Volume 19, Issue 1

In this paper, we consider an inverse problem for a system of evolution equations. The inverse problem can also be interpreted as a exact control problem in the transition of a substance W ( x, t ) from a nonlocal state W 1 ( x ) to another nonlocal state W 2 ( x ). The control element is assumed to have a nonlocal form. We show the existence of the inverse problem in classes of entire functions with respect to the spatial variable by the constructive method. Explicit formulas of the unknown functions W ( x, t ) and λ ( x ) are presented.

In the present paper we consider a system of equations given for one unknown function that have generalized D'Alambert representation of solutions. We establish necessary and sufficient conditions for nontrivial solvability of the Dirichlet problem for such system in the unit ball. A simple necessary condition is obtained which says: for any i and j the angle between the vectors a i and a j must be a π -rational number.

The main objective of this paper is to compare – analytically as well as numerically – different approaches for obtaining optimal input currents in impedance tomography. Following the approaches described in, e.g. [Cheney and Isaacson, IEEE Trans. Biomed. Eng. 39: 852–860, 1992, Isaacson, IEEE Trans. Med. Imag. 5: 91–95, 1986, Ito and Kunisch, SIAM J. Contr. Optim. 33: 643–666., 1995, Knowles, An optimal current functional for electrical impedance tomography, 2004], we aim at constructing input currents j , which contain the most information about the difference between the unknown physical conductivity σ * and a given approximation σ 0 . The differences can be measured by different discrepancy functionals and the optimal input currents which maximize these functionals depend on the function spaces chosen for defining j and on the norm for measuring the discrepancy. Moreover, the definition of the appropriately weighted Sobolev spaces depends on σ and this subsequently influences the iteration for maximizing the functionals. Numerical experiments illustrate features of the optimal input currents obtained for several combinations of function spaces. The reconstructions with these optimal currents are compared with those with standard input currents (sinusoid and dipole). The differences between the optimal currents obtained by different function space settings are significant. Two newly developed optimal currents can yield qualitatively better reconstructions.

The aim of the present paper is to generalize the results in [Denisov, J. Inverse Ill-Posed Probl. 16: 837–848, 2008.] devoted to a one-dimensional semilinear wave equation and consisting of recovering a time-dependent function α representing the transformed argument. More exactly: (i) we will deal with a general semilinear integro-differential hyperbolic d -dimensional equation in divergence form ( d = 1, 2, 3); (ii) the space-time set considered here is a smooth cylinder where surface boundary conditions are prescribed; (iii) the term with transformed arguments is allowed to contain integral operators; (iv) the additional information is of integral type. The existence and uniqueness results for our specific problem are deduced as consequences of similar results for an operator integro-differential identification problem in a Hilbert space.

Suppose that a uniqueness theorem is valid for an ill-posed problem. It is shown then that the distance between the exact solution and terms of a minimizing sequence of the Tikhonov functional is less than the distance between the exact solution and the first guess. Unlike the classical case when the regularization parameter tends to zero, only a single value of this parameter is used. Indeed, the latter is always the case in computations. Next, this result is applied to a specific coefficient inverse problem. A uniqueness theorem for this problem is based on the method of Carleman estimates. In particular, the importance of obtaining an accurate first approximation for the correct solution follows from Theorems 7 and 8. The latter points towards the importance of the development of globally convergent numerical methods as opposed to conventional locally convergent ones. A numerical example is presented.

Suppose q i ( x ), i = 1, 2 are smooth functions on and U i ( x, t ) the solutions of the initial value problem , U i ( x, t ) = 0 for t < 0. Pick R, T so that 0 < R < T and let C be the vertical cylinder {( x, t ) : | x | = R, R ≤ t ≤ T }. We show that if ( U 1 , U 1 r ) = ( U 2 , U 2 r ) on C then q 1 = q 2 on the annular region R ≤ | x | ≤ ( R + T )/2 provided there is a γ > 0, independent of r , so that ∫ | x | = r | Δ S ( q 1 – q 2 )| 2 dS x ≤ γ ∫ | x | = r | q 1 – q 2 | 2 dS x for all r ∈ [ R , ( R + T /2)]. Here Δ S is the spherical Laplacian on | x | = r .

Let D be a bounded domain in the n -dimensional Euclidian space ( n ≥ 2) having smooth boundary ∂D . We indicate appropriate Sobolev spaces with negative smoothness in D in order to consider the non-homogeneous ill-posed Cauchy problem for an overdetermined operator A with injective symbol. We prove that elements of the indicated Sobolev spaces have traces on the boundary. This easily leads to a weak formulation of the Cauchy problem and to the corresponding uniqueness theorem. We also describe solvability conditions of the problem and construct its exact and approximate solutions. Namely, we obtain the Carleman formula recovering a vector-function u from the indicated negative Sobolev class via its Cauchy data on an open connected set Γ ⊂ ∂D and values of Au on the domain D . Some instructive examples are considered.

A new numerical model of a galaxy interaction in gasodynamical approach is considered. In this model the equations of a gas dynamics with a cold function and the Poisson equation for a gravity potential are solved. A total system of equations is illposed. It allows us to describe a gravitational instability of astrophysical processes. As an example, the results of the collapse simulation are presented. The gas dynamic system of equations is solved by the Fluids-in-Cell method with the energy balance correction. Using the 3D Cartesian simulation a rotation, a self-consistent gravitational field, a complex central body geometry and a gas temperature were taken into account. A parallel implementation of the method for the simulation of various galaxy interaction scenarios was used in the model. Also, a research of the numerical model application range for various collision velocities and a mass relation of gas and star components was made here.