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This work investigates the industrial applicability of graphics and stream processors in the field of fluid simulations. For this purpose, an explicit Runge-Kutta discontinuous Galerkin method in arbitrarily high order is implemented completely for the hardware architecture of GPUs. The same functionality is simultaneously realized for CPUs and compared to GPUs. Explicit time steppings as well as established implicit methods are under consideration for the CPU. This work aims at the simulation of inviscid, transsonic flows over the ONERA M6 wing. The discontinuities which typically arise in hyperbolic equations are treated with an artificial viscosity approach. It is further investigated how this approach fits into the explicit time stepping and works together with the special architecture of the GPU. Since the treatment of artificial viscosity is close to the simulation of the Navier-Stokes equations, it is reviewed how GPU-accelerated methods could be applied for computing viscous flows. This work is based on a nodal discontinuous Galerkin approach for linear hyperbolic problems. Here, it is extended to non-linear problems, which makes the application of numerical quadrature obligatory. Moreover, the representation of complex geometries is realized using isoparametric mappings. Higher order methods are typically very sensitive with respect to boundaries which are not properly resolved. For this purpose, an approach is presented which fits straight-sided DG meshes to curved geometries which are described by NURBS surfaces. The mesh is modeled as an elastic body and deformed according to the solution of closest point problems in order to minimize the gap to the original spline surface. The sensitivity with respect to geometry representations is reviewed in the end of this work in the context of shape optimization. Here, the aerodynamic drag of the ONERA M6 wing is minimized according to the shape gradient which is implicitly smoothed within the mesh deformation approach. In this context a comparison to the classical Laplace-Beltrami operator is made in a Stokes flow situation.
In this thesis we focus on the development and investigation of methods for the computation of confluent hypergeometric functions. We point out the relations between these functions and parabolic boundary value problems and demonstrate applications to models of heat transfer and fluid dynamics. For the computation of confluent hypergeometric functions on compact (real or complex) intervals we consider a series expansion based on the Hadamard product of power series. It turnes out that the partial sums of this expansion are easily computable and provide a better rate of convergence in comparison to the partial sums of the Taylor series. Regarding the computational accuracy the problem of cancellation errors is reduced considerably. Another important tool for the computation of confluent hypergeometric functions are recurrence formulae. Although easy to implement, such recurrence relations are numerically unstable e.g. due to rounding errors. In order to circumvent these problems a method for computing recurrence relations in backward direction is applied. Furthermore, asymptotic expansions for large arguments in modulus are considered. From the numerical point of view the determination of the number of terms used for the approximation is a crucial point. As an application we consider initial-boundary value problems with partial differential equations of parabolic type, where we use the method of eigenfunction expansion in order to determine an explicit form of the solution. In this case the arising eigenfunctions depend directly on the geometry of the considered domain. For certain domains with some special geometry the eigenfunctions are of confluent hypergeometric type. Both a conductive heat transfer model and an application in fluid dynamics is considered. Finally, the application of several heat transfer models to certain sterilization processes in food industry is discussed.
In this thesis, we study the convergence behavior of an efficient optimization method used for the identification of parameters for underdetermined systems. The research is motivated by optimization problems arising from the estimation of parameters in neural networks as well as in option pricing models. In the first application, we are concerned with neural networks used to forecasting stock market indices. Since neural networks are able to describe extremely complex nonlinear structures they are used to improve the modelling of the nonlinear dependencies occurring in the financial markets. Applying neural networks to the forecasting of economic indicators, we are confronted with a nonlinear least squares problem of large dimension. Furthermore, in this application the number of parameters of the neural network to be determined is usually much larger than the number of patterns which are available for the determination of the unknowns. Hence, the residual function of our least squares problem is underdetermined. In option pricing, an important but usually not known parameter is the volatility of the underlying asset of the option. Assuming that the underlying asset follows a one-factor continuous diffusion model with nonconstant drift and volatility term, the value of an European call option satisfies a parabolic initial value problem with the volatility function appearing in one of the coefficients of the parabolic differential equation. Using this system equation, the estimation of the volatility function is described by a nonlinear least squares problem. Since the adaption of the volatility function is based only on a small number of observed market data these problems are naturally ill-posed. For the solution of these large-scale underdetermined nonlinear least squares problems we use a fully iterative inexact Gauss-Newton algorithm. We show how the structure of a neural network as well as that of the European call price model can be exploited using iterative methods. Moreover, we present theoretical statements for the convergence of the inexact Gauss-Newton algorithm applied to the less examined case of underdetermined nonlinear least squares problems. Finally, we present numerical results for the application of neural networks to the forecasting of stock market indices as well as for the construction of the volatility function in European option pricing models. In case of the latter application, we discretize the parabolic differential equation using a finite difference scheme and we elucidate convergence problems of the discrete scheme when the initial condition is not everywhere differentiable.
The present work considers the normal approximation of the binomial distribution and yields estimations of the supremum distance of the distribution functions of the binomial- and the corresponding standardized normal distribution. The type of the estimations correspond to the classical Berry-Esseen theorem, in the special case that all random variables are identically Bernoulli distributed. In this case we state the optimal constant for the Berry-Esseen theorem. In the proof of these estimations several inequalities regarding the density as well as the distribution function of the binomial distribution are presented. Furthermore in the estimations mentioned above the distribution function is replaced by the probability of arbitrary, not only unlimited intervals and in this new situation we also present an upper bound.
Large scale non-parametric applied shape optimization for computational fluid dynamics is considered. Treating a shape optimization problem as a standard optimal control problem by means of a parameterization, the Lagrangian usually requires knowledge of the partial derivative of the shape parameterization and deformation chain with respect to input parameters. For a variety of reasons, this mesh sensitivity Jacobian is usually quite problematic. For a sufficiently smooth boundary, the Hadamard theorem provides a gradient expression that exists on the surface alone, completely bypassing the mesh sensitivity Jacobian. Building upon this, the gradient computation becomes independent of the number of design parameters and all surface mesh nodes are used as design unknown in this work, effectively allowing a free morphing of shapes during optimization. Contrary to a parameterized shape optimization problem, where a smooth surface is usually created independently of the input parameters by construction, regularity is not preserved automatically in the non-parametric case. As part of this work, the shape Hessian is used in an approximative Newton method, also known as Sobolev method or gradient smoothing, to ensure a certain regularity of the updates, and thus a smooth shape is preserved while at the same time the one-shot optimization method is also accelerated considerably. For PDE constrained shape optimization, the Hessian usually is a pseudo-differential operator. Fourier analysis is used to identify the operator symbol both analytically and discretely. Preconditioning the one-shot optimization by an appropriate Hessian symbol is shown to greatly accelerate the optimization. As the correct discretization of the Hadamard form usually requires evaluating certain surface quantities such as tangential divergence and curvature, special attention is also given to discrete differential geometry on triangulated surfaces for evaluating shape gradients and Hessians. The Hadamard formula and Hessian approximations are applied to a variety of flow situations. In addition to shape optimization of internal and external flows, major focus lies on aerodynamic design such as optimizing two dimensional airfoils and three dimensional wings. Shock waves form when the local speed of sound is reached, and the gradient must be evaluated correctly at discontinuous states. To ensure proper shock resolution, an adaptive multi-level optimization of the Onera M6 wing is conducted using more than 36, 000 shape unknowns on a standard office workstation, demonstrating the applicability of the shape-one-shot method to industry size problems.
This work is concerned with arbitrage bounds for prices of contingent claims under transaction costs, but regardless of other conceivable market frictions. Assumptions on the underlying market are held as weak as convenient for the deduction of meaningful results that make good economic sense. In discrete time we also allow for underlying price processes with uncountable state space. In continuous time the underlying price process is modeled by a semimartingale. For the most part we could avoid any stronger assumptions. The main problems with which we deal in this work are the modelling of (proportional) transaction costs, Fundamental Theorems of Asset Pricing under transaction costs, dual characterizations of arbitrage bounds under transaction costs, Quantile-Hedging under transaction costs, alternatives to the Black-Scholes model in continuous time (under transaction costs). The results apply to stock and currency markets.
Recently, optimization has become an integral part of the aerodynamic design process chain. However, because of uncertainties with respect to the flight conditions and geometrical uncertainties, a design optimized by a traditional design optimization method seeking only optimality may not achieve its expected performance. Robust optimization deals with optimal designs, which are robust with respect to small (or even large) perturbations of the optimization setpoint conditions. The resulting optimization tasks become much more complex than the usual single setpoint case, so that efficient and fast algorithms need to be developed in order to identify, quantize and include the uncertainties in the overall optimization procedure. In this thesis, a novel approach towards stochastic distributed aleatory uncertainties for the specific application of optimal aerodynamic design under uncertainties is presented. In order to include the uncertainties in the optimization, robust formulations of the general aerodynamic design optimization problem based on probabilistic models of the uncertainties are discussed. Three classes of formulations, the worst-case, the chance-constrained and the semi-infinite formulation, of the aerodynamic shape optimization problem are identified. Since the worst-case formulation may lead to overly conservative designs, the focus of this thesis is on the chance-constrained and semi-infinite formulation. A key issue is then to propagate the input uncertainties through the systems to obtain statistics of quantities of interest, which are used as a measure of robustness in both robust counterparts of the deterministic optimization problem. Due to the highly nonlinear underlying design problem, uncertainty quantification methods are used in order to approximate and consequently simplify the problem to a solvable optimization task. Computationally demanding evaluations of high dimensional integrals resulting from the direct approximation of statistics as well as from uncertainty quantification approximations arise. To overcome the curse of dimensionality, sparse grid methods in combination with adaptive refinement strategies are applied. The reduction of the number of discretization points is an important issue in the context of robust design, since the computational effort of the numerical quadrature comes up in every iteration of the optimization algorithm. In order to efficiently solve the resulting optimization problems, algorithmic approaches based on multiple-setpoint ideas in combination with one-shot methods are presented. A parallelization approach is provided to overcome the amount of additional computational effort involved by multiple-setpoint optimization problems. Finally, the developed methods are applied to 2D and 3D Euler and Navier-Stokes test cases verifying their industrial usability and reliability. Numerical results of robust aerodynamic shape optimization under uncertain flight conditions as well as geometrical uncertainties are presented. Further, uncertainty quantification methods are used to investigate the influence of geometrical uncertainties on quantities of interest in a 3D test case. The results demonstrate the significant effect of uncertainties in the context of aerodynamic design and thus the need for robust design to ensure a good performance in real life conditions. The thesis proposes a general framework for robust aerodynamic design attacking the additional computational complexity of the treatment of uncertainties, thus making robust design in this sense possible.
Industrial companies mainly aim for increasing their profit. That is why they intend to reduce production costs without sacrificing the quality. Furthermore, in the context of the 2020 energy targets, energy efficiency plays a crucial role. Mathematical modeling, simulation and optimization tools can contribute to the achievement of these industrial and environmental goals. For the process of white wine fermentation, there exists a huge potential for saving energy. In this thesis mathematical modeling, simulation and optimization tools are customized to the needs of this biochemical process and applied to it. Two different models are derived that represent the process as it can be observed in real experiments. One model takes the growth, division and death behavior of the single yeast cell into account. This is modeled by a partial integro-differential equation and additional multiple ordinary integro-differential equations showing the development of the other substrates involved. The other model, described by ordinary differential equations, represents the growth and death behavior of the yeast concentration and development of the other substrates involved. The more detailed model is investigated analytically and numerically. Thereby existence and uniqueness of solutions are studied and the process is simulated. These investigations initiate a discussion regarding the value of the additional benefit of this model compared to the simpler one. For optimization, the process is described by the less detailed model. The process is identified by a parameter and state estimation problem. The energy and quality targets are formulated in the objective function of an optimal control or model predictive control problem controlling the fermentation temperature. This means that cooling during the process of wine fermentation is controlled. Parameter and state estimation with nonlinear economic model predictive control is applied in two experiments. For the first experiment, the optimization problems are solved by multiple shooting with a backward differentiation formula method for the discretization of the problem and a sequential quadratic programming method with a line search strategy and a Broyden-Fletcher-Goldfarb-Shanno update for the solution of the constrained nonlinear optimization problems. Different rounding strategies are applied to the resulting post-fermentation control profile. Furthermore, a quality assurance test is performed. The outcomes of this experiment are remarkable energy savings and tasty wine. For the next experiment, some modifications are made, and the optimization problems are solved by using direct transcription via orthogonal collocation on finite elements for the discretization and an interior-point filter line-search method for the solution of the constrained nonlinear optimization problems. The second experiment verifies the results of the first experiment. This means that by the use of this novel control strategy energy conservation is ensured and production costs are reduced. From now on tasty white wine can be produced at a lower price and with a clearer conscience at the same time.
In this thesis, global surrogate models for responses of expensive simulations are investigated. Computational fluid dynamics (CFD) have become an indispensable tool in the aircraft industry. But simulations of realistic aircraft configurations remain challenging and computationally expensive despite the sustained advances in computing power. With the demand for numerous simulations to describe the behavior of an output quantity over a design space, the need for surrogate models arises. They are easy to evaluate and approximate quantities of interest of a computer code. Only a few number of evaluations of the simulation are stored for determining the behavior of the response over a whole range of the input parameter domain. The Kriging method is capable of interpolating highly nonlinear, deterministic functions based on scattered datasets. Using correlation functions, distinct sensitivities of the response with respect to the input parameters can be considered automatically. Kriging can be extended to incorporate not only evaluations of the simulation, but also gradient information, which is called gradient-enhanced Kriging. Adaptive sampling strategies can generate more efficient surrogate models. Contrary to traditional one-stage approaches, the surrogate model is built step-by-step. In every stage of an adaptive process, the current surrogate is assessed in order to determine new sample locations, where the response is evaluated and the new samples are added to the existing set of samples. In this way, the sampling strategy learns about the behavior of the response and a problem-specific design is generated. Critical regions of the input parameter space are identified automatically and sampled more densely for reproducing the response's behavior correctly. The number of required expensive simulations is decreased considerably. All these approaches treat the response itself more or less as an unknown output of a black-box. A new approach is motivated by the assumption that for a predefined problem class, the behavior of the response is not arbitrary, but rather related to other instances of the mutual problem class. In CFD, for example, responses of aerodynamic coefficients share structural similarities for different airfoil geometries. The goal is to identify the similarities in a database of responses via principal component analysis and to use them for a generic surrogate model. Characteristic structures of the problem class can be used for increasing the approximation quality in new test cases. Traditional approaches still require a large number of response evaluations, in order to achieve a globally high approximation quality. Validating the generic surrogate model for industrial relevant test cases shows that they generate efficient surrogates, which are more accurate than common interpolations. Thus practical, i.e. affordable surrogates are possible already for moderate sample sizes. So far, interpolation problems were regarded as separate problems. The new approach uses the structural similarities of a mutual problem class innovatively for surrogate modeling. Concepts from response surface methods, variable-fidelity modeling, design of experiments, image registration and statistical shape analysis are connected in an interdisciplinary way. Generic surrogate modeling is not restricted to aerodynamic simulation. It can be applied, whenever expensive simulations can be assigned to a larger problem class, in which structural similarities are expected.
In a paper of 1996 the british mathematician Graham R. Allan posed the question, whether the product of two stable elements is again stable. Here stability describes the solvability of a certain infinite system of equations. Using a method from the theory of homological algebra, it is proved that in the case of topological algebras with multiplicative webs, and thus in all common locally convex topological algebras that occur in standard analysis, the answer of Allan's question is affirmative.