Posteriori Strategies Research Articles

Simultaneous Determination of the Space-Dependent Source and the Initial Distribution in a Heat Equation by Regularizing Fourier Coefficients of the Given Measurements

We consider an inverse problem for simultaneously determining the space-dependent source and the initial distribution in heat conduction equation. First, we study the ill-posedness of the inverse problem. Then, we construct a regularization problem to approximate the originally inverse problem and obtain the regularization solutions with their stability and convergence results. Furthermore, convergence rates of the regularized solutions are presented under a prior and a posteriori strategies for selecting regularization parameters. Results of numerical examples show that the proposed regularization method is stable and effective for the considered inverse problem.

Parameter Choice Strategies for Least-squares Approximation of Noisy Smooth Functions on the Sphere

We consider a polynomial reconstruction of smooth functions from their noisy values at discrete nodes on the unit sphere by a variant of the regularized least-squares method of An et al. [SIAM J. Numer. Anal., 50 (2012), pp. 1513--1534]. As nodes we use the points of a positive-weight cubature formula that is exact for all spherical polynomials of degree up to 2M, where M is the degree of the reconstructing polynomial. We first obtain a reconstruction error bound in terms of the regularization parameter and the penalization parameters in the regularization operator. Then we discuss a priori and a posteriori strategies for choosing these parameters. Finally, we give numerical examples illustrating the theoretical results.

Convergence rates for the iteratively regularized Gauss–Newton method in Banach spaces**Research has been partly conducted during the Mini Special Semester on Inverse Problems, 18 May–15 July 2009, organized by RICAM (Austrian Academy of Sciences), Linz, Austria, and moreover supported by Deutsche Forschungsgemeinschaft (DFG) under Grant HO1454/7-2 as well as within the Cluster of Excellence SimTech, University of Stuttgart.

In this paper we consider the iteratively regularized Gauss–Newton method (IRGNM) in a Banach space setting and prove optimal convergence rates under approximate source conditions. These are related to the classical concept of source conditions that is available only in Hilbert space. We provide results in the framework of general index functions, which include, e.g. Hölder and logarithmic rates. Concerning the regularization parameters in each Newton step as well as the stopping index, we provide both a priori and a posteriori strategies, the latter being based on the discrepancy principle.

A posteriori versus a priori access strategies in slotted all-optical WDM rings

The paper presents the concept of “a posteriori” access strategy, as opposed to the previously studied concept of “a priori” strategy, to provide a fair and efficient access technique in slotted wavelength division multiplexing (WDM) rings under a large variety of traffic patterns. Using a posteriori strategies, the source node selects the packet for transmission based on the state of the WDM channels in the arriving slot, thus adapting the access decision to the instantaneous traffic in the various channels. The packet selection process of a posteriori strategies, when contrasted with a priori strategies, yields fair and efficient utilization of the ring bandwidth without requiring the equalization of the ring latency, nor imposing restrictions on the traffic patterns allowed in the system. Since the same features cannot be provided by a priori access strategies, a posteriori strategies offers a more versatile access control that justifies the higher implementation complexity due to the multi-channel sensing and on-line packet selection.