This paper discusses the solution of nonlinear integral equations with noisy integral kernels as they appear in nonparametric instrumental regression. We propose a regularized Newton-type iteration ...and establish convergence and convergence rate results. A particular emphasis is on instrumental regression models where the usual conditional mean assumption is replaced by a stronger independence assumption. We demonstrate for the case of a binary instrument that our approach allows the correct estimation of regression functions which are not identifiable with the standard model. This is illustrated in computed examples with simulated data.
We study the convergence of regularized Newton methods applied to nonlinear operator equations in Hilbert spaces if the data are perturbed by random noise. It is shown that the expected square error ...is bounded by a constant times the minimax rates of the corresponding linearized problem if the stopping index is chosen using a priori knowledge of the smoothness of the solution. For unknown smoothness the stopping index can be chosen adaptively based on Lepskiĭ's balancing principle. For this stopping rule we establish an oracle inequality, which implies order optimal rates for deterministic errors, and optimal rates up to a logarithmic factor for random noise. The performance and the statistical properties of the proposed method are illustrated by Monte Carlo simulations.
We consider the inverse problem of recovering the spherically symmetric sound speed, density and attenuation in the Sun from the observations of the acoustic field randomly excited by turbulent ...convection. We show that observations at two heights above the photosphere and at two frequencies above the acoustic cutoff frequency uniquely determine the solar parameters. We also present numerical simulations which confirm this theoretical result.
For many wave propagation problems with random sources it has been demonstrated that cross correlations of wave fields are proportional to the imaginary part of the Green function of the underlying ...wave equation. This leads to the inverse problem to recover coefficients of a wave equation from the imaginary part of the Green function on some measurement manifold. In this paper we prove, in particular, local uniqueness results for the Schrödinger equation with one frequency and for the acoustic wave equation with unknown density and sound speed and two frequencies. As the main tool of our analysis, we establish new algebraic identities between the real and the imaginary part of Green's function, which in contrast to the well-known Kramers–Kronig relations, involve only one frequency.
The purpose of this work is to develop an automatic method for the scaling of unknowns in model‐based nonlinear inverse reconstructions and to evaluate its application to real‐time phase‐contrast ...(RT‐PC) flow magnetic resonance imaging (MRI). Model‐based MRI reconstructions of parametric maps which describe a physical or physiological function require the solution of a nonlinear inverse problem, because the list of unknowns in the extended MRI signal equation comprises multiple functional parameters and all coil sensitivity profiles. Iterative solutions therefore rely on an appropriate scaling of unknowns to numerically balance partial derivatives and regularization terms. The scaling of unknowns emerges as a self‐adjoint and positive‐definite matrix which is expressible by its maximal eigenvalue and solved by power iterations. The proposed method is applied to RT‐PC flow MRI based on highly undersampled acquisitions. Experimental validations include numerical phantoms providing ground truth and a wide range of human studies in the ascending aorta, carotid arteries, deep veins during muscular exercise and cerebrospinal fluid during deep respiration. For RT‐PC flow MRI, model‐based reconstructions with automatic scaling not only offer velocity maps with high spatiotemporal acuity and much reduced phase noise, but also ensure fast convergence as well as accurate and precise velocities for all conditions tested, i.e. for different velocity ranges, vessel sizes and the simultaneous presence of signals with velocity aliasing. In summary, the proposed automatic scaling of unknowns in model‐based MRI reconstructions yields quantitatively reliable velocities for RT‐PC flow MRI in various experimental scenarios.
This work describes an automatic method for the scaling of unknowns in model‐based nonlinear inverse reconstructions which numerically balances partial derivatives and regularization terms during iterative optimization. The method emerges as a self‐adjoint and positive‐definite matrix expressible by its maximal eigenvalue and solved by power iterations. Applications to real‐time phase‐contrast flow magnetic resonance imaging ensure fast convergence and offer velocity maps with much reduced phase noise, high spatiotemporal acuity and accurate and precise velocities for all conditions tested.
We derive rates of convergence and asymptotic normality of the least squares estimator for a large class of parametric inverse regression models Y = (Φf)(X)+ε. Our theory provides a unified ...asymptotic tretament for estimation of f with discontinuities of certain order, including piecewise polynomials and piecewise kink functions. Our results cover several classical and new examples, including splines with free knots or the estimation of piecewise linear functions with indirect observations under a nonlinear Hammerstein integral operator. Furthermore, we show that ℓ0-penalisation leads to a consistent model selection, using techniques from empirical process theory. The asymptotic normality is used to provide confidence bands for f. Simulation studies and a data example from rheology illustrate the results.
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