49 research outputs found
Numerical studies of the Lagrangian approach for reconstruction of the conductivity in a waveguide
We consider an inverse problem of reconstructing the conductivity function in
a hyperbolic equation using single space-time domain noisy observations of the
solution on the backscattering boundary of the computational domain. We
formulate our inverse problem as an optimization problem and use Lagrangian
approach to minimize the corresponding Tikhonov functional. We present a
theorem of a local strong convexity of our functional and derive error
estimates between computed and regularized as well as exact solutions of this
functional, correspondingly. In numerical simulations we apply domain
decomposition finite element-finite difference method for minimization of the
Lagrangian. Our computational study shows efficiency of the proposed method in
the reconstruction of the conductivity function in three dimensions
Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data
We study Newton type methods for inverse problems described by nonlinear
operator equations in Banach spaces where the Newton equations
are regularized variationally using a general
data misfit functional and a convex regularization term. This generalizes the
well-known iteratively regularized Gauss-Newton method (IRGNM). We prove
convergence and convergence rates as the noise level tends to 0 both for an a
priori stopping rule and for a Lepski{\u\i}-type a posteriori stopping rule.
Our analysis includes previous order optimal convergence rate results for the
IRGNM as special cases. The main focus of this paper is on inverse problems
with Poisson data where the natural data misfit functional is given by the
Kullback-Leibler divergence. Two examples of such problems are discussed in
detail: an inverse obstacle scattering problem with amplitude data of the
far-field pattern and a phase retrieval problem. The performence of the
proposed method for these problems is illustrated in numerical examples
Weakly Consistent Regularisation Methods for Ill-Posed Problems
This Chapter takes its origin in the lecture notes for a 9 h course at the Institut Henri Poincaré in September 2016. The course was divided in three parts. In the first part, which is not included herein, the aim was to first recall some basic aspects of stabilised finite element methods for convection-diffusion problems. We focus entirely on the second and third parts which were dedicated to ill-posed problems and their approximation using stabilised finite element methods. First we introduce the concept of conditional stability. Then we consider the elliptic Cauchy-problem and a data assimilation problem in a unified setting and show how stabilised finite element methods may be used to derive error estimates that are consistent with the stability properties of the problem and the approximation properties of the finite element space. Finally, we extend the result to a data assimilation problem subject to the heat equation