Volume 13, Issue 1

We compare parametric and nonparametric estimation methods in the context of PBPK modeling using simulation studies. We implement a Monte Carlo Markov Chain simulation technique in the parametric method, and a functional analytical approach to estimate the probability distribution function directly in the non-parametric method. The simulation results suggest an advantage for the parametric method when the underlying model can capture the true population distribution. On the other hand, our calculations demonstrate some advantages for a nonparametric approach since it is a more cautious (and hence safer) way to assess the distribution when one does not have sufficient knowledge to assume a population distribution form or parametrization. The parametric approach has obvious advantages when one has significant a priori information on the distributions sought, although when used in the nonparametric method, prior information can also significantly facilitate estimation.

We describe a new method for solving an inverse Dirichlet problem for harmonic functions that arises in the mathematical modelling of electrostatic and thermal imaging methods. This method may be interpreted as a hybrid of a decomposition method, in the spirit of a method developed by Kirsch and Kress, and a regularized Newton method for solving a nonlinear ill-posed operator equation, in terms of the solution operator that maps the unknown boundary onto the solution of the direct problem. As opposed to the Newton iterations the new method does not require a forward solver. Its feasibility is demonstrated through numerical examples.

We investigate the applicability of the method of maximum entropy regularization (MER) to a specific nonlinear ill-posed inverse problem (SIP) in a purely time-dependent model of option pricing, introduced and analyzed for an L 2 -setting in [9]. In order to include the identification of volatility functions with a weak pole, we extend the results of [12, 13], concerning convergence and convergence rates of regularized solutions in L 1 , in some details. Numerical case studies illustrate the chances and limitations of (MER) versus Tikhonov regularization (TR) for smooth solutions and solutions with a sharp peak. A particular paragraph is devoted to the singular case of at-the-money options, where derivatives of the forward operator degenerate.

By means of the Laplace transform method sufficient conditions for the existence of exponentially decaying memory kernels in heat flow and viscoelasticity are derived solving corresponding inverse problems. The observation functionals of the inverse problems are built up by n eigenfunctions of the related elliptic equation or the data of the direct problems possess n non-vanishing Fourier coefficients, only. In the special cases n = 1 and n = 2 the Laplace transforms of the memory kernel are given in explicit form.

We give a new derivation of the inversion formula of Bukhgeim and Kazantsev for the attenuated vectorial Radon transform. We also study what happens if the attenuation tends to zero.