Local Analysis of Inverse Problems: H\"{o}lder Stability and Iterative Reconstruction

We consider a class of inverse problems defined by a nonlinear map from parameter or model functions to the data. We assume that solutions exist. The space of model functions is a Banach space which is smooth and uniformly convex; however, the data space can be an arbitrary Banach space. We study sequences of parameter functions generated by a nonlinear Landweber iteration and conditions under which these strongly converge, locally, to the solutions within an appropriate distance. We express the conditions for convergence in terms of H\"{o}lder stability of the inverse maps, which ties naturally to the analysis of inverse problems

Nonparametric instrumental regression with non-convex constraints

This paper considers the nonparametric regression model with an additive error that is dependent on the explanatory variables. As is common in empirical studies in epidemiology and economics, it also supposes that valid instrumental variables are observed. A classical example in microeconomics considers the consumer demand function as a function of the price of goods and the income, both variables often considered as endogenous. In this framework, the economic theory also imposes shape restrictions on the demand function, like integrability conditions. Motivated by this illustration in microeconomics, we study an estimator of a nonparametric constrained regression function using instrumental variables by means of Tikhonov regularization. We derive rates of convergence for the regularized model both in a deterministic and stochastic setting under the assumption that the true regression function satisfies a projected source condition including, because of the non-convexity of the imposed constraints, an additional smallness condition

Discretization of variational regularization in Banach spaces

Consider a nonlinear ill-posed operator equation $F(u)=y$ where $F$ is defined on a Banach space $X$. In general, for solving this equation numerically, a finite dimensional approximation of $X$ and an approximation of $F$ are required. Moreover, in general the given data \yd of $y$ are noisy. In this paper we analyze finite dimensional variational regularization, which takes into account operator approximations and noisy data: We show (semi-)convergence of the regularized solution of the finite dimensional problems and establish convergence rates in terms of Bregman distances under appropriate sourcewise representation of a solution of the equation. The more involved case of regularization in nonseparable Banach spaces is discussed in detail. In particular we consider the space of finite total variation functions, the space of functions of finite bounded deformation, and the $L^\infty$--space

Regularization of Linear Ill-posed Problems by the Augmented Lagrangian Method and Variational Inequalities

We study the application of the Augmented Lagrangian Method to the solution of linear ill-posed problems. Previously, linear convergence rates with respect to the Bregman distance have been derived under the classical assumption of a standard source condition. Using the method of variational inequalities, we extend these results in this paper to convergence rates of lower order, both for the case of an a priori parameter choice and an a posteriori choice based on Morozov's discrepancy principle. In addition, our approach allows the derivation of convergence rates with respect to distance measures different from the Bregman distance. As a particular application, we consider sparsity promoting regularization, where we derive a range of convergence rates with respect to the norm under the assumption of restricted injectivity in conjunction with generalized source conditions of H\"older type

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 $F(u)=g$ in Banach spaces where the Newton equations $F'(u_n;u_{n+1}-u_n) = g-F(u_n)$ 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

Convergence rates in expectation for Tikhonov-type regularization of Inverse Problems with Poisson data

In this paper we study a Tikhonov-type method for ill-posed nonlinear operator equations \gdag = F( ag) where \gdag is an integrable, non-negative function. We assume that data are drawn from a Poisson process with density t\gdag where $t>0$ may be interpreted as an exposure time. Such problems occur in many photonic imaging applications including positron emission tomography, confocal fluorescence microscopy, astronomic observations, and phase retrieval problems in optics. Our approach uses a Kullback-Leibler-type data fidelity functional and allows for general convex penalty terms. We prove convergence rates of the expectation of the reconstruction error under a variational source condition as $t\to\infty$ both for an a priori and for a Lepski{\u\i}-type parameter choice rule

Necessary conditions for variational regularization schemes

We study variational regularization methods in a general framework, more precisely those methods that use a discrepancy and a regularization functional. While several sets of sufficient conditions are known to obtain a regularization method, we start with an investigation of the converse question: How could necessary conditions for a variational method to provide a regularization method look like? To this end, we formalize the notion of a variational scheme and start with comparison of three different instances of variational methods. Then we focus on the data space model and investigate the role and interplay of the topological structure, the convergence notion and the discrepancy functional. Especially, we deduce necessary conditions for the discrepancy functional to fulfill usual continuity assumptions. The results are applied to discrepancy functionals given by Bregman distances and especially to the Kullback-Leibler divergence.Comment: To appear in Inverse Problem

Water Futures and Solution - Fast Track Initiative (Final Report)

The Water Futures and Solutions Initiative (WFaS) is a cross-sector, collaborative global water project. Its objective is to apply systems analysis, develop scientific evidence and identify water-related policies and management practices, working together consistently across scales and sectors to improve human well-being through water security. The approach is a stakeholder-informed, scenario-based assessment of water resources and water demand that employs ensembles of state-of-the-art socio-economic and hydrological models, examines possible futures and tests the feasibility, sustainability and robustness of options that can be implemented today and can be sustainable and robust across a range of possible futures and associated uncertainties. This report aims at assessing the global current and future water situation

Impact of GBA1 variants on long-term clinical progression and mortality in incident Parkinson’s disease

Funder: Foundation for the National Institutes of Health; FundRef: http://dx.doi.org/10.13039/100000009Funder: Van Geest FoundationFunder: Patrick Berthoud Charitable Trust; FundRef: http://dx.doi.org/10.13039/501100004218Funder: Cure Parkinson's TrustFunder: Michael J Fox FoundationFunder: Innovate UK; FundRef: http://dx.doi.org/10.13039/501100006041Funder: Dooley LLCFunder: American Parkinson's disease associationFunder: Medical Research Council; FundRef: http://dx.doi.org/10.13039/501100000265Funder: Cambridge Centre for Parkinson-PlusFunder: Parkinson's UK; FundRef: http://dx.doi.org/10.13039/501100000304Funder: John Black charitable foundationFunder: Wellcome Trust; FundRef: http://dx.doi.org/10.13039/100004440Funder: National Institute for Health Research; FundRef: http://dx.doi.org/10.13039/501100000272Funder: Van Andel Research Institute; FundRef: http://dx.doi.org/10.13039/100006019Introduction: Variants in the GBA1 gene have been identified as a common risk factor for Parkinson’s disease (PD). In addition to pathogenic mutations (those associated with Gaucher disease), a number of ‘non-pathogenic’ variants also occur at increased frequency in PD. Previous studies have reported that pathogenic variants adversely affect the clinical course of PD. The role of ‘non-pathogenic’ GBA1 variants on PD course is less clear. In this study, we report the effect of GBA1 variants in incident PD patients with long-term follow-up. Methods: The study population consisted of patients in the Cambridgeshire Incidence of Parkinson’s disease from General Practice to Neurologist and Parkinsonism: Incidence, Cognition and Non-motor heterogeneity in Cambridgeshire cohorts. Patients were grouped into non-carriers, carriers of ‘non-pathogenic’ GBA1 variants and carriers of pathogenic GBA1 mutations. Survival analyses for time to development of dementia, postural instability and death were carried out. Cox regression analysis controlling for potential confounders were used to determine the impact of GBA1 variants on these outcome measures. Results: GBA1 variants were identified in 14.4% of patients. Pathogenic and ‘non-pathogenic’ GBA1 variants were associated with the accelerated development of dementia and a more aggressive motor course. Pathogenic GBA1 variants were associated with earlier mortality in comparison with non-carriers, independent of the development of dementia. Discussion: GBA1 variants, including those not associated with Gaucher disease, are common in PD and result in a more aggressive disease course