Volume 16, Issue 5

In a model situation the problem of parallel identification of two parameters in an elliptic equation was examined. Here, coefficients were considered which are piecewise constant. In a numerical study we deal with various questions. The main focal point is on additional ill-posedness phenomena and numerical effects which are caused by the simultaneous determination of more than one coefficients in differential equations. Different known approaches for solving the underlying problem are investigated with respect to their efficiency and reliability. Moreover, the effect of noise in the given data is analyzed as well as the influence of the specific choice of the boundary conditions on the unique solvability of the identification problem.

We introduce and analyze moduli of continuity for specific classes of Nemytskii operators on spaces of continuous functions, which are given by kernels, strictly monotone in their second argument. Such operators occur as non-linear (outer) mappings for certain problems of option pricing within the Black–Scholes model for time-dependent volatility. This nonlinear mapping can be seen to be continuous, however its convergence properties are poor. Our general results allow to bound the related moduli of continuity, both for the forward and backward non-linear mappings. In particular we explain the observed ill-conditioning of the nonlinear backward problem. The analysis uses some abstract local analysis of index functions, which may be of independent interest.

This paper addresses the regularization by sparsity constraints by means of weighted ℓ p penalties for 0 ≤ p ≤ 2. For 1 ≤ p ≤ 2 special attention is payed to convergence rates in norm and to source conditions. As main results it is proven that one gets a convergence rate of in the 2-norm for 1 < p ≤ 2 and in the 1-norm for p = 1 as soon as the unknown solution is sparse. The case p = 1 needs a special technique where not only Bregman distances but also a so-called Bregman-Taylor distance has to be employed. For p < 1 only preliminary results are shown. These results indicate that, different from p ≥ 1, the regularizing properties depend on the interplay of the operator and the basis of sparsity. A counterexample for p = 0 shows that regularization need not to happen.

In this article we investigate and prove relationships between metric and Bregman projections induced by powers of the norm of a Banach space. We consider Bregman projections onto affine subspaces of Banach spaces and deduce some interesting analogies to results which are well known for Hilbert spaces. Using these concepts as well as ideas from sequential subspace optimization techniques we construct efficient iterative methods to compute Bregman projections onto affine subspaces that are connected to linear, bounded operators between Banach spaces. Especially these methods can be used to compute minimum-norm solutions of linear operator equations or best approximations in the range of a linear operator. Numerical experiments illuminate the performance of our iterative algorithms and demonstrate a significant acceleration compared to the Landweber method.

For solving linear ill-posed problems regularization methods are required when the right hand side and the operator is with some noise. In the present paper regularized approximations are obtained by Tikhonov regularization, by regularized total least squares (RTLS) and by dual regularized total least squares (DRTLS). We discuss computational aspects and provide order optimal error bounds under two assumptions characterizing the smoothness of the unknown solution and the smoothing properties of the forward operator. The derived error bounds extend results from [S. Lu, S.V. Pereverzev, and U. Tautenhahn, Regularized total least squares: computational aspects and error bounds , Johann Radon Institute for Computational and Applied Mathematics, Report no. 2007-30, 2007.] where the above regularization methods are studied for finitely smoothing forward operators. We illustrate our theory by a special inverse heat conduction problem.