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The goal of this thesis is to transfer the logarithmic barrier approach, which led to very efficient interior-point methods for convex optimization problems in recent years, to convex semi-infinite programming problems. Based on a reformulation of the constraints into a nondifferentiable form this can be directly done for convex semi- infinite programming problems with nonempty compact sets of optimal solutions. But, by means of an involved max-term this reformulation leads to nondifferentiable barrier problems which can be solved with an extension of a bundle method of Kiwiel. This extension allows to deal with inexact objective values and subgradient information which occur due to the inexact evaluation of the maxima. Nevertheless we are able to prove similar convergence results as for the logarithmic barrier approach in the finite optimization. In the further course of the thesis the logarithmic barrier approach is coupled with the proximal point regularization technique in order to solve ill-posed convex semi-infinite programming problems too. Moreover this coupled algorithm generates sequences converging to an optimal solution of the given semi-infinite problem whereas the pure logarithmic barrier only produces sequences whose accumulation points are such optimal solutions. If there are certain additional conditions fulfilled we are further able to prove convergence rate results up to linear convergence of the iterates. Finally, besides hints for the implementation of the methods we present numerous numerical results for model examples as well as applications in finance and digital filter design.
This work is concerned with the numerical solution of optimization problems that arise in the context of ground water modeling. Both ground water hydraulic and quality management problems are considered. The considered problems are discretized problems of optimal control that are governed by discretized partial differential equations. Aspects of special interest in this work are inaccurate function evaluations and the ensuing numerical treatment within an optimization algorithm. Methods for noisy functions are appropriate for the considered practical application. Also, block preconditioners are constructed and analyzed that exploit the structure of the underlying linear system. Specifically, KKT systems are considered, and the preconditioners are tested for use within Krylov subspace methods. The project was financed by the foundation Stiftung Rheinland-Pfalz für Innovation and carried out in joint work with TGU GmbH, a company of consulting engineers for ground water and water resources.
The discretization of optimal control problems governed by partial differential equations typically leads to large-scale optimization problems. We consider flow control involving the time-dependent Navier-Stokes equations as state equation which is stamped by exactly this property. In order to avoid the difficulties of dealing with large-scale (discretized) state equations during the optimization process, a reduction of the number of state variables can be achieved by employing a reduced order modelling technique. Using the snapshot proper orthogonal decomposition method, one obtains a low-dimensional model for the computation of an approximate solution to the state equation. In fact, often a small number of POD basis functions suffices to obtain a satisfactory level of accuracy in the reduced order solution. However, the small number of degrees of freedom in a POD based reduced order model also constitutes its main weakness for optimal control purposes. Since a single reduced order model is based on the solution of the Navier-Stokes equations for a specified control, it might be an inadequate model when the control (and consequently also the actual corresponding flow behaviour) is altered, implying that the range of validity of a reduced order model, in general, is limited. Thus, it is likely to meet unreliable reduced order solutions during a control problem solution based on one single reduced order model. In order to get out of this dilemma, we propose to use a trust-region proper orthogonal decomposition (TRPOD) approach. By embedding the POD based reduced order modelling technique into a trust-region framework with general model functions, we obtain a mechanism for updating the reduced order models during the optimization process, enabling the reduced order models to represent the flow dynamics as altered by the control. In fact, a rigorous convergence theory for the TRPOD method is obtained which justifies this procedure also from a theoretical point of view. Benefiting from the trust-region philosophy, the TRPOD method guarantees to save a lot of computational work during the control problem solution, since the original state equation only has to be solved if we intend to update our model function in the trust-region framework. The optimization process itself is completely based on reduced order information only.
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.
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.
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 optimal control of fluid flows described by the Navier-Stokes equations requires massive computational resources, which has led researchers to develop reduced-order models, such as those derived from proper orthogonal decomposition (POD), to reduce the computational complexity of the solution process. The object of the thesis is the acceleration of such reduced-order models through the combination of POD reduced-order methods with finite element methods at various discretization levels. Special stabilization methods required for high-order solution of flow problems with dominant convection on coarse meshes lead to numerical data that is incompatible with standard POD methods for reduced-order modeling. We successfully adapt the POD method for such problems by introducing the streamline diffusion POD method (SDPOD). Using the novel SDPOD method, we experiment with multilevel recursive optimization at Reynolds numbers of Re=400 and Re=10,000.
The subject of this thesis is hypercyclic, mixing, and chaotic C0-semigroups on Banach spaces. After introducing the relevant notions and giving some examples the so called hypercyclicity criterion and its relation with weak mixing is treated. Some new equivalent formulations of the criterion are given which are used to derive a very short proof of the well-known fact that a C0-semigroup is weakly mixing if and only if each of its operators is. Moreover, it is proved that under some "regularity conditions" each hypercyclic C0-semigroup is weakly mixing. Furthermore, it is shown that for a hypercyclic C0-semigroup there is always a dense set of hypercyclic vectors having infinitely differentiable trajectories. Chaotic C0-semigroups are also considered. It is proved that they are always weakly mixing and that in certain cases chaoticity is already implied by the existence of a single periodic point. Moreover, it is shown that strongly elliptic differential operators on bounded C^1-domains never generate chaotic C0-semigroups. A thorough investigation of transitivity, weak mixing, and mixing of weighted compositioin operators follows and complete characterisations of these properties are derived. These results are then used to completely characterise hypercyclicity, weak mixing, and mixing of C0-semigroups generated by first order partial differential operators. Moreover, a characterisation of chaos for these C0-semigroups is attained. All these results are achieved on spaces of p-integrable functions as well as on spaces of continuous functions and illustrated by various concrete examples.
In this thesis, we investigate the quantization problem of Gaussian measures on Banach spaces by means of constructive methods. That is, for a random variable X and a natural number N, we are searching for those N elements in the underlying Banach space which give the best approximation to X in the average sense. We particularly focus on centered Gaussians on the space of continuous functions on [0,1] equipped with the supremum-norm, since in that case all known methods failed to achieve the optimal quantization rate for important Gauss-processes. In fact, by means of Spline-approximations and a scheme based on the Best-Approximations in the sense of the Kolmogorov n-width we were able to attain the optimal rate of convergence to zero for these quantization problems. Moreover, we established a new upper bound for the quantization error, which is based on a very simple criterion, the modulus of smoothness of the covariance function. Finally, we explicitly constructed those quantizers numerically.
Considering the numerical simulation of mathematical models it is necessary to have efficient methods for computing special functions. We will focus our considerations in particular on the classes of Mittag-Leffler and confluent hypergeometric functions. The PhD Thesis can be structured in three parts. In the first part, entire functions are considered. If we look at the partial sums of the Taylor series with respect to the origin we find that they typically only provide a reasonable approximation of the function in a small neighborhood of the origin. The main disadvantages of these partial sums are the cancellation errors which occur when computing in fixed precision arithmetic outside this neighborhood. Therefore, our aim is to quantify and then to reduce this cancellation effect. In the next part we consider the Mittag-Leffler and the confluent hypergeometric functions in detail. Using the method we developed in the first part, we can reduce the cancellation problems by "modifying" the functions for several parts of the complex plane. Finally, in in the last part two other approaches to compute Mittag-Leffler type and confluent hypergeometric functions are discussed. If we want to evaluate such functions on unbounded intervals or sectors in the complex plane, we have to consider methods like asymptotic expansions or continued fractions for large arguments z in modulus.
The thesis studies the question how universal behavior is inherited by the Hadamard product. The type of universality that is considered here is universality by overconvergence; a definition will be given in chapter five. The situation can be described as follows: Let f be a universal function, and let g be a given function. Is the Hadamard product of f and g universal again? This question will be studied in chapter six. Starting with the Hadamard product for power series, a definition for a more general context must be provided. For plane open sets both containing the origin this has already been done. But in order to answer the above question, it becomes necessary to have a Hadamard product for functions that are not holomorphic at the origin. The elaboration of such a Hadamard product and its properties are the second central part of this thesis; chapter three will be concerned with them. The idea of the definition of such a Hadamard product will follow the case already known: The Hadamard product will be defined by a parameter integral. Crucial for this definition is the choice of appropriate integration curves; these will be introduced in chapter two. By means of the Hadamard product- properties it is possible to prove the Hadamard multiplication theorem and the Borel-Okada theorem. A generalization of these theorems will be presented in chapter four.
The subject of this thesis is a homological approach to the splitting theory of PLS-spaces, i.e. to the question for which topologically exact short sequences 0->X->Y->Z->0 of PLS-spaces X,Y,Z the right-hand map admits a right inverse. We show that the category (PLS) of PLS-spaces and continuous linear maps is an additive category in which every morphism admits a kernel and a cokernel, i.e. it is pre-abelian. However, we also show that it is neither quasi-abelian nor semi-abelian. As a foundation for our homological constructions we show the more general result that every pre-abelian category admits a largest exact structure in the sense of Quillen. In the pre-abelian category (PLS) this exact structure consists precisely of the topologically exact short sequences of PLS-spaces. Using a construction of Ext-functors due to Yoneda, we show that one can define for each PLS-space A and every natural number k the k-th abelian-group valued covariant and contravariant Ext-functors acting on the category (PLS) of PLS-spaces, which induce for every topologically exact short sequence of PLS-spaces a long exact sequence of abelian groups and group morphisms. These functors are studied in detail and we establish a connection between the Ext-functors of PLS-spaces and the Ext-functors for LS-spaces. Through this connection we arrive at an analogue of a result for Fréchet spaces which connects the first derived functor of the projective limit with the first Ext-functor and also gives sufficient conditions for the vanishing of the higher Ext-functors. Finally, we show that Ext^k(E,F) = 0 for a k greater or equal than 1, whenever E is a closed subspace and F is a Hausdorff-quotient of the space of distributions, which generalizes a result of Wengenroth that is itself a generalization of results due to Domanski and Vogt.
This thesis introduces a calibration problem for financial market models based on a Monte Carlo approximation of the option payoff and a discretization of the underlying stochastic differential equation. It is desirable to benefit from fast deterministic optimization methods to solve this problem. To be able to achieve this goal, possible non-differentiabilities are smoothed out with an appropriately chosen twice continuously differentiable polynomial. On the basis of this so derived calibration problem, this work is essentially concerned about two issues. First, the question occurs, if a computed solution of the approximating problem, derived by applying Monte Carlo, discretizing the SDE and preserving differentiability is an approximation of a solution of the true problem. Unfortunately, this does not hold in general but is linked to certain assumptions. It will turn out, that a uniform convergence of the approximated objective function and its gradient to the true objective and gradient can be shown under typical assumptions, for instance the Lipschitz continuity of the SDE coefficients. This uniform convergence then allows to show convergence of the solutions in the sense of a first order critical point. Furthermore, an order of this convergence in relation to the number of simulations, the step size for the SDE discretization and the parameter controlling the smooth approximation of non-differentiabilites will be shown. Additionally the uniqueness of a solution of the stochastic differential equation will be analyzed in detail. Secondly, the Monte Carlo method provides only a very slow convergence. The numerical results in this thesis will show, that the Monte Carlo based calibration indeed is feasible if one is concerned about the calculated solution, but the required calculation time is too long for practical applications. Thus, techniques to speed up the calibration are strongly desired. As already mentioned above, the gradient of the objective is a starting point to improve efficiency. Due to its simplicity, finite differences is a frequently chosen method to calculate the required derivatives. However, finite differences is well known to be very slow and furthermore, it will turn out, that there may also occur severe instabilities during optimization which may lead to the break down of the algorithm before convergence has been reached. In this manner a sensitivity equation is certainly an improvement but suffers unfortunately from the same computational effort as the finite difference method. Thus, an adjoint based gradient calculation will be the method of choice as it combines the exactness of the derivative with a reduced computational effort. Furthermore, several other techniques will be introduced throughout this thesis, that enhance the efficiency of the calibration algorithm. A multi-layer method will be very effective in the case, that the chosen initial value is not already close to the solution. Variance reduction techniques are helpful to increase accuracy of the Monte Carlo estimator and thus allow for fewer simulations. Storing instead of regenerating the random numbers required for the Brownian increments in the SDE will be efficient, as deterministic optimization methods anyway require to employ the identical random sequence in each function evaluation. Finally, Monte Carlo is very well suited for a parallelization, which will be done on several central processing units (CPUs).
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.
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.
Extension of inexact Kleinman-Newton methods to a general monotonicity preserving convergence theory
(2011)
The thesis at hand considers inexact Newton methods in combination with algebraic Riccati equation. A monotone convergence behaviour is proven, which enables a non-local convergence. Above relation is transferred to a general convergence theory for inexact Newton methods securing the monotonicity of the iterates for convex or concave mappings. Several application prove the pratical benefits of the new developed theory.
Variational inequality problems constitute a common basis to investigate the theory and algorithms for many problems in mathematical physics, in economy as well as in natural and technical sciences. They appear in a variety of mathematical applications like convex programming, game theory and economic equilibrium problems, but also in fluid mechanics, physics of solid bodies and others. Many variational inequalities arising from applications are ill-posed. This means, for example, that the solution is not unique, or that small deviations in the data can cause large deviations in the solution. In such a situation, standard solution methods converge very slowly or even fail. In this case, so-called regularization methods are the methods of choice. They have the advantage that an ill-posed original problem is replaced by a sequence of well-posed auxiliary problems, which have better properties (like, e.g., a unique solution and a better conditionality). Moreover, a suitable choice of the regularization term can lead to unconstrained auxiliary problems that are even equivalent to optimization problems. The development and improvement of such methods are a focus of current research, in which we take part with this thesis. We suggest and investigate a logarithmic-quadratic proximal auxiliary problem (LQPAP) method that combines the advantages of the well-known proximal-point algorithm and the so-called auxiliary problem principle. Its exploration and convergence analysis is one of the main results in this work. The LQPAP method continues the recent developments of regularization methods. It includes different techniques presented in literature to improve the numerical stability: The logarithmic-quadratic distance function constitutes an interior point effect which allows to treat the auxiliary problems as unconstrained ones. Furthermore, outer operator approximations are considered. This simplifies the numerical solution of variational inequalities with multi-valued operators since, for example, bundle-techniques can be applied. With respect to the numerical practicability, inexact solutions of the auxiliary problems are allowed using a summable-error criterion that is easy to implement. As a further advantage of the logarithmic-quadratic distance we verify that it is self-concordant (in the sense of Nesterov/Nemirovskii). This motivates to apply the Newton method for the solution of the auxiliary problems. In the numerical part of the thesis the LQPAP method is applied to linearly constrained, differentiable and nondifferentiable convex optimization problems, as well as to nonsymmetric variational inequalities with co-coercive operators. It can often be observed that the sequence of iterates reaches the boundary of the feasible set before being close to an optimal solution. Against this background, we present the strategy of under-relaxation, which robustifies the LQPAP method. Furthermore, we compare the results with an appropriate method based on Bregman distances (BrPAP method). For differentiable, convex optimization problems we describe the implementation of the Newton method to solve the auxiliary problems and carry out different numerical experiments. For example, an adaptive choice of the initial regularization parameter and a combination of an Armijo and a self-concordance step size are evaluated. Test examples for nonsymmetric variational inequalities are hardly available in literature. Therefore, we present a geometric and an analytic approach to generate test examples with known solution(s). To solve the auxiliary problems in the case of nondifferentiable, convex optimization problems we apply the well-known bundle technique. The implementation is described in detail and the involved functions and sequences of parameters are discussed. As far as possible, our analysis is substantiated by new theoretical results. Furthermore, it is explained in detail how the bundle auxiliary problems are solved with a primal-dual interior point method. Such investigations have by now only been published for Bregman distances. The LQPAP bundle method is again applied to several test examples from literature. Thus, this thesis builds a bridge between theoretical and numerical investigations of solution methods for variational inequalities.
The main topic of this treatise is the solution of two problems from the general theory of linear partial differential equations with constant coefficients. While surjectivity criteria for linear partial differential operators in spaces of smooth functions over an open subset of euclidean space and distributions were proved by B. Malgrange and L. Hörmander in 1955, respectively 1962, concrete evaluation of these criteria is still a highly non-trivial task. In particular, it is well-known that surjectivity in the space of smooth functions over an open subset of euclidean space does not automatically imply surjectivity in the space of distributions. Though, examples for this fact all live in three or higher dimensions. In 1966, F. Trèves conjectured that in the two dimensional setting surjectivity of a linear partial differential operator on the smooth functions indeed implies surjectivity on the space of distributions. An affirmative solution to this problem is presented in this treatise. The second main result solves the so-called problem of (distributional) parameter dependence for solutions of linear partial differential equations with constant coefficients posed by J. Bonet and P. Domanski in 2006. It is shown that, in dimensions three or higher, this problem in general has a negative solution even for hypoelliptic operators. Moreover, it is proved that the two dimensional case is again an exception, because in this setting the problem of parameter dependence always has a positive solution.