On the Linearization of Operators Related to the Full Waveform Inversion in Seismology

We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients

Seismic tomography is locally ill-posed

We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients

On the stable estimation of flow geometry and wall shear stress from magnetic resonance images

We consider the stable reconstruction of flow geometry and wall shear stress from measurements obtained by magnetic resonance imaging (MRI). As noted in a review article by Petersson, most approaches considered so far in the literature seem to not be satisfactory. We therefore propose a systematic reconstruction procedure that allows us to obtain stable estimates of flow geometry and wall shear stress and we are able to quantify the reconstruction errors in terms of bounds for the measurement errors under reasonable smoothness assumptions. A complete analysis of our approach is given in the framework of regularization methods. In addition, we briefly discuss the implementation of our method and we demonstrate its viability, accuracy, and regularizing properties for experimental data

An Inexact Newton Regularization in Banach Spaces based on the Nonstationary Iterated Tikhonov Method

We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients

Resolution-Controlled Conductivity Discretization in Electrical Impedance Tomography

We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients

Up-to-date Interval Arithmetic From Closed Intervals to Connected Sets of Real Numbers

We consider biperiodic integral equations of the second kind with weakly singular kernels such as they arise in boundary integral equation methods. The equations are solved numerically using a collocation scheme based on trigonometric polynomials. The weak singularity is removed by a local change to polar coordinates. The resulting operators have smooth kernels and are discretized using the tensor product composite trapezodial rule. We prove stability and convergence of the scheme under suitable parameter choices, achieving algebraic convergence of any order under appropriate regularity assumptions. The method can be applied to typical boundary value problems such as potential and scattering problems both for bounded obstacles and for periodic surfaces. We present numerical results demonstrating that the expected convergence rates can be observed in practice

Model-Aware Newton-Type Inversion Scheme For Electrical Impedance Tomography

We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients

A Collocation Method for Integral Equations with Super-Algebraic Convergence Rate

We consider biperiodic integral equations of the second kind with weakly singular kernels such as they arise in boundary integral equation methods. The equations are solved numerically using a collocation scheme based on trigonometric polynomials. The weak singularity is removed by a local change to polar coordinates. The resulting operators have smooth kernels and are discretized using the tensor product composite trapezodial rule. We prove stability and convergence of the scheme under suitable parameter choices, achieving algebraic convergence of any order under appropriate regularity assumptions. The method can be applied to typical boundary value problems such as potential and scattering problems both for bounded obstacles and for periodic surfaces. We present numerical results demonstrating that the expected convergence rates can be observed in practice