Volume 15, Issue 4

Linear integral equations of the third kind usually lead to ill-posed inverse problems if the normed solution space X and the normed data space Y are required to be equal. In the present paper we develop a two-step regularization method—called RPMO method—to regularize such inverse problems. This method does not require Hilbert space properties. Convergence results are presented indicating that there is no general theoretical upper bound less than one for the convergence rates if the corresponding exact data solutions are sufficiently smooth. Moreover, we illustrate the RPMO method by applying its discretized version to an implicit linear boundary value problem which can be transformed into an equivalent third-kind integral equation.

In the paper, physical foundation of a functional for solving discontinuous stationary heat conduction problems with a kind of regularisation parameter has been presented. Then, the noncontinuous FEM with Trefftz base functions and a wide range of the regularisation parameter values has been applied to solving direct and inverse problem of linear heat conduction. 2D test problem solution has been discussed. Then a problem of finding heat transfer coefficient and temperature on the inner boundary of a turbine blade has been considered. To solve the problem the numerical entropy production intensity and energy dissipation function have been minimized on the boundary of the blade cross-section. In order to simplify the numerical calculation the Bernstein polynomials have been used to approximate the boundary temperature. Increasing the number of base functions in the finite element substantially decreases the inaccuracies of direct and inverse problem solution. It seems that the only disadvantage of minimising the functionals of entropy production intensity or energy dissipation lead to nonlinear system of algebraic equation.

We examine nonstationary inverse problems in which the time evolution of the unknown quantity is modelled by a stochastic partial differential equation. We consider the problem as a state estimation problem. The time discrete state evolution equation is exact since the solution is given by an analytic semigroup. For the practical reasons the space discretization of the time discrete state estimation system must be performed. However, space discretization causes an error and inverse problems are known to be very intolerant to both measurement errors and errors in models. In this paper we analyse the discretization error so that the statistics of the discretization error can be taken into account in the estimation. We are interested in the related filtering problem. A suitable filtering method is presented. We also verify the method using numerical simulation.

We propose a new algorithm for computing regularized solutions to inverse problems where the unknown functions is a characteristic function and where the forward operator is linear. Our approach can be seen as an alternative to the level-set method and is based on an efficient computation of minimizers for a Tikhonov functional. One main ingredient is to use surrogate functionals to define an iteration which has a subsequence converging to a limit that can be interpreted as a critical point. For the minimization of the surrogate functionals we propose the method of exact relaxation, which allows us to compute exact minimizer for the corresponding binary optimization problem.

This paper presents a novel, direct method for detecting surface cracks in a magnetic plate. In the leakage magnetic field method, a crack is modeled as being full of magnetic dipoles aligned along the crack, which is to be reconstructed from measurement of the leaked field. First, we introduce a source model of an equivalent magnetic dipole (EMD) whose location and moment coincide with the centroid of a crack and total moment distributed along the crack, respectively. Then, under the multiple EMDs model, using the multipole expansion of the leakage field, we derive algebraic equations relating the EMD parameters to data. They are formulated as a so-called 'moment problem' so that the centroid positions are reconstructed as generalized eigenvalues of Hankel matrices consisting of the multipole moments. The advantage of our method is that it requires neither initial values of the EMD parameters nor computing forward solution iteratively so that the essential parameters of the cracks are obtained by quite simple, algebraic computation. Numerical simulations show validity of our method under noisy condition.

Usual way to characterize the quality of the different a posteriori parameter choices is to prove their order-optimality on the different sets of solutions. In this paper we introduce the property of the quasioptimality to characterize the quality of the rule of the a posteriori regularization parameter choice for concrete problem Au = f . If the method P with a priori parameter choice is order optimal on the set of solutions M , then it is order-optimal with quasioptimal a posteriori parameter choice on the set M also. We consider two concepts of the quasioptimality and discuss the quasioptimality of different well-known rules for the a posteriori parameter choice as discrepancy principle, the modification of the discrepancy principle, Lepskii principle and monotone error rule. We consider also the family of weakly quasioptimal rules for a posteriori parameter choice.