Volume 17, Issue 2

We present a mathematical and statistical computational framework for inverse problems involving delay or hysteretic differential equations. We demonstrate efficacy of the methodology in the context of models for insect maturation and mortality due to insecticide exposure.

The forward scattering problem is to find the solution u = u f ( x, t ) of the dynamical system where l ≥ 0 is an integer, f ∈ L 2 (0, ∞) is a control, q is a smooth perturbation of the potential obeying sup supp q < ∞. With the problem one associates a response operator R acting by the rule where the kernel r is a smooth function obeying sup supp r < ∞. By hyperbolicity of the wave equation, the kernel depends locally on the potential: for any ξ > 0, its part r | [2ξ,∞) is determined by q | [ξ,∞) . Our setup of the inverse problem takes into account such a locality: let ξ > 0 be fixed; given r | [2ξ,∞) to determine q | [ξ,∞) . The paper proposes a procedure determining q and provides the characterization of inverse data , i.e., the necessary and sufficient conditions for a function r to be the response kernel of a dynamical system of the above-mentioned type. The boundary control method is in use and the character of controllability of the system plays a key role. It depends on l : for l = 0 the system is locally controllable, whereas for l ≥ 1 a certain lack of controllability occurs. To recover the controllability we extend the space of controls by adding a finite dimensional subspace of polynomials. Thereafter, the standard devices of the BC-method ( M -transform, visualization of waves, etc) are applied for determination of the potential.

We consider an axisymmetric inverse heat conduction problem of determining the surface temperature from a fixed location inside a cylinder. The problem is ill-posed: the solution (if it exists) does not depend continuously on the data. A modified Tikhonov regularization method and a Fourier regularization method are applied to formulate regularized solutions, which are stably convergent to the exact ones with order optimal error estimates.

A new second order Newton method for reconstructing the shape of a sound soft scatterer from the measured far-field pattern for scattering of time harmonic plane waves is presented. This method extends a hybrid between regularized Newton iterations and decomposition methods that has been suggested and analyzed in a number of papers by Kress and Serranho [Inverse Problems 19: 91–104, 2003, Inverse Problems 21: 773–784, 2005, J. Comput. Appl. Math. 204: 418–427, 2007, Inverse Problems 22: 663–680, 2006, Inverse Problems and Imaging 1: 691–712, 2007] and has some features in common with the second degree method for ill-posed nonlinear problems as considered by Hettlich and Rundell [SIAM J.Numer. Anal. 37: 587–620, 2000]. The main idea of our iterative method is to use Huygen's principle, i.e., represent the scattered field as a single-layer potential. Given an approximation for the boundary of the scatterer, this leads to an ill-posed integral equation of the first kind that is solved via Tikhonov regularization. Then, in a second order Taylor expansion, the sound soft boundary condition is employed to update the boundary approximation. In an iterative procedure, these two steps are alternated until some stopping criterium is satisfied. We describe the method in detail and illustrate its feasibility through examples with exact and noisy data.

We address the efficiency issue for the globally convergent convexification algorithm for coefficient inverse problems. By properly choosing the upper limit for pseudo-frequency and with quadratic polynomial approximations of the quantities which depend on the pseudo-frequency, we show that the algorithm can be made dramatically faster relative to the previous “tail-free” implementation. Numerical results from imaging and the mathematical modeling of antipersonnel land mines demonstrate that the algorithm can detect the location(s) of the inclusion(s) from the background medium, as well as correctly identify the material property of the inclusion(s) and the background. This indicates that the convexification algorithm may be applied in real-time to detect and image mine-like targets in the field.