In this paper, we investigate the question of the spectrally equivalence of the so- called Pseudo-Laplacian to the usual discrete Laplacian in order to use hierarchical preconditioners for this more complicate matrix. The spectral equivalence is shown to be equivalent to a Brezzi-type inequality, which is fulfilled for the finite element spaces considered here.

Mathcad gilt in den unterschiedlichsten Bereichen, z.B. in den Ingenieurwissenschaften, der Mathematik, der Physik, der Biologie oder sogar der Qualitätssicherung als hervorragendes Werkzeug zur übersichtlichen Darstellung komplexer Berechnungen. Sollten die enthaltenen Funktionalitäten nicht mehr ausreichen, besteht die Möglichkeit, Mathcad mit Hilfe von User-DLLs zu erweitern. Diese Erweiterung kann perfekt als Schnittstelle zwischen Mathcad und anderen Softwarepaketen genutzt werden. Die von der inuTech GmbH entwickelte Klassenbibliothek Diffpack zur Simulation und numerischen Lösung von Differentialgleichungen aus den verschiedensten Bereichen eignet sich hervorragend, erforderliche Funktionalitäten für Mathcad zu implementieren. Mathcad kann somit zur Parametrisierung, für Berechnungen und zur Darstellung der Ergebnisse verwendet werden, während Diffpack die Lösung der partiellen Differentialgleichung, z.B. mittels FEM, übernimmt.

Pulse Detonation Engines (PDEs) are propulsion systems that use repeated detonations to generate thrust. Currently in early stages of development, PDEs have been theorised to have advantages over current deflagration based engines. Air-breathing PDEs could attain higher specific impulse values and operate at higher Mach numbers than today's air-breathing engines, while Pulse Detonation Rocket Engines (PDREs) could become a lighter, cheaper, and more reliable alternative to traditional rocket engines. There are still however, many technological hurdles to overcome before PDEs can be developed into practical propulsion systems, one major barrier being management of their immense heat loads. This thesis outlines the development of a numerical model for determining thrust performance and heat load characteristics of PDEs. The model is based on a set of analytical equations which characterise the gas dynamics inside the engine throughout it's cyclic process. Being numerically light -when compared to CFD analysis- the model allows for fast turnaround of results and the ability to sweep through parameters to determine optimum operating conditions to maximise engine performance and reduce heat load. In this study, the working principles of the model are described and it's outputs are validated against data from published experimental and numerical studies. The model is then used to conduct a comprehensive parametric study on the effects of various reactant combinations, operating conditions, and engine geometries on engine thrust, specific impulse and heat load. Lastly, a brief study is conducted on the feasibility of regenerative cooling for PDEs, using model outputs to determine if a heat balance can be achieved and the performance losses and complications that would result.

This thesis considers classical methods to solve stochastic control problems and valuation problems from financial mathematics numerically. To this end, (linear) partial differential equations (PDEs) in non-divergence form or the optimality conditions known as the (nonlinear) Hamilton-Jacobi-Bellman (HJB) equations are solved by means of finite differences, volumes and elements. We consider all of these three approaches in detail after a thorough introduction to stochastic control problems and discuss various solution terms including classical solutions, strong solutions, weak solutions and viscosity solutions. A particular role in this thesis play degenerate problems. Here, a new model for the optimal control of an energy storage facility is developed which extends the model introduced in [Chen, Forsyth (2007)]. This four-dimensional HJB equation is solved by the classical finite difference Kushner-Dupuis scheme [Kushner, Dupuis (2001)] and a semi-Lagrangian variant which are both discussed in detail. Additionally, a convergence proof of the standard scheme in the setting of parabolic HJB equations is given. Finite volume schemes are another classical method to solve partial differential equations numerically. Sharing similarities to both finite difference and finite element schemes we develop a vertex-centered dual finite volume scheme. We discuss convergence properties and apply the scheme to the solution of HJB equations, which has not been done in such a broad context, to the best of our knowledge. Astonishingly, this is one of the first times the finite volume approach is systematically discussed for the solution of HJB equations. Furthermore, we give many examples which show advantages and disadvantages of the approach. Finally, we investigate novel tailored non-conforming finite element approximations of second-order PDEs in non-divergence form, utilizing finite-element Hessian recovery strategies to approximate second derivatives in the equation. We study approximations with both continuous and discontinuous trial functions. Of particular interest are a-priori and a-posteriori error estimates as well as adaptive finite element methods. In numerical experiments our method is compared with other approaches known from the literature. We discuss implementations of all three approaches in MATLAB (finite differences and volumes) and FEniCS (finite elements) publicly available in GitHub repositories under https://github.com/janblechschmidt. Many numerical experiments show convergence properties as well as pros and cons of the respective approach. Additionally, a new postprocessing procedure for policies obtained from numerical solutions of HJB equations is developed which improves the accuracy of control laws and their incurred values.

This dissertation addresses the flow control problem in hyperbolic heat transfer mechanisms. It raises in concentrated distributed solar collectors to enhance their production efficiency under the unpredictable variations of the solar energy and the external disturbances. These factors which are either locally measured (the solar irradiance) or inaccessible for measurement (the collectors’ cleanliness) affect the source term of the distributed model and represent a major difficulty for the control design. Moreover, the temperature in the collector can only be measured at the boundaries. In this dissertation, we propose new adaptive control approaches to provide the adequate level of heat while coping with the unpredictable varying disturbances. First, we design model based control strategies for a better efficiency, in terms of accuracy and response time, with a relatively reduced complexity. Second, we enhance the controllers with on-line adaptation laws to continuously update the efficient value of the external conditions. In this study, we approach the control problem using both, the infinite dimensional model (late lumping) and a finite dimensional approximate representation (early lumping). For the early lumping approach, we introduce a new reduced order bilinear approximate model for system analysis and control design. This approximate state representation is then used to derive a nonlinear state feedback resorting to Lyapunov stability theory. To compensate for the external disturbances and the approximation uncertainties, an adaptive controller is developed based on a phenomenological representation of the system dynamics. For the late lumping approach, we propose two PDE based controllers by stabilization of the reference tracking error distributed profile. The control laws are explicitly defined as functions of the available measurement. The first one is obtained using a direct approach for error stabilization while the second one is derived through a nonlinear mapping. Furthermore, we endow the nonlinear controllers with an adaptation law to cope with the unpredictable unmeasured disturbances. The proposed adaptation law is based on a Proportional plus Integral correction feedback. We show that the control objectives with the required performance can be achieved following both approaches, but yet are conditioned with the physical limitations of the system.

This thesis studies novel and efficient computational sampling methods for appli- cations in three types of stochastic inversion problems: seismic waveform inversion, filtering problems, and static parameter estimation. A primary goal of a large class of seismic inverse problems is to detect parameters that characterize an earthquake. We are interested to solve this task by analyzing the full displacement time series at a given set of seismographs, but approaching the full waveform inversion with the standard Monte Carlo (MC) method is prohibitively expensive. So we study tools that can make this computation feasible. As part of the inversion problem, we must evaluate the misfit between recorded and synthetic seismograms efficiently. We employ as misfit function the Wasserstein metric origi- nally suggested to measure the distance between probability distributions, which is becoming increasingly popular in seismic inversion. To compute the expected values of the misfits, we use a sampling algorithm called Multi-Level Monte Carlo (MLMC). MLMC performs most of the sampling at a coarse space-time resolution, with only a few corrections at finer scales, without compromising the overall accuracy. We further investigate the Wasserstein metric and MLMC method in the context of filtering problems for partially observed diffusions with observations at periodic time intervals. Particle filters can be enhanced by considering hierarchies of discretizations to reduce the computational effort to achieve a given tolerance. This methodology is called Multi-Level Particle Filter (MLPF). However, particle filters, and consequently MLPFs, suffer from particle ensemble collapse, which requires the implementation of a resampling step. We suggest for one-dimensional processes a resampling procedure based on optimal Wasserstein coupling. We show that it is beneficial in terms of computational costs compared to standard resampling procedures. Finally, we consider static parameter estimation for a class of continuous-time state-space models. Unbiasedness of the gradient of the log-likelihood is an important property for gradient ascent (descent) methods to ensure their convergence. We propose a novel unbiased estimator of the gradient of the log-likelihood based on a double-randomization scheme. We use this estimator in the stochastic gradient ascent method to recover unknown parameters of the dynamics.

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A method of sensor location selection is introduced for distributed parameter systems. In this method, the sensitivities of spatial outputs to model parameters are computed by a model and transformed via continuous wavelet transforms into the time-scale domain to characterize the shape attributes of output sensitivities and accentuate their differences. Regions are then sought in the time-scale plane wherein the wavelet coefficient of an output-sensitivity surpasses all the others’ as indication of the output sensitivity’s uniqueness. This method yields a comprehensive account of identifiability each output provides for the model parameters as the basis of output selection. This output selection strategy is evaluated for a numerical case of pollutant dispersion by advection and discussion in a two-dimensional area.

We give a concise explication of the theory of level set methods for modeling motion of an interface as well as the numerical implementation of these methods. We then introduce the geometry of a camera and the mathematical models for 3D reconstruction with a few examples both simulated and from a real camera. We finally describe the model for 3D surface reconstruction from n-camera views using level set methods.