solutions to the named problems in the case of the weak degeneration.

We investigate a two-dimensional radially symmetric inverse heat conduction problem, which is ill-posed in the sense that the solution does not depend continuously on input data. By generalizing the idea of kernel approximation, we devise a modified kernel in the frequency domain to reconstruct a numerical solution for the inverse heat conduction problem from the given noisy data. For the stability of the numerical approximation, we develop seven regularization techniques with some stability and convergence error estimates to reconstruct the unknown solution. Numerical experiments illustrate that the proposed numerical algorithm with regularization techniques provides a feasible and effective approximation to the solution of the inverse and ill-posed problem.

In this article we assess variability in cell proliferation dynamics observed for CD4+ and CD8+ T cells collected from two healthy donors. We review a recently developed class of models that incorporates the so-called “cyton model” for cell numbers into a conservation-based PDE model for cell population dynamics and describe a statistical model that relates CFSE-based flow cytometry data to such models. A parameter estimation scheme is summarized and then applied to a large body of data to assess experimental variability (variation in parameter estimates as identical experiments are replicated) and biological variability (differences in parameter estimates obtained for different donors and cell types) in the context of these models. Variability in the data obtained from replicated experiments is also discussed. The results of this study indicate that many of the cyton model parameters for describing cell proliferation can be reliably estimated using our approach; however, they also show that substantial changes to our mathematical model and/or experimental procedures may be required to ensure identifiability of the remaining cell proliferation parameters.

The X -ray computerized tomography is concerned to the reconstruction of a function f from line (ray) integrals of f . In general, we may have the parallel ray or divergent ray geometry. In this paper we develop an analytical singular value decomposition method for the cone-beam transform in divergent ray geometry. For the parallel ray transform a singular value decomposition has been derived by P. Maass in 1987.

Imaging of nano structures is necessary for the quality control in their manufacturing. In the case when X-rays probe the medium, only the modulus of the complex valued scattered wave field can be measured. The phase cannot be measured. In the case of the Born approximation, we obtain an explicit reconstruction formula for the unknown coefficient from the phaseless scattering data.