08 Fakultät Mathematik und Physik
Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/9
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Item Open Access Long wave approximation over and beyond the natural time scale(2024) Hofbauer, Sarah; Schneider, Guido (Prof. Dr.)Item Open Access From simplicial groups to crossed squares(2022) Asiki, Natalia-MariaSimplicial groups are defined to be contravariant functors from the simplex category to the category of groups. The truncation functor maps a simplicial group to a [2,0]-simplicial group, satisfying the Conduché condition. A functor from the category of [2,0]-simplicial groups to the category of crossed squares is constructed, following Porter. It is shown that the latter functor is not an equivalence of categories. In addition, Loday's variant of the resulting crossed square is constructed and shown to be isomorphic to Porter's variant.Item Open Access Stability of compact symmetric spaces(2022) Semmelmann, Uwe; Weingart, GregorIn this article, we study the stability problem for the Einstein-Hilbert functional on compact symmetric spaces following and completing the seminal work of Koiso on the subject. We classify in detail the irreducible representations of simple Lie algebras with Casimir eigenvalue less than the Casimir eigenvalue of the adjoint representation and use this information to prove the stability of the Einstein metrics on both the quaternionic and Cayley projective plane. Moreover, we prove that the Einstein metrics on quaternionic Grassmannians different from projective spaces are unstable.Item Open Access Gendo-Frobenius algebras and comultiplication(2022) Yırtıcı, ÇiğdemGendo-Frobenius algebras are a common generalisation of Frobenius algebras and of gendo-symmetric algebras. A comultiplication is constructed for gendo-Frobenius algebras, which specialises to the known comultiplications on Frobenius and on gendo-symmetric algebras. In addition, Frobenius algebras are shown to be precisely those gendo-Frobenius algebras that have a counit compatible with this comultiplication. Moreover, a new characterisation of gendo-Frobenius algebras is given. This new characterisation is a key for constructing the comultiplication of gendo-Frobenius algebras.Item Open Access Learning with high-dimensional data(2022) Fischer, Simon; Steinwart, Ingo (Prof. Dr.)This thesis is divided into three parts. In the first part we introduce a framework that allows us to investigate learning scenarios with restricted access to the data. We use this framework to model high-dimensional learning scenarios as an infinite-dimensional one in which the learning algorithm has only access to some finite-dimensional projections of the data. Finally, we provide a prototypical example of such an infinite-dimensional classification problem in which histograms can achieve polynomial learning rates. In the second part we present some individual results that might by useful for the investigation of kernel-based learning methods using Gaussian kernels in high- or infinite-dimensional learning problems. To be more precise, we present log-covering number bounds for Gaussian reproducing kernel Hilbert spaces on general bounded subsets of the Euclidean space. Unlike previous results in this direction we focus on small explicit constants and their dependence on crucial parameters such as the kernel width as well as the size and dimension of the underlying space. Afterwards, we generalize these bounds to Gaussian kernels defined on special infinite-dimensional compact subsets of the sequence space ℓ_2. More precisely, the considered domains are given by the image of the unit ℓ_∞-ball under some diagonal operator. In the third part we contribute some new insights to the compactness properties of diagonal operators from ℓ_p to ℓ_q for p ≠ q.Item Open Access Existence and non-existence of breather solutions on necklace graphs(2023) Kielwein, Tobias; Schneider, Guido (Prof. Dr.)Item Open Access Algebraic analogues of resolution of singularities, quasi-hereditary covers and Schur algebras(2021) Santos Cruz, Tiago Miguel dos; Koenig, Steffen (Prof. Dr.)In algebraic geometry, a resolution of singularities is, roughly speaking, a replacement of a local commutative Noetherian ring of infinite global dimension by a local commutative Noetherian ring of finite global dimension. In representation theory, an analogous problem is asking to resolve algebras of infinite global dimension by algebras of finite global dimension. In addition, such resolutions should have nicer properties to help us study the representation theory of algebras of infinite global dimension. This motivates us to take split quasi-hereditary covers as these algebraic analogues of resolutions of singularities and measure their quality using generalisations of dominant dimension and deformation results based on change of rings techniques. The Schur algebra, SR(n,d) with n ≥ d, together with the Schur functor is a classical example of a split quasi-hereditary cover of the group algebra of the symmetric group, RSd , for every commutative Noetherian ring R. The block algebras of the classical category O, together with their projective-injective module, are split quasi-hereditary covers of subalgebras of coinvariant algebras. In this thesis, we study split quasi-hereditary covers, and their quality, of some cellular algebras over commutative Noetherian rings. The quality of a split quasi-hereditary cover can be measured by the fully faithfulness of the Schur functor on standard modules and on m-fold extensions of standard modules. Over fields, the dominant dimension controls the quality of the split quasi-hereditary cover of KSd formed by the Schur algebra SK(n,d) and the Schur functor. In particular, this quality improves by increasing the characteristic of the ground field. To understand the integral cases, the classical concept of dominant dimension is not useful since in most cases there are no projective-injective modules. Using relative homological algebra, we develop and study a new concept of dominant dimension, which we call relative dominant dimension, for Noetherian algebras which are projective over the ground ring making this concept suitable for the integral setup. For simplicity, we call Noetherian algebras which are projective over the ground ring just projective Noetherian algebras. While developing the theory of relative dominant dimension, we generalize the Morita-Tachikawa correspondence for projective Noetherian algebras and we prove that computations of relative dominant dimension over projective Noetherian algebras can be reduced to computations of dominant dimension over finite-dimensional algebras over algebraically closed fields. Using relative dominant dimension, concepts like Morita algebras and gendo-symmetric algebras can be defined for Noetherian algebras. We compute the relative dominant dimension of Schur algebras SR(n,d) for every commutative Noetherian ring R. Using such computations together with deformation results that involve the spectrum of the ground ring R we determine the quality of the split quasi-hereditary covers of RSd , (SR(n,d), V⊗d) formed by the Schur algebra SR(n,d) and the Schur functor HomSR(n,d)(V⊗d,−): SR(n,d)-mod → RSd-mod for all regular Noetherian rings. Over local commutative regular rings R, the quality of (SR(n,d), V⊗d) depends only on the relative dominant dimension and on R containing a field or not. For this cover, the quality improves compared with the finite-dimensional case whenever the local commutative Noetherian ring does not contain a field. This theory is also applied to q-Schur algebras and Iwahori-Hecke algebras of the symmetric group. In full generality, we prove that the quality of a split quasi-hereditary cover of a finite-dimensional algebra B is bounded above by the number of non-isomorphic simple B-modules. Other split quasi-hereditary algebras that we study in this thesis are deformations of block algebras of the Bernstein-Gelfand-Gelfand category O of a semi-simple Lie algebra. These deformations provide split quasi-hereditary covers of deformations of subalgebras of coinvariant algebras. We compute the relative dominant dimensions of these block algebras and we determine the quality of these covers. In these deformations, the quality dramatically improves compared with the finite-dimensional case. Using approximation theory to generalize once more the concept of dominant dimension to relative dominant dimension with respect to direct summands of the characteristic tilting module, we find new split quasi-hereditary covers. In particular, the relative dominant dimension of a characteristic tilting module of SR(n,d) with respect to V⊗d is a lower bound of the quality of a split quasi-hereditary cover of the cellular algebra EndSR(n,d)(V⊗d)op, independent of the natural numbers n and d. This split quasi-hereditary cover involves the Ringel dual of the Schur algebra SR(n,d). Using this technology for deformations of block algebras of the classical BGG category O of a semi-simple Lie algebra, we obtain a new proof for Ringel self-duality of the blocks of the classical BGG category O of a complex semi-simple Lie algebra. Here, the uniqueness of split quasi-hereditary covers of deformations of subalgebras of coinvariant algebras with higher quality is the crucial factor to deduce Ringel self-duality.Item Open Access Regression from linear models to neural networks : double descent, active learning, and sampling(2023) Holzmüller, David; Steinwart, Ingo (Prof. Dr.)Regression, that is, the approximation of functions from (noisy) data, is a ubiquitous task in machine learning and beyond. In this thesis, we study regression in three different settings. First, we study the double descent phenomenon in non-degenerate unregularized linear regression models, proving that these models are always very noise-sensitive when the number of parameters is close to the number of samples. Second, we study batch active learning algorithms for neural network regression from a more applied perspective: We introduce a framework for building existing and new algorithms and provide a large-scale benchmark showing that a new algorithm can achieve state-of-the-art performance. Third, we study convergence rates for non-log-concave sampling and log-partition estimation algorithms, including approximation-based methods, and prove many results on optimal rates, efficiently achievable rates, multi-regime behaviors, reductions, and the relation to optimization.Item Open Access Efficient simulation of challenging PDE problems on CPU and GPU clusters(2021) Schirwon, Malte; Göddeke, Dominik (Prof. Dr.)The main contribution of this dissertation is to show how efficient parallelization techniques for numerical simulations of partial differential equations (PDEs) can be developed and which aspects have to be considered in order to obtain the best possible performance. For this purpose, the target platforms range from high-performance workstations to small clusters and up to supercomputers. In particular, we focus on platforms accelerated by graphics cards. We emphasize that the efficient numerical simulation of PDE problems comprises and combines, in novel ways, aspects from numerical analysis, numerical methods (algorithmics, data structures and other areas more related to computer science) and hardware details. Many models in science, engineering and economics are based on systems of PDEs. The choice of modeling techniques, the implementation of numerical solution techniques, as well as the chosen target platform limit the accuracy and the duration of the simulation. Increasing the accuracy and/or reducing the duration of the simulation is usually not possible without efficient software. Based on three application scenarios, we adapt already existing methodologies and algorithms to the target platforms or change the way they are implemented in order to achieve optimal efficiency. As a guiding scheme, we consider the challenging case of unstructured data and schemes. The first application is the wave propagation in optical fibers. We present an MPI-parallel implementation that is particularly suitable for small clusters. %Here, we change the numerical method and the implementation technique to increase efficiency and decrease runtime. The second application scenario is the flow in porous media. Based on both applications, we develop implementation techniques that increase their efficiency. Furthermore, we present an adapted version of a neighborhood algorithm that further increases the efficiency for current graphics cards. The increased efficiency and reduced runtime allows to perform more complex simulations. %For example, higher resolutions can be simulated or more physical parameters can be included. One of theses applications is considered to be the third application, which is seismic wave propagation and waveform inversion. The feasibility of developing efficient implementations for computationally powerful target platforms permits us to consider the inversion of seismic waves in viscoelastic materials. In particular, we present an inversion scheme that also allows us to determine the damping parameters of the viscoelastic material. In addition, regularization methods and a modified solver method are presented, which can be used for a more efficient solution of such problems.Item Unknown Analysis of target data-dependent greedy kernel algorithms : convergence rates for f-, f· P- and f/P-greedy(2022) Wenzel, Tizian; Santin, Gabriele; Haasdonk, BernardData-dependent greedy algorithms in kernel spaces are known to provide fast converging interpolants, while being extremely easy to implement and efficient to run. Despite this experimental evidence, no detailed theory has yet been presented. This situation is unsatisfactory, especially when compared to the case of the data-independent P-greedy algorithm, for which optimal convergence rates are available, despite its performances being usually inferior to the ones of target data-dependent algorithms. In this work, we fill this gap by first defining a new scale of greedy algorithms for interpolation that comprises all the existing ones in a unique analysis, where the degree of dependency of the selection criterion on the functional data is quantified by a real parameter. We then prove new convergence rates where this degree is taken into account, and we show that, possibly up to a logarithmic factor, target data-dependent selection strategies provide faster convergence. In particular, for the first time we obtain convergence rates for target data adaptive interpolation that are faster than the ones given by uniform points, without the need of any special assumption on the target function. These results are made possible by refining an earlier analysis of greedy algorithms in general Hilbert spaces. The rates are confirmed by a number of numerical examples.