## Abstract

In this work we consider a specific problem of optimal planning of maritime transportation of multiproduct cargo by ships of one (corporate strategy) or several (partially corporate strategy) companies: the core of the problem consists of the existence of the network of intermediate seaports (i.e. transitional seaports), where for every ship arrived the cargo handling is done, and which are situated between the starting and the finishing seaports. In this work, there are mathematical models built from scratch in the form of multicriteria optimization problem; then the goal attainment method of Gembicki is used for reducing the built models to a one-criterion problem of linear programming.

## Regularization Parameter Selection in Discrete Ill-Posed Problems — The Use of the U-Curve

To obtain smooth solutions to ill-posed problems, the standard Tikhonov regularization method is most often used. For the practical choice of the regularization parameter α we can then employ the well-known L-curve criterion, based on the L-curve which is a plot of the norm of the regularized solution versus the norm of the corresponding residual for all valid regularization parameters. This paper proposes a new criterion for choosing the regularization parameter α, based on the so-called U-curve. A comparison of the two methods made on numerical examples is additionally included.

## Abstract

This paper deals with the problem of determining an unknown source and an unknown initial condition in a abstract final value parabolic problem. This problem is ill-posed in the sense that the solutions do not depend continuously on the data. To solve the considered problem a modified Tikhonov regularization method is proposed. Using this method regularized solutions are constructed and under boundary conditions assumptions, convergence estimates between the exact solutions and their regularized approximations are obtained. Moreover numerical results are presented to illustrate the accuracy and efficiency of the proposed method.

resolve integer ambiguity in rapid positioning using single frequency GPS receivers. Chin Sci Bull 49(2):196-200 Shagimuratov, I. I., Baran, L. W., Wielgosz, P., and Yakimova, G. A., (2002), The structure of mid- and high-latitude ionosphere during September 1999 storm event obtained from GPS observations, Annales Geophysicae, Vol. 20, No 6, 665-671 Shen Y, Li B (2007) Regularized solution to fast GPS ambiguity resolution. J Surv Eng 133(4):168-172 Teunissen PJG (1993) Least squares estimation of the integer GPS ambiguities. Invited lecture, Section IV: theory and

[ {{\text{J}}_{\text{k}}} \right]}^{\text{T}}}\left[ {{\text{J}}_{\text{k}}} \right]+\lambda \text{I} \right]}^{-1}}\left[ {{\left[ {{\text{J}}_{\text{k}}} \right]}^{\text{T}}}{{\left( \Delta \text{V} \right)}_{\text{k}}}-\lambda \text{I}{{\sigma }_{\text{k}}} \right]$$ Where J k and ( ΔV ) k are the Jacobian and voltage difference matrix respectively at the k th iteration. Thus the Gauss-Newton method based inverse solver algorithm gives a regularized solution of the conductivity distribution for the k th iteration as: (13) σ k+l = σ k + [ [ J k ] T [ J k ] + λ I

the solution. Correspondingly, an LS-optimized Tikhonov-regularized solution of the linearized MITS inverse problem using a one-step Gauss-Newton formulation was applied in image reconstruction according to the following equation: (18) I m ( Δ σ # ) = S T ( S S T + λ I m ) − 1 I m ( Δ v # ) $$Im\left( \Delta {{\mathbf{\sigma }}_{\#}} \right)\mathbf{=}{{\mathbf{S}}^{T}}{{\left( \mathbf{S}{{\mathbf{S}}^{T}}+\lambda {{\mathbf{I}}_{m}} \right)}^{-1}}Im\left( \Delta {{\mathbf{v}}_{\#}} \right)$$ where S is the sensitivity (Jacobian) matrix representing the rate at

small amount of noise in the boundary data can lead to enormous errors in the estimates. Hence, EIT needs a regularization technique [ 39 , 40 , 41 , 42 , 43 ] with a suitable regularization parameter (λ) to constrain its solution space as well as to convert the ill-posed problem into a well-posed one. A regularized solution of the inverse problem not only decreases the ill-posed characteristics of the inverse matrix but also, it improves the reconstructed image quality. Considering V m as the measured voltage matrix and f as a function mapping an E