Volume 15, Issue 1

We review the asymptotic theory for standard errors in classical ordinary least squares (OLS) inverse or parameter estimation problems involving general nonlinear dynamical systems where sensitivity matrices can be used to compute the asymptotic covariance matrices. We discuss possible pitfalls in computing standard errors in regions of low parameter sensitivity and/or near a steady state solution of the underlying dynamical system.

This paper deals with the construction of orthogonal polynomial bases for particular subspaces of vector fields defined in the unit ball of ℝ 3 . Our approach uses vector spherical harmonics to construct orthogonal sets of specific solenoidal and potential vector fields by means of ridge functions. It is shown that the approach leads to bases according to the subspaces induced by the Helmholtz–Hodge decomposition of square integrable vector fields.

In this paper we study a time harmonic inverse acoustic scattering problem involving an obstacle that when hit by an incoming acoustic wave tries to appear in a location in space different from its actual location eventually with a shape and an acoustic boundary impedance different from its actual ones. We refer to this problem as inverse ghost obstacle problem and to the scatterers of this type as a special class of smart obstacles. The term "ghost" comes from the fact that the smart obstacle, when hit by the incoming wave, tries to appear as a ghost obstacle, that is as a virtual obstacle in a location in space where no real obstacle is present. We assume that the intersection between the obstacle and the ghost obstacle is empty. The obstacle tries to produce this virtual image of itself circulating on its boundary a suitable pressure current. A pressure current is a quantity whose physical dimension is pressure divided by time. The following time harmonic inverse ghost obstacle scattering problem is considered: from the knowledge of several far fields generated by the smart obstacle when hit by known time harmonic waves, of its acoustic boundary impedance and of the acoustic boundary impedance of the ghost obstacle find the location of the smart obstacle, that is find a point in the interior of the smart obstacle and eventually find the shape of the smart obstacle. The method proposed to solve this problem is a generalization of the well known Herglotz function method. We present some numerical experiments based on synthetic data that are used to test the numerical method introduced here. Some material that helps the understanding of this paper and some animations relative to the numerical experiments can be found in the website: http://www.econ.univpm.it/recchioni/w15.

The paper is devoted to the analysis of linear ill-posed operator equations Ax = y with solution x 0 in a Hilbert space setting. In an introductory part, we recall assertions on convergence rates based on general source conditions for wide classes of linear regularization methods. The source conditions are formulated by using index functions. Error estimates for the regularization methods are developed by exploiting the concept of Mathé and Pereverzev that assumes the qualification of such a method to be an index function. In the main part of the paper we show that convergence rates can also be obtained based on distance functions d ( R ) depending on radius R > 0 and expressing for x 0 the violation of a benchmark source condition. This paper is focused on the moderate source condition x 0 = A ∗ v . The case of distance functions with power type decay rates d ( R ) = ( R –η/(1–η) ) as R → ∞ for exponents 0 < η < 1 is especially discussed. For the integration operator in L 2 (0, 1) aimed at finding the antiderivative of a square-integrable function the distance function can be verified in a rather explicit way by using the Lagrange multiplier method and by solving the occurring Fredholm integral equations of the second kind. The developed theory is illustrated by an example, where the optimal decay order of d ( R ) → 0 for some specific solution x 0 can be derived directly from explicit solutions of associated integral equations.

In this paper we apply the new regularization method a daptive grid regularization , which is well-suited for ill-posed problems with discontinuous solutions and which was introduced in an earlier paper [5], to 2D linear integral equations of the first kind. After describing the method we present some numerical examples showing that this method is a powerful tool to identify discontinuities in ill-posed problems.