Volume 8, Issue 3

We analyze the optimal superconvergence properties of piecewise polynomial collocation solutions on uniform meshes for Volterra integral and integrodifferential equations with multiple (vanishing) proportional delays. It is shown that for delay integro-differential equations the recently obtained optimal order is also attainable. For integral equations with multiple vanishing delays this is no longer true.

The application of a parallel multiblock geometric multigrid is consid-ered. It is applied to solve a two-dimensional poroelastic model. This system of PDEs is approximated by a special stabilized monotone finite-difference scheme. The obtained system of linear algebraic equations is solved by a multigrid method, when a domain is partitioned into structured blocks. A new strategy for the solution of the discrete problem on the coarsest grid is proposed and the efficiency of the obtained algorithm is investigated. The geometrical structure of the sequential multigrid method is used to develop a parallel version of the multigrid algorithm. The convergence properties of several smoothers are investigated and some computational results are presented.

We consider linear ill-posed problems in Hilbert spaces with a noisy right hand side and a given noise level. To solve non-self-adjoint problems by the (it-erated) Tikhonov method, one effective rule for choosing the regularization parameter is the monotone error rule (Tautenhahn and Hamarik, Inverse Problems, 1999, 15, 1487– 1505). In this paper we consider the solution of self-adjoint problems by the (iterated) Lavrentiev method and propose for parameter choice an analog of the monotone error rule. We prove under certain mild assumptions the quasi-optimality of the proposed rule guaranteeing convergence and order optimal error estimates. Numerical examples show for the proposed rule and its modifications much better performance than for the modified discrepancy principle.

In this paper we continue our study of solving ill-posed problems with a noisy right-hand side and a noisy operator. Regularized approximations are obtained by Tikhonov regularization with differential operators and by dual regularized total least squares (dual RTLS) which can be characterized as a special multi-parameter regularization method where one of the two regularization parameters is negative. We report on order optimality results for both regularized approximations, discuss compu-tational aspects, provide special algorithms and show by experiments that dual RTLS is competitive to Tikhonov regularization with differential operators.

The technical successes in radio navigation and the availability of nu-merical algorithms have promoted the implementation of GPS-technology to atmo-spheric sciences. The tomographical contribution of Global Satellite Navigation Sys-tems (GNSS) is possible due to the methods of high precision detection of tropospheric delays of navigation signals from satellites to receivers. The principal specific char-acter in initial constraints, data collection and assimilation methods, the obtaining of final numerical results and their interpretation make the continuation of the success story for GPS-tomography very challenging. The authors use numerical simulation as the most time- and cost-efficient way to study different processes related to tro-pospheric water vapor tomography. This paper tends to give a short overview about some known methods in GPS-tomography for detection, monitoring and modeling of the tropospheric water vapor. The possible mathematical approach to the construc-tion of virtual network of ground-based sensors (GPS-receivers) for a real geographical location and discretization of the troposphere, also some aspects of raw data filtering and analysis are described. Output of tomographical modelling of the troposphere can be used to improve the results of large-scale numerical weather prediction models and also real-time navigation. The questions of voxel geometry and methods of data processing are supposed to be the key questions in constructing an effective network of GPS-receivers for water vapor tomography.

For solving linear ill-posed problems with noisy data regularization methods are required. We analyze a simplified regularization scheme in Hilbert scales for operator equations with nonnegative self-adjoint operators. By exploiting the op-erator monotonicity of certain functions, order-optimal error bounds are derived that characterize the accuracy of the regularized approximations. These error bounds have been obtained under general smoothness conditions.

Approximations to a solution and its derivatives of a boundary value problem of an nth order linear Fredholm integro-differential equation with weakly sin-gular or other nonsmooth kernels have been determined. These approximations are piecewise polynomial functions on special graded grids. To find them, a fully discrete version of the Galerkin method has been constructed. This version is based on a dis-crete inner product concept and some suitable product integration techniques. Optimal global convergence estimates have been derived and a collection of numerical results of a test problem is given.