Volume 23, Issue 6

We consider a boundary value problem for a Lamé type operator, which corresponds to a linearisation of the Navier–Stokes' equations for compressible flow of Newtonian fluids in the case where pressure is known. It consists of recovering a vector function, satisfying the parabolic Lamé type system in a cylindrical domain, via its values and the values of the boundary stress tensor on a given part of the lateral surface of the cylinder. We prove that the problem is ill-posed in the natural spaces of smooth functions and in the corresponding Hölder spaces; besides, additional initial data do not turn the problem to a well-posed one. Using the integral representation's method we obtain a uniqueness theorem and solvability conditions for the problem. We also describe conditions, providing dense solvability of the problem.

Several generalizations of the traditional Tikhonov–Phillips regularization method have been proposed during the last two decades. Many of these generalizations are based upon inducing stability throughout the use of different penalizers which allow the capturing of diverse properties of the exact solution (e.g. edges, discontinuities, borders, etc.). However, in some problems in which it is known that the regularity of the exact solution is heterogeneous and/or anisotropic, it is reasonable to think that a much better option could be the simultaneous use of two or more penalizers of different nature. Such is the case, for instance, in some image restoration problems in which preservation of edges, borders or discontinuities is an important matter. In this work we present some results on the simultaneous use of penalizers of L 2 and of bounded variation (BV) type. For particular cases, existence and uniqueness results are proved. Open problems are discussed and results to signal restoration problems are presented.

In the present paper, we consider the inverse problem of a Schrödinger equation defined on an open, bounded, connected set Ω of a complete, n -dimensional Riemannian manifold M with metric g . It is subject to a non-homogenous Dirichlet boundary term. We aim at determining the potential coefficient by means of a Neumann boundary measurement on a portion Γ 1 of the boundary Γ of Ω. Under sharp conditions on the complementary part Γ 0 = Γ∖Γ 1 , and under weak regularity requirements on the data, we establish two canonical results of the inverse problem: (i) global uniqueness and (ii) global Lipschitz stability. The lower bound inequality corresponding to the upper bound inequality contained in (ii) is also given. Our proofs rely on four main ingredients: (a) sharp Carleman estimate at the H 1 -level for Schrödinger equations on Riemannian manifolds [Control Methods in PDE-Dynamical Systems (Snowbird 2005), Contemp. Math. 426, American Mathematical Society, Providence (2007), 339–404], (b) related continuous observability inequality at the H 1 -level [Control Methods in PDE-Dynamical Systems (Snowbird 2005), Contemp. Math. 426, American Mathematical Society, Providence (2007), 339–404], (c) a continuous observability inequality at the L 2 -level [J. Inverse Ill-Posed Probl. 12 (2004), 43–123], [Functional Analysis and Evolution Equations, Birkhäuser, Basel (2008), 613–636], (d) optimal regularity theory for Schrödinger equations with Dirichlet boundary data ([Differential Integral Equations 5 (1992), 521–535], [Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Encyclopedia Math. Appl. 2, Cambridge University Press, Cambridge, 2000] as well as the new Theorem 3.6).

It is well known that Tikhonov regularization in standard form may determine approximate solutions that are too smooth, i.e., the approximate solution may lack many details that the desired exact solution might possess. Two different approaches, both referred to as fractional Tikhonov methods have been introduced to remedy this shortcoming. This paper investigates the convergence properties of these methods by reviewing results published previously by various authors. We show that both methods are order optimal when the regularization parameter is chosen according to the discrepancy principle. The theory developed suggests situations in which the fractional methods yield approximate solutions of higher quality than Tikhonov regularization in standard form. Computed examples that illustrate the behavior of the methods are presented.

We present formulas and equations for finding scattering data from the Dirichlet-to-Neumann map for a time-harmonic wave equation with first order perturbation with compactly supported coefficients. We assume that the coefficients are matrix-valued in general. To our knowledge, these results are new even for the general scalar case.

We study inverse spectral problems for ordinary differential equations with regular singularities on compact star-type graphs when differential equations have different orders on different edges. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are generalizations of the Weyl function for the classical Sturm-Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.

A simplified model of a laser beam leads to an ill-posed Cauchy problem for the Helmholtz equation on an infinite strip. In the case of large wave numbers, the problem corresponds to an operator equation with an unbounded operator. The first problem considered concerns optimality of a spectral type regularization method for reconstructing the radiation field from measurements given only on a part of the boundary. The optimal order of convergence, previously known for particular cases, is proved for an arbitrary wave number and for nonzero Dirichlet and Neumann conditions under a priori and a posteriori choice of a regularization parameter. In the second part, the regularized solutions are employed to describe the geometrical properties of the beam and, in particular, to find approximate beam quality parameters such as the waist of the axial profile of the beam and its position. The mathematical considerations are preceded by a section where the main notions of laser beam optics are introduced and briefly explained.

We suggest stochastic simulation techniques for solving two classes of linear and nonlinear inverse and ill-posed problems: (1) recovering the particle nanosize distribution from diffusion battery measurements, and (2) retrieving the step structure of the epitaxial films from the x-ray diffraction analysis. To solve these problems we develop three stochastic based methods: (1) the random projection method, a stochastic version of the Kaczmarz method, (2) a randomized SVD method, and (3) stochastic genetic algorithm. Results of comparative simulations of the three methods are also presented.

The coefficient inverse problem for the acoustic equation is considered. We propose the method for reconstructing the density based on the N -approximation by the finite system of one-dimensional problems and the two-dimensional M. G. Krein approach. The two-dimensional analogue of the M. G. Krein approach is applied to reduce the non-linear inverse problem to a family of linear integral equations. We consider the fast algorithm for solving the relevant linear system, based on using the block-Toeplitz structure of the matrix. The algorithm applied to the M. G. Krein equation allows to obtain the solution of the whole family of the integral equations by solving only one linear system. Results of numerical calculations are presented.