Volume 21, Issue 1

Abstract. We consider seismic data regularization problems in this paper. Gaussian beam decomposition model is proposed for seismic data representation. To solve the representation problem, an l 0 quasi-norm minimization model with different smooth approximations is proposed. To solve the l 0 quasi-norm minimization problem, a projected gradient method with nonmonotone choice of iterative steps is developed. Numerical simulations on one-dimensional and two-dimensional seismic imaging problems are performed to verify the feasibility of our methods.

Abstract. Non-invasive detection, localization and characterization of an arterial stenosis (a blockage or partial blockage in the artery) continues to be an important problem in medicine. Partial blockage stenoses are known to generate disturbances in blood flow which generate shear waves in the chest cavity. We examine a one-dimensional viscoelastic model that incorporates Kelvin–Voigt damping and internal variables, and develop a proof-of-concept methodology using simulated data. We first develop an estimation procedure for the material parameters. We use this procedure to determine confidence intervals for the estimated parameters, which indicates the efficacy of finding parameter estimates in practice. Confidence intervals are computed using asymptotic error theory as well as bootstrapping. We then develop a model comparison test to be used in determining if a particular data set came from a low input amplitude or a high input amplitude; this we anticipate will aid in determining when stenosis is present. These two thrusts together will serve as the methodological basis for our continuing analysis using experimental data currently being collected.

Abstract. We investigate the idea of solving calibration problems for stochastic dynamical systems using statistical tests. We consider a specific stochastic dynamical system: the normal SABR model. The SABR model has been introduced in mathematical finance in 2002 by Hagan, Kumar, Lesniewski and Woodward. The model is a system of two stochastic differential equations whose independent variable is time and whose dependent variables are the forward prices/rates and the associated stochastic volatility. The normal SABR model is a special case of the SABR model. The calibration problem for the normal SABR model is an inverse problem that consists in determining the values of the parameters of the model from a set of data. We consider as set of data two different sets of forward prices/rates and we study the resulting calibration problems. Ad hoc statistical tests are developed to solve these calibration problems. Estimates with statistical significance of the parameters of the model are obtained. Let be a constant, we consider multiple independent trajectories of the normal SABR model associated to given initial conditions assigned at time . In the first calibration problem studied the set of the forward prices/rates observed at time in this set of trajectories is used as data sample of a statistical test. The statistical test used to solve this calibration problem is based on some new formulae for the moments of the forward prices/rates variable of the normal SABR model. The second calibration problem studied uses as data sample the observations of the forward prices/rates made on a discrete set of given time values along a single trajectory of the normal SABR model. The statistical test used to solve this second calibration problem is based on the numerical evaluation of some high-dimensional integrals. The results obtained in the study of the normal SABR model are easily extended from mathematical finance to other contexts in science and engineering where stochastic models involving stochastic volatility or stochastic state space models are used.

Abstract. For a semilinear elliptic equation, we prove uniqueness results in determining potentials and semilinear terms from partial Cauchy data on an arbitrary subboundary.

Abstract. A problem of iterative approximation is investigated for a nonlinear operator equation regularized by the Tikhonov method. The Levenberg–Marquardt method, its modified analogue, and the steepest descent method are used. For the first and second methods the regularizing properties of iterations are established and the error of approximate solution is given. For the third method it was proved that iterations are stabilized in a neighborhood of the required solution and satisfy the strong Fejér property.

Abstract. This paper is concerned with a 3D linear integral equation of the first kind which arises when solving nonlinear coefficient inverse problems for second order hyperbolic PDEs. We prove that the integral equation has at most one solution when its right part is given on appropriate analytic 2D and 1D manifolds. As compared to known uniqueness theorems, the dimension of a smooth manifold supporting problem data is decreased here from four to one. As auxiliary results, we give examples of 1D analytic manifolds of uniqueness and examples of minimal countable sets of uniqueness for harmonic functions in 3D domains.

Abstract. The two inverse problems of determining an unknown parameter in a nonhomogeneous part of the equation for an abstract second-order elliptic equation in a Banach space with boundary conditions of Bitsadze–Samarski type are considered. For the first problem we use the conditions of Dirichlet, and for the second problem we use the conditions of Neumann. Theorems of existence and uniqueness of solutions for both direct and inverse problems are proved. Explicit formulas for the solutions are obtained.

Abstract. We investigate the global asymptotical stability of dynamical systems with cylindrical phase space, which appear in the synchronization problem of various control systems. New effective criteria of the global asymptotic stability of a countable equilibrium position are proposed. The technique of improper integrals evaluation along solutions is used.