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Institut
- Fachbereich 4 (13)
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Nonlocal operators are used in a wide variety of models and applications due to many natural phenomena being driven by nonlocal dynamics. Nonlocal operators are integral operators allowing for interactions between two distinct points in space. The nonlocal models investigated in this thesis involve kernels that are assumed to have a finite range of nonlocal interactions. Kernels of this type are used in nonlocal elasticity and convection-diffusion models as well as finance and image analysis. Also within the mathematical theory they arouse great interest, as they are asymptotically related to fractional and classical differential equations.
The results in this thesis can be grouped according to the following three aspects: modeling and analysis, discretization and optimization.
Mathematical models demonstrate their true usefulness when put into numerical practice. For computational purposes, it is important that the support of the kernel is clearly determined. Therefore nonlocal interactions are typically assumed to occur within an Euclidean ball of finite radius. In this thesis we consider more general interaction sets including norm induced balls as special cases and extend established results about well-posedness and asymptotic limits.
The discretization of integral equations is a challenging endeavor. Especially kernels which are truncated by Euclidean balls require carefully designed quadrature rules for the implementation of efficient finite element codes. In this thesis we investigate the computational benefits of polyhedral interaction sets as well as geometrically approximated interaction sets. In addition to that we outline the computational advantages of sufficiently structured problem settings.
Shape optimization methods have been proven useful for identifying interfaces in models governed by partial differential equations. Here we consider a class of shape optimization problems constrained by nonlocal equations which involve interface-dependent kernels. We derive the shape derivative associated to the nonlocal system model and solve the problem by established numerical techniques.
In this thesis, we aim to study the sampling allocation problem of survey statistics under uncertainty. We know that the stratum specific variances are generally not known precisely and we have no information about the distribution of uncertainty. The cost of interviewing each person in a stratum is also a highly uncertain parameter as sometimes people are unavailable for the interview. We propose robust allocations to deal with the uncertainty in both stratum specific variances and costs. However, in real life situations, we can face such cases when only one of the variances or costs is uncertain. So we propose three different robust formulations representing these different cases. To the best of our knowledge robust allocation in the sampling allocation problem has not been considered so far in any research.
The first robust formulation for linear problems was proposed by Soyster (1973). Bertsimas and Sim (2004) proposed a less conservative robust formulation for linear problems. We study these formulations and extend them for the nonlinear sampling allocation problem. It is very unlikely to happen that all of the stratum specific variances and costs are uncertain. So the robust formulations are in such a way that we can select how many strata are uncertain which we refer to as the level of uncertainty. We prove that an upper bound on the probability of violation of the nonlinear constraints can be calculated before solving the robust optimization problem. We consider various kinds of datasets and compute robust allocations. We perform multiple experiments to check the quality of the robust allocations and compare them with the existing allocation techniques.
We consider a linear regression model for which we assume that some of the observed variables are irrelevant for the prediction. Including the wrong variables in the statistical model can either lead to the problem of having too little information to properly estimate the statistic of interest, or having too much information and consequently describing fictitious connections. This thesis considers discrete optimization to conduct a variable selection. In light of this, the subset selection regression method is analyzed. The approach gained a lot of interest in recent years due to its promising predictive performance. A major challenge associated with the subset selection regression is the computational difficulty. In this thesis, we propose several improvements for the efficiency of the method. Novel bounds on the coefficients of the subset selection regression are developed, which help to tighten the relaxation of the associated mixed-integer program, which relies on a Big-M formulation. Moreover, a novel mixed-integer linear formulation for the subset selection regression based on a bilevel optimization reformulation is proposed. Finally, it is shown that the perspective formulation of the subset selection regression is equivalent to a state-of-the-art binary formulation. We use this insight to develop novel bounds for the subset selection regression problem, which show to be highly effective in combination with the proposed linear formulation.
In the second part of this thesis, we examine the statistical conception of the subset selection regression and conclude that it is misaligned with its intention. The subset selection regression uses the training error to decide on which variables to select. The approach conducts the validation on the training data, which oftentimes is not a good estimate of the prediction error. Hence, it requires a predetermined cardinality bound. Instead, we propose to select variables with respect to the cross-validation value. The process is formulated as a mixed-integer program with the sparsity becoming subject of the optimization. Usually, a cross-validation is used to select the best model out of a few options. With the proposed program the best model out of all possible models is selected. Since the cross-validation is a much better estimate of the prediction error, the model can select the best sparsity itself.
The thesis is concluded with an extensive simulation study which provides evidence that discrete optimization can be used to produce highly valuable predictive models with the cross-validation subset selection regression almost always producing the best results.
Harvesting of silage maize in late autumn on waterlogged soils may result in several ecological problems such as soil compaction and may subsequently be a major threat to soil fertility in Europe. It was hypothesized that perennial energy crops might reduce the vulnerability for soil compaction through earlier harvest dates and improved soil stability. However, the performance of such crops to be grown on soil that are periodically waterlogged and implications for soil chemical and microbial properties are currently an open issue. Within the framework of a two-year pot experiment we investigated the potential of the cup plant (Silphium perfoliatum L.), Jerusalem artichoke (Helianthus tuberosus), giant knotweed (Fallopia japonicum X bohemica), tall wheatgrass (Agropyron elongatum), and reed canary grass (Phalaris arundinacea) for cultivation under periodically waterlogged soil conditions during the winter half year and implications for soil chemical and biological properties. Examined perennial energy crops coped with periodical waterlogging and showed yields 50% to 150% higher than in the control which was never faced with waterlogging. Root formation was similar in waterlogged and non-waterlogged soil layers. Soil chemical and microbial properties clearly responded to different soil moisture treatments. For example, dehydrogenase activity was two to four times higher in the periodically waterlogged treatment compared to the control. Despite waterlogging, aerobic microbial activity was significantly elevated indicating morphological and metabolic adaptation of the perennial crops to withstand waterlogged conditions. Thus, our results reveal first evidence of a site-adapted biomass production on periodical waterlogged soils through the cultivation of perennial energy crops and for intense plant microbe interactions.
A satellite-based climatology of wind-induced surface temperature anomalies for the Antarctic
(2019)
It is well-known that katabatic winds can be detected as warm signatures in the surface temperature over the slopes of the Antarctic ice sheets. For appropriate synoptic forcing and/or topographic channeling, katabatic surges occur, which result in warm signatures also over adjacent ice shelves. Moderate Resolution Imaging Spectroradiometer (MODIS) ice surface temperature (IST) data are used to detect warm signatures over the Antarctic for the winter periods 2002–2017. In addition, high-resolution (5 km) regional climate model data is used for the years of 2002 to 2016. We present a case study and a climatology of wind-induced IST anomalies for the Ross Ice Shelf and the eastern Weddell Sea. The IST anomaly distributions show maxima around 10–15K for the slopes, but values of more than 25K are also found. Katabatic surges represent a strong climatological signal with a mean warm anomaly of more than 5K on more than 120 days per winter for the Byrd Glacier and the Nimrod Glacier on the Ross Ice Shelf. The mean anomaly for the Brunt Ice Shelf is weaker, and exceeds 5K on about 70 days per winter. Model simulations of the IST are compared to the MODIS IST, and show a very good agreement. The model data show that the near-surface stability is a better measure for the response to the wind than the IST itself.
Abstract: Thermal infrared (TIR) multi-/hyperspectral and sun-induced fluorescence (SIF) approaches together with classic solar-reflective (visible, near-, and shortwave infrared reflectance (VNIR)/SWIR) hyperspectral remote sensing form the latest state-of-the-art techniques for the detection of crop water stress. Each of these three domains requires dedicated sensor technology currently in place for ground and airborne applications and either have satellite concepts under development (e.g., HySPIRI/SBG (Surface Biology and Geology), Sentinel-8, HiTeSEM in the TIR) or are subject to satellite missions recently launched or scheduled within the next years (i.e., EnMAP and PRISMA (PRecursore IperSpettrale della Missione Applicativa, launched on March 2019) in the VNIR/SWIR, Fluorescence Explorer (FLEX) in the SIF). Identification of plant water stress or drought is of utmost importance to guarantee global water and food supply. Therefore, knowledge of crop water status over large farmland areas bears large potential for optimizing agricultural water use. As plant responses to water stress are numerous and complex, their physiological consequences affect the electromagnetic signal in different spectral domains. This review paper summarizes the importance of water stress-related applications and the plant responses to water stress, followed by a concise review of water-stress detection through remote sensing, focusing on TIR without neglecting the comparison to other spectral domains (i.e., VNIR/SWIR and SIF) and multi-sensor approaches. Current and planned sensors at ground, airborne, and satellite level for the TIR as well as a selection of commonly used indices and approaches for water-stress detection using the main multi-/hyperspectral remote sensing imaging techniques are reviewed. Several important challenges are discussed that occur when using spectral emissivity, temperature-based indices, and physically-based approaches for water-stress detection in the TIR spectral domain. Furthermore, challenges with data processing and the perspectives for future satellite missions in the TIR are critically examined. In conclusion, information from multi-/hyperspectral TIR together with those from VNIR/SWIR and SIF sensors within a multi-sensor approach can provide profound insights to actual plant (water) status and the rationale of physiological and biochemical changes. Synergistic sensor use will open new avenues for scientists to study plant functioning and the response to environmental stress in a wide range of ecosystems.
Background
In light of the current biodiversity crisis, DNA barcoding is developing into an essential tool to quantify state shifts in global ecosystems. Current barcoding protocols often rely on short amplicon sequences, which yield accurate identification of biological entities in a community but provide limited phylogenetic resolution across broad taxonomic scales. However, the phylogenetic structure of communities is an essential component of biodiversity. Consequently, a barcoding approach is required that unites robust taxonomic assignment power and high phylogenetic utility. A possible solution is offered by sequencing long ribosomal DNA (rDNA) amplicons on the MinION platform (Oxford Nanopore Technologies).
Findings
Using a dataset of various animal and plant species, with a focus on arthropods, we assemble a pipeline for long rDNA barcode analysis and introduce a new software (MiniBar) to demultiplex dual indexed Nanopore reads. We find excellent phylogenetic and taxonomic resolution offered by long rDNA sequences across broad taxonomic scales. We highlight the simplicity of our approach by field barcoding with a miniaturized, mobile laboratory in a remote rainforest. We also test the utility of long rDNA amplicons for analysis of community diversity through metabarcoding and find that they recover highly skewed diversity estimates.
Conclusions
Sequencing dual indexed, long rDNA amplicons on the MinION platform is a straightforward, cost-effective, portable, and universal approach for eukaryote DNA barcoding. Although bulk community analyses using long-amplicon approaches may introduce biases, the long rDNA amplicons approach signifies a powerful tool for enabling the accurate recovery of taxonomic and phylogenetic diversity across biological communities.
A huge number of clinical studies and meta-analyses have shown that psychotherapy is effective on average. However, not every patient profits from psychotherapy and some patients even deteriorate in treatment. Due to this result and the restricted generalization of clinical studies to clinical practice, a more patient-focused research strategy has emerged. The question whether a particular treatment works for an individual case is the focus of this paradigm. The use of repeated assessments and the feedback of this information to therapists is a major ingredient of patient-focused research. Improving patient outcomes and reducing dropout rates by the use of psychometric feedback seems to be a promising path. Therapists seem to differ in the degree to which they make use of and profit from such feedback systems. This dissertation aims to better understand therapist differences in the context of patient-focused research and the impact of therapists on psychotherapy. Three different studies are included, which focus on different aspects within the field:
Study I (Chapter 5) investigated how therapists use psychometric feedback in their work with patients and how much therapists differ in their usage. Data from 72 therapists treating 648 patients were analyzed. It could be shown that therapists used the psychometric feedback for most of their patients. Substantial variance in the use of feedback (between 27% and 52%) was attributable to therapists. Therapists were more likely to use feedback when they reported being satisfied with the graphical information they received. The results therefore indicated that not only patient characteristics or treatment progress affected the use of feedback.
Study II (Chapter 6) picked up on the idea of analyzing systematic differences in therapists and applied it to the criterion of premature treatment termination (dropout). To answer the question whether therapist effects occur in terms of patients’ dropout rates, data from 707 patients treated by 66 therapists were investigated. It was shown that approximately six percent of variance in dropout rates could be attributed to therapists, even when initial impairment was controlled for. Other predictors of dropout were initial impairment, sex, education, personality styles, and treatment expectations.
Study III (Chapter 7) extends the dissertation by investigating the impact of a transfer from one therapist to another within ongoing treatments. Data from 124 patients who agreed to and experienced a transfer during their treatment were analyzed. A significant drop in patient-rated as well as therapist-rated alliance levels could be observed after a transfer. On average, there seemed to be no difficulties establishing a good therapeutic alliance with the new therapist, although differences between patients were observed. There was no increase in symptom severity due to therapy transfer. Various predictors of alliance and symptom development after transfer were investigated. Impacts on clinical practice were discussed.
Results of the three studies are discussed and general conclusions are drawn. Implications for future research as well as their utility for clinical practice and decision-making are presented.
In this thesis, we consider the solution of high-dimensional optimization problems with an underlying low-rank tensor structure. Due to the exponentially increasing computational complexity in the number of dimensions—the so-called curse of dimensionality—they present a considerable computational challenge and become infeasible even for moderate problem sizes.
Multilinear algebra and tensor numerical methods have a wide range of applications in the fields of data science and scientific computing. Due to the typically large problem sizes in practical settings, efficient methods, which exploit low-rank structures, are essential. In this thesis, we consider an application each in both of these fields.
Tensor completion, or imputation of unknown values in partially known multiway data is an important problem, which appears in statistics, mathematical imaging science and data science. Under the assumption of redundancy in the underlying data, this is a well-defined problem and methods of mathematical optimization can be applied to it.
Due to the fact that tensors of fixed rank form a Riemannian submanifold of the ambient high-dimensional tensor space, Riemannian optimization is a natural framework for these problems, which is both mathematically rigorous and computationally efficient.
We present a novel Riemannian trust-region scheme, which compares favourably with the state of the art on selected application cases and outperforms known methods on some test problems.
Optimization problems governed by partial differential equations form an area of scientific computing which has applications in a variety of areas, ranging from physics to financial mathematics. Due to the inherent high dimensionality of optimization problems arising from discretized differential equations, these problems present computational challenges, especially in the case of three or more dimensions. An even more challenging class of optimization problems has operators of integral instead of differential type in the constraint. These operators are nonlocal, and therefore lead to large, dense discrete systems of equations. We present a novel solution method, based on separation of spatial dimensions and provably low-rank approximation of the nonlocal operator. Our approach allows the solution of multidimensional problems with a complexity which is only slightly larger than linear in the univariate grid size; this improves the state of the art for a particular test problem problem by at least two orders of magnitude.
Because EU water quality policy can result in infrastructure creation or adaptation at the local level across member states, compliance cases are worth examining critically from a sustainable spatial planning perspective. In this study, the 2000 EU Water Framework Directive’s (WFD) reach to local implementation efforts in average towns and cities is shown through the case study of nonconforming household wastewater infrastructure in the German state of Rhineland Palatinate. Seeing wastewater as a socio-technical infrastructure, we ask how the WFD implementation can be understood in the context of local infrastructure development, sustainability, and spatial planning concepts. In particular, this study examines what compliance meant for the centralization or decentralization of local wastewater infrastructure systems—and the sustainability implications for cities
from those choices.
When do anorexic patients perceive their body as too fat? Aggravating and ameliorating factors
(2019)
Objective
Our study investigated body image representations in female patients with anorexia nervosa
and healthy controls using a size estimation with pictures of their own body. We also
explored a method to reduce body image distortions through right hemispheric activation.
Method
Pictures of participants’ own bodies were shown on the left or right visual fields for 130 ms
after presentation of neutral, positive, or negative word primes, which could be self-relevant
or not, with the task of classifying the picture as “thinner than”, “equal to”, or “fatter than”
one’s own body. Subsequently, activation of the left- or right hemispheric through right- or
left-hand muscle contractions for 3 min., respectively. Finally, participants completed the
size estimation task again.
Results
The distorted “fatter than” body image was found only in patients and only when a picture of
their own body appeared on the right visual field (left hemisphere) and was preceded by
negative self-relevant words. This distorted perception of the patients’ body image was
reduced after left-hand muscle contractions (right hemispheric activation).
Discussion
To reduce body image distortions it is advisable to find methods that help anorexia nervosa
patients to increase their self-esteem. The body image distortions were ameliorated after
right hemispheric activation. A related method to prevent distorted body-image representations
in these patients may be Eye Movement Desensitization and Reprocessing (EMDR)
therapy.
Background: Increasing exposure to engineered inorganic nanoparticles takes actually place in both terrestric and aquatic ecosystems worldwide. Although we already know harmful effects of AgNP on the soil bacterial community, information about the impact of the factors functionalization, concentration, exposure time, and soil texture on the AgNP effect expression are still rare. Hence, in this study, three soils of different grain size were exposed for up to 90 days to bare and functionalized AgNP in concentrations ranging from 0.01 to 1.00 mg/kg soil dry weight. Effects on soil microbial community were quantified by various biological parameters, including 16S rRNA gene, photometric, and fluorescence analyses.
Results: Multivariate data analysis revealed significant effects of AgNP exposure for all factors and factor combinations investigated. Analysis of individual factors (silver species, concentration, exposure time, soil texture) in the unifactorial ANOVA explained the largest part of the variance compared to the error variance. In depth analysis of factor combinations revealed even better explanation of variance. For the biological parameters assessed in this study, the matching of soil texture and silver species, and the matching of soil texture and exposure time were the two most relevant factor combinations. The factor AgNP concentration contributed to a lower extent to the effect expression compared to silver species, exposure time and physico–chemical composition of soil.
Conclusions: The factors functionalization, concentration, exposure time, and soil texture significantly impacted the effect expression of AgNP on the soil microbial community. Especially long-term exposure scenarios are strongly needed for the reliable environmental impact assessment of AgNP exposure in various soil types.
Many combinatorial optimization problems on finite graphs can be formulated as conic convex programs, e.g. the stable set problem, the maximum clique problem or the maximum cut problem. Especially NP-hard problems can be written as copositive programs. In this case the complexity is moved entirely into the copositivity constraint.
Copositive programming is a quite new topic in optimization. It deals with optimization over the so-called copositive cone, a superset of the positive semidefinite cone, where the quadratic form x^T Ax has to be nonnegative for only the nonnegative vectors x. Its dual cone is the cone of completely positive matrices, which includes all matrices that can be decomposed as a sum of nonnegative symmetric vector-vector-products.
The related optimization problems are linear programs with matrix variables and cone constraints.
However, some optimization problems can be formulated as combinatorial problems on infinite graphs. For example, the kissing number problem can be formulated as a stable set problem on a circle.
In this thesis we will discuss how the theory of copositive optimization can be lifted up to infinite dimension. For some special cases we will give applications in combinatorial optimization.