Volume 20, Issue 1

Abstract. Optimal design methods (designed to choose optimal sampling distributions through minimization of a specific cost function related to the resulting error in parameter estimates) for inverse or parameter estimation problems are considered. We compare a recent design criteria, SE-optimal design (standard error optimal design) with the more traditional D-optimal and E-optimal designs. The optimal sampling distributions from each design are used to compute and compare standard errors; here the standard errors for parameters are computed using the optimal mesh along with Monte Carlo simulations as compared to asymptotic theory based standard errors. We illustrate ideas with two examples: the Verhulst–Pearl logistic population model and the standard harmonic oscillator model.

Abstract. A nonlinear inverse problem of pharmacokinetics for the dual-compartment model with extravascular route of the drug administration is considered. The problem is significantly ill-posed: its solution is not unique and is unstable. We discuss nonuniqueness of the inverse problem and the degree of ill-posedness. The initial problem is reduced to a linear operator equation by linearization. It is demonstrated that the resolving ability of the inverse problem can be improved by varying of the location of measurement data points. Results of numerical experiments are presented. The question of choosing the initial approximations is considered.

Abstract. We prove existence and uniqueness results for mild solutions to direct and inverse Cauchy problems related to linear first-order partial differential equations in Banach spaces. Some applications are given to ultraparabolic differential and integrodifferential problems.

Abstract. We consider the inverse multidimensional problem for wave equation with unknown variable speed and absorption. A linear procedure based on the BC method for absorption reconstruction is proposed. The speed stays unknown.

Abstract. We study inverse spectral problems for arbitrary order ordinary differential equations on noncompact trees. As the main spectral characteristics we introduce and investigate the so-called Weyl matrices which are a generalization of the Weyl function for the classical Sturm–Liouville operator. We provide a procedure for constructing the solution of the inverse problem from the Weyl matrices and prove its uniqueness.