1,204 research outputs found
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 where is
defined on a Banach space . In general, for solving this equation
numerically, a finite dimensional approximation of and an approximation of
are required. Moreover, in general the given data \yd of 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
--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 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
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 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 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
- …