A Bregman forward-backward linesearch algorithm for nonconvex composite optimization: superlinear convergence to nonisolated local minima

We introduce Bella, a locally superlinearly convergent Bregman forward backward splitting method for minimizing the sum of two nonconvex functions, one of which satisfying a relative smoothness condition and the other one possibly nonsmooth. A key tool of our methodology is the Bregman forward-backward envelope (BFBE), an exact and continuous penalty function with favorable first- and second-order properties, and enjoying a nonlinear error bound when the objective function satisfies a Lojasiewicz-type property. The proposed algorithm is of linesearch type over the BFBE along candidate update directions, and converges subsequentially to stationary points, globally under a KL condition, and owing to the given nonlinear error bound can attain superlinear convergence rates even when the limit point is a nonisolated minimum, provided the directions are suitably selected

Assessment of a Variable Projection Algorithm for Trace Gas Retrieval in the Short-Wave Infrared

An important part of atmospheric remote sensing is the monitoring of its composition, which can be retrieved from radiance measurements, e.g. in the short-wave infrared (SWIR). For deriving trace gas concentrations in the SWIR spectral region a radiative transfer model is fitted to observations by least squares optimization. The aim of this thesis is to present the well-established variable projection method for solving separable nonlinear least squares problems and to examine and configure it for trace gas retrieval. For this, a Python implementation of the algorithm, called varpro.py, will be outlined and later utilized in retrievals with real satellite observations. These are meant to assess the efficiency, accuracy and robustness of three iterative algorithms for nonlinear least squares problems which have been built into varpro.py. Furthermore, a new feature - applying bounds to the non-linear fit parameters - will be included in the implementation and evaluated for its quality and usefulness. As a result of these tests, a new 'default' configuration will be suggested based on the algorithm with the best performance for trace gas retrieval. Also, ideas for analysing and testing strategies which could lead to even more insights will be proposed. Finally, possible future applications for trace gas retrieval will be motivated and suggestions for further research and modifications of varpro.py will be made

Estimation of Soil Water Dynamics Based on Hydrogeophysical Measurements

Quantitative understanding of soil water movement is essential to develop methods that allow for a more sustainable use of limited freshwater resources. In this study, methods are developed that allow to estimate the spatial distribution of these materials, their effective soil hydraulic material properties, and the effect of unrepresented model errors on these properties. To acquire the necessary data, a 2D subsurface architecture (ASSESS) was forced with a fluctuating groundwater table. The resulting hydraulic dynamics were essentially monitored with time domain reflectometry (TDR) and ground-penetrating radar (GPR). Based on the TDR data, the effect of unrepresented model errors on estimated soil hydraulic material properties is analyzed with a structural error analysis. This method compares inversions of increasingly complex models, since the required model complexity for a consistent description of the measurement data is application-dependent and unknown a priori. It is demonstrated that the method can indicate unrepresented model errors and quantify their effects on the resulting material properties. Based on the GPR data, a new heuristic event detection and association algorithm was developed that allows to identify and to extract relevant information from GPR data. It is demonstrated for synthetic and measured data that this approach provides reasonably accurate estimates for the spatial distribution of materials and their soil hydraulic material properties

Retrieval of Non-Spherical Dust Aerosol Properties from Satellite Observations

An accurate and generalized global retrieval algorithm from satellite observations is a prerequisite to understand the radiative effect of atmospheric aerosols on the climate system. Current operational aerosol retrieval algorithms are limited by the inversion schemes and suffering from the non-uniqueness problem. In order to solve these issues, a new algorithm is developed for the retrieval of non-spherical dust aerosol over land using multi-angular radiance and polarized measurements of the POLDER (POLarization and Directionality of the Earth’s Reflectances) and wide spectral high-resolution measurements of the MODIS (MODerate resolution Imaging Spectro-radiometer). As the first step to account for the non-sphericity of irregularly shaped dust aerosols in the light scattering problem, the spheroidal model is introduced. To solve the basic electromagnetic wave scattering problem by a single spheroid, we developed an algorithm, by transforming the transcendental infinite-continued-fraction-formeigen equation into a symmetric tri-diagonal linear system, for the calculation of the spheroidal angle function, radial functions of the first and second kind, as well as the corresponding first order derivatives. A database is developed subsequently to calculate the bulk scattering properties of dust aerosols for each channel of the satellite instruments. For the purpose of simulation of satellite observations, a code is developed to solve the VRTE (Vector Radiative Transfer Equation) for the coupled atmosphere-surface system using the adding-doubling technique. An alternative fast algorithm, where all the solid angle integrals are converted to summations on an icosahedral grid, is also proposed to speed-up the code. To make the model applicable to various land and ocean surfaces, a surface BRDF (Bidirectional Reflectance Distribution Function) library is embedded into the code. Considering the complimentary features of the MODIS and the POLDER, the collocated measurements of these two satellites are used in the retrieval process. To reduce the time spent on the simulation of dust aerosol scattering properties, a single-scattering property database of tri-axial ellipsoid is incorporated. In addition, atmospheric molecule correction is considered using the LBLRTM (Line-By-Line Ra- diative Transfer Model). The Levenberg-Marquardt method was employed to retrieve all the interested dust aerosol parameters and surface parameters simultaneously. As an example, dust aerosol properties retrieved over the Sahara Desert are presented

Hierarchical structure-and-motion recovery from uncalibrated images

This paper addresses the structure-and-motion problem, that requires to find camera motion and 3D struc- ture from point matches. A new pipeline, dubbed Samantha, is presented, that departs from the prevailing sequential paradigm and embraces instead a hierarchical approach. This method has several advantages, like a provably lower computational complexity, which is necessary to achieve true scalability, and better error containment, leading to more stability and less drift. Moreover, a practical autocalibration procedure allows to process images without ancillary information. Experiments with real data assess the accuracy and the computational efficiency of the method.Comment: Accepted for publication in CVI