In this paper, we study three interconnected inverse problems in shift invariant spaces: 1) the convolution/deconvolution problem; 2) the uniformly sampled convolution and the reconstruction problem; 3) the sampled convolution followed by sampling on irregular grid and the reconstruction problem. In all three cases, we study both the stable reconstruction as well as ill-posed reconstruction pro...

Consider the inverse diffraction problem to determine a two-dimensional periodic structure from scattered elastic waves measured above the structure. We formulate the inverse problem as a least squares optimization problem, following the two-step algorithm by G. Bruckner and J. Elschner (2003 Inverse Problems 19 315-329) for electromagnetic diffraction gratings. Such a method is based on the Ki...

Ill posed quadratic optimization frequently occurs in control and inverse problems and are not covered by the Lax-Milgram-Riesz theory. Typically small changes in the input data can produce very large oscillations on the output. We investigate the conditions under which the minimum value of the cost function is finite and we explore the ‘hidden connection’ between the optimization problem and t...

In this paper, the Trefftz collocation method is applied to solve the inverse Cauchy problem of anisotropic elasticity, wherein both tractions as well as displacements are prescribed at a small part of the boundary of an arbitrary simply/multiply connected anisotropic elastic domain. The Stroh formalism is used to construct the Trefftz basis functions. Negative and positive power series are use...

Inversion of MT data is an inherently nonunique and unstable problem due to the ill-posedness of the electromagnetic inverse problem. A variety of models may fit the data very well. To overcome this illposed nature of the inverse problem, we use Tikhonov’s regularization in which the ill-posed problem is replaced by a family of well-posed problems. We also analyze the behavior of the Tikhonov r...

We present new numerical methods for the solution of inverse spectral problem to determine the dielectric constants of core and cladding in optical fibers. These methods use measurements of propagation constants. Our algorithms are based on approximate solution of a nonlinear nonselfadjoint eigenvalue problem for a system of weakly singular integral equations.We study three inverse problems and...