Relativistic LCAO with Minimax Principle and New Balanced Basis Sets

Relativistic density functional theory is widely applied in molecular calculations with heavy atoms, where relativistic and correlation effects are on the same footing. Variational stability of the Dirac Hamiltonian is a very important field of research from the beginning of relativistic molecular calculations on, among efforts for accuracy, efficiency, and density functional formulation, etc. Approximations of one- or two-component methods and searching for suitable basis sets are two major means for good projection power against the negative continuum. The minimax two-component spinor linear combination of atomic orbitals (LCAO) is applied in the present work for both light and super-heavy one-electron systems, providing good approximations in the whole energy spectrum, being close to the benchmark minimax finite element method (FEM) values and without spurious and contaminated states, in contrast to the presence of these artifacts in the traditional four-component spinor LCAO. The variational stability assures that minimax LCAO is bounded from below. New balanced basis sets, kinetic and potential defect balanced (TVDB), following the minimax idea, are applied with the Dirac Hamiltonian. Its performance in the same super-heavy one-electron quasi-molecules shows also very good projection capability against variational collapse, as the minimax LCAO is taken as the best projection to compare with. The TVDB method has twice as many basis coefficients as four-component spinor LCAO, which becomes now linear and overcomes the disadvantage of great time-consumption in the minimax method. The calculation with both the TVDB method and the traditional LCAO method for the dimers with elements in group 11 of the periodic table investigates their difference. New bigger basis sets are constructed than in previous research, achieving high accuracy within the functionals involved. Their difference in total energy is much smaller than the basis incompleteness error, showing that the traditional four-spinor LCAO keeps enough projection power from the numerical atomic orbitals and is suitable in research on relativistic quantum chemistry. In scattering investigations for the same comparison purpose, the failure of the traditional LCAO method of providing a stable spectrum with increasing size of basis sets is contrasted to the TVDB method, which contains no spurious states already without pre-orthogonalization of basis sets. Keeping the same conditions including the accuracy of matrix elements shows that the variational instability prevails over the linear dependence of the basis sets. The success of the TVDB method manifests its capability not only in relativistic quantum chemistry but also for scattering and under the influence of strong external electronic and magnetic fields. The good accuracy in total energy with large basis sets and the good projection property encourage wider research on different molecules, with better functionals, and on small effects.

On the relationship between the Method of Least Squares and Gram-Schmidt orthogonalization

The method of Least Squares is due to Carl Friedrich Gauss. The Gram-Schmidt orthogonalization method is of much younger date. A method for solving Least Squares Problems is developed which automatically results in the appearance of the Gram-Schmidt orthogonalizers. Given these orthogonalizers an induction-proof is available for solving Least Squares Problems.

Echtzeitsimulation großer Grafikmodelle

Die vorliegende Arbeit entstand während meiner Zeit als wissenschaftlicher Mitarbeiter im Fachgebiet Technische Informatik an der Universität Kassel. Im Rahmen dieser Arbeit werden der Entwurf und die Implementierung eines Cluster-basierten verteilten Szenengraphen gezeigt. Bei der Implementierung des verteilten Szenengraphen wurde von der Entwicklung eines eigenen Szenengraphen abgesehen. Stattdessen wurde ein bereits vorhandener Szenengraph namens OpenSceneGraph als Basis für die Entwicklung des verteilten Szenengraphen verwendet. Im Rahmen dieser Arbeit wurde eine Clusterunterstützung in den vorliegenden OpenSceneGraph integriert. Bei der Erweiterung des OpenSceneGraphs wurde besonders darauf geachtet den vorliegenden Szenengraphen möglichst nicht zu verändern. Zusätzlich wurde nach Möglichkeit auf die Verwendung und Integration externer Clusterbasierten Softwarepakete verzichtet. Für die Verteilung des OpenSceneGraphs wurde auf Basis von Sockets eine eigene Kommunikationsschicht entwickelt und in den OpenSceneGraph integriert. Diese Kommunikationsschicht wurde verwendet um Sort-First- und Sort-Last-basierte Visualisierung dem OpenSceneGraph zur Verfügung zu stellen. Durch die Erweiterung des OpenScenGraphs um die Cluster-Unterstützung wurde eine Ansteuerung beliebiger Projektionssysteme wie z.B. einer CAVE ermöglicht. Für die Ansteuerung einer CAVE wurden mittels VRPN diverse Eingabegeräte sowie das Tracking in den OpenSceneGraph integriert. Durch die Anbindung der Geräte über VRPN können diese Eingabegeräte auch bei den anderen Cluster-Betriebsarten wie z.B. einer segmentierten Anzeige verwendet werden. Die Verteilung der Daten auf den Cluster wurde von dem Kern des OpenSceneGraphs separat gehalten. Damit kann eine beliebige OpenSceneGraph-basierte Anwendung jederzeit und ohne aufwendige Modifikationen auf einem Cluster ausgeführt werden. Dadurch ist der Anwender in seiner Applikationsentwicklung nicht behindert worden und muss nicht zwischen Cluster-basierten und Standalone-Anwendungen unterscheiden.

Downscaling and Modeling the Effects of Climate Change on Hydrology and Water Resources in the Upper Blue Nile River Basin, Ethiopia

The Upper Blue Nile River Basin (UBNRB) located in the western part of Ethiopia, between 7° 45’ and 12° 45’N and 34° 05’ and 39° 45’E has a total area of 174962 km2 . More than 80% of the population in the basin is engaged in agricultural activities. Because of the particularly dry climate in the basin, likewise to most other regions of Ethiopia, the agricultural productivity depends to a very large extent on the occurrence of the seasonal rains. This situation makes agriculture highly vulnerable to the impact of potential climate hazards which are about to inflict Africa as a whole and Ethiopia in particular. To analyze these possible impacts of future climate change on the water resources in the UBNRB, in the first part of the thesis climate projection for precipitation, minimum and maximum temperatures in the basin, using downscaled predictors from three GCMs (ECHAM5, GFDL21 and CSIRO-MK3) under SRES scenarios A1B and A2 have been carried out. The two statistical downscaling models used are SDSM and LARS-WG, whereby SDSM is used to downscale ECHAM5-predictors alone and LARS-WG is applied in both mono-model mode with predictors from ECHAM5 and in multi-model mode with combined predictors from ECHAM5, GFDL21 and CSIRO-MK3. For the calibration/validation of the downscaled models, observed as well as NCEP climate data in the 1970 - 2000 reference period is used. The future projections are made for two time periods; 2046-2065 (2050s) and 2081-2100 (2090s). For the 2050s future time period the downscaled climate predictions indicate rise of 0.6°C to 2.7°C for the seasonal maximum temperatures Tmax, and of 0.5°C to 2.44°C for the minimum temperatures Tmin. Similarly, during the 2090s the seasonal Tmax increases by 0.9°C to 4.63°C and Tmin by 1°C to 4.6°C, whereby these increases are generally higher for the A2 than for the A1B scenario. For most sub-basins of the UBNRB, the predicted changes of Tmin are larger than those of Tmax. Meanwhile, for the precipitation, both downscaling tools predict large changes which, depending on the GCM employed, are such that the spring and summer seasons will be experiencing decreases between -36% to 1% and the autumn and winter seasons an increase of -8% to 126% for the two future time periods, regardless of the SRES scenario used. In the second part of the thesis the semi-distributed, physically based hydrologic model, SWAT (Soil Water Assessment Tool), is used to evaluate the impacts of the above-predicted future climate change on the hydrology and water resources of the UBNRB. Hereby the downscaled future predictors are used as input in the SWAT model to predict streamflow of the Upper Blue Nile as well as other relevant water resources parameter in the basin. Calibration and validation of the streamflow model is done again on 1970-2000 measured discharge at the outlet gage station Eldiem, whereby the most sensitive out the numerous “tuneable” calibration parameters in SWAT have been selected by means of a sophisticated sensitivity analysis. Consequently, a good calibration/validation model performance with a high NSE-coefficient of 0.89 is obtained. The results of the future simulations of streamflow in the basin, using both SDSM- and LARS-WG downscaled output in SWAT reveal a decline of -10% to -61% of the future Blue Nile streamflow, And, expectedly, these obviously adverse effects on the future UBNRB-water availibiliy are more exacerbated for the 2090’s than for the 2050’s, regardless of the SRES.

Multisensor Modeling Underwater with Uncertain Information

This thesis develops an approach to the construction of multidimensional stochastic models for intelligent systems exploring an underwater environment. It describes methods for building models by a three- dimensional spatial decomposition of stochastic, multisensor feature vectors. New sensor information is incrementally incorporated into the model by stochastic backprojection. Error and ambiguity are explicitly accounted for by blurring a spatial projection of remote sensor data before incorporation. The stochastic models can be used to derive surface maps or other representations of the environment. The methods are demonstrated on data sets from multibeam bathymetric surveying, towed sidescan bathymetry, towed sidescan acoustic imagery, and high-resolution scanning sonar aboard a remotely operated vehicle.

Statistical Object Recognition

Two formulations of model-based object recognition are described. MAP Model Matching evaluates joint hypotheses of match and pose, while Posterior Marginal Pose Estimation evaluates the pose only. Local search in pose space is carried out with the Expectation--Maximization (EM) algorithm. Recognition experiments are described where the EM algorithm is used to refine and evaluate pose hypotheses in 2D and 3D. Initial hypotheses for the 2D experiments were generated by a simple indexing method: Angle Pair Indexing. The Linear Combination of Views method of Ullman and Basri is employed as the projection model in the 3D experiments.

Geometry and Photometry in 3D Visual Recognition

The report addresses the problem of visual recognition under two sources of variability: geometric and photometric. The geometric deals with the relation between 3D objects and their views under orthographic and perspective projection. The photometric deals with the relation between 3D matte objects and their images under changing illumination conditions. Taken together, an alignment-based method is presented for recognizing objects viewed from arbitrary viewing positions and illuminated by arbitrary settings of light sources.

Stereo-Based Head Pose Tracking Using Iterative Closest Point and Normal Flow Constraint

In this text, we present two stereo-based head tracking techniques along with a fast 3D model acquisition system. The first tracking technique is a robust implementation of stereo-based head tracking designed for interactive environments with uncontrolled lighting. We integrate fast face detection and drift reduction algorithms with a gradient-based stereo rigid motion tracking technique. Our system can automatically segment and track a user's head under large rotation and illumination variations. Precision and usability of this approach are compared with previous tracking methods for cursor control and target selection in both desktop and interactive room environments. The second tracking technique is designed to improve the robustness of head pose tracking for fast movements. Our iterative hybrid tracker combines constraints from the ICP (Iterative Closest Point) algorithm and normal flow constraint. This new technique is more precise for small movements and noisy depth than ICP alone, and more robust for large movements than the normal flow constraint alone. We present experiments which test the accuracy of our approach on sequences of real and synthetic stereo images. The 3D model acquisition system we present quickly aligns intensity and depth images, and reconstructs a textured 3D mesh. 3D views are registered with shape alignment based on our iterative hybrid tracker. We reconstruct the 3D model using a new Cubic Ray Projection merging algorithm which takes advantage of a novel data structure: the linked voxel space. We present experiments to test the accuracy of our approach on 3D face modelling using real-time stereo images.

Image Based Rendering Using Algebraic Techniques

This paper presents an image-based rendering system using algebraic relations between different views of an object. The system uses pictures of an object taken from known positions. Given three such images it can generate "virtual'' ones as the object would look from any position near the ones that the two input images were taken from. The extrapolation from the example images can be up to about 60 degrees of rotation. The system is based on the trilinear constraints that bind any three view so fan object. As a side result, we propose two new methods for camera calibration. We developed and used one of them. We implemented the system and tested it on real images of objects and faces. We also show experimentally that even when only two images taken from unknown positions are given, the system can be used to render the object from other view points as long as we have a good estimate of the internal parameters of the camera used and we are able to find good correspondence between the example images. In addition, we present the relation between these algebraic constraints and a factorization method for shape and motion estimation. As a result we propose a method for motion estimation in the special case of orthographic projection.

On Convergence Properties of the EM Algorithm for Gaussian Mixtures

"Expectation-Maximization'' (EM) algorithm and gradient-based approaches for maximum likelihood learning of finite Gaussian mixtures. We show that the EM step in parameter space is obtained from the gradient via a projection matrix $P$, and we provide an explicit expression for the matrix. We then analyze the convergence of EM in terms of special properties of $P$ and provide new results analyzing the effect that $P$ has on the likelihood surface. Based on these mathematical results, we present a comparative discussion of the advantages and disadvantages of EM and other algorithms for the learning of Gaussian mixture models.

Priors Stabilizers and Basis Functions: From Regularization to Radial, Tensor and Additive Splines

We had previously shown that regularization principles lead to approximation schemes, as Radial Basis Functions, which are equivalent to networks with one layer of hidden units, called Regularization Networks. In this paper we show that regularization networks encompass a much broader range of approximation schemes, including many of the popular general additive models, Breiman's hinge functions and some forms of Projection Pursuit Regression. In the probabilistic interpretation of regularization, the different classes of basis functions correspond to different classes of prior probabilities on the approximating function spaces, and therefore to different types of smoothness assumptions. In the final part of the paper, we also show a relation between activation functions of the Gaussian and sigmoidal type.

A Nonparametric Approach to Pricing and Hedging Derivative Securities via Learning Networks

We propose a nonparametric method for estimating derivative financial asset pricing formulae using learning networks. To demonstrate feasibility, we first simulate Black-Scholes option prices and show that learning networks can recover the Black-Scholes formula from a two-year training set of daily options prices, and that the resulting network formula can be used successfully to both price and delta-hedge options out-of-sample. For comparison, we estimate models using four popular methods: ordinary least squares, radial basis functions, multilayer perceptrons, and projection pursuit. To illustrate practical relevance, we also apply our approach to S&P 500 futures options data from 1987 to 1991.

Uncertainty Propagation in Model-Based Recognition

Building robust recognition systems requires a careful understanding of the effects of error in sensed features. Error in these image features results in a region of uncertainty in the possible image location of each additional model feature. We present an accurate, analytic approximation for this uncertainty region when model poses are based on matching three image and model points, for both Gaussian and bounded error in the detection of image points, and for both scaled-orthographic and perspective projection models. This result applies to objects that are fully three- dimensional, where past results considered only two-dimensional objects. Further, we introduce a linear programming algorithm to compute the uncertainty region when poses are based on any number of initial matches. Finally, we use these results to extend, from two-dimensional to three- dimensional objects, robust implementations of alignmentt interpretation- tree search, and ransformation clustering.

Certified Rapid Solution of Parametrized Linear Elliptic Equations: Application to Parameter Estimation

We present a technique for the rapid and reliable evaluation of linear-functional output of elliptic partial differential equations with affine parameter dependence. The essential components are (i) rapidly uniformly convergent reduced-basis approximations — Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N (optimally) selected points in parameter space; (ii) a posteriori error estimation — relaxations of the residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs; and (iii) offline/online computational procedures — stratagems that exploit affine parameter dependence to de-couple the generation and projection stages of the approximation process. The operation count for the online stage — in which, given a new parameter value, we calculate the output and associated error bound — depends only on N (typically small) and the parametric complexity of the problem. The method is thus ideally suited to the many-query and real-time contexts. In this paper, based on the technique we develop a robust inverse computational method for very fast solution of inverse problems characterized by parametrized partial differential equations. The essential ideas are in three-fold: first, we apply the technique to the forward problem for the rapid certified evaluation of PDE input-output relations and associated rigorous error bounds; second, we incorporate the reduced-basis approximation and error bounds into the inverse problem formulation; and third, rather than regularize the goodness-of-fit objective, we may instead identify all (or almost all, in the probabilistic sense) system configurations consistent with the available experimental data — well-posedness is reflected in a bounded "possibility region" that furthermore shrinks as the experimental error is decreased.

An Immersed Interface Method for the Incompressible Navier-Stokes Equations in Irregular Domains

We present an immersed interface method for the incompressible Navier Stokes equations capable of handling rigid immersed boundaries. The immersed boundary is represented by a set of Lagrangian control points. In order to guarantee that the no-slip condition on the boundary is satisfied, singular forces are applied on the fluid at the immersed boundary. The forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are interpolated using cubic splines. The strength of singular forces is determined by solving a small system of equations at each time step. The Navier-Stokes equations are discretized on a staggered Cartesian grid by a second order accurate projection method for pressure and velocity.