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92. Annual Meeting of the International Association of Applied Mathematics and Mechanics, GAMM 2022, Aachen, Germany, 15 Aug 2022 - 19 Aug 2022; Proceedings in applied mathematics and mechanics 22(1), e202200093 (2023). doi:10.1002/pamm.202200093 special issue: "Special Issue: 92nd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) / Issue Edited by: Ch. Böhm, K. Mang, B. Markert, S. Reese, M. Schmidtchen, J. Waimann, M. Kaliske", Published by Wiley-VCH, Weinheim

To overcome topological constraints and improve the expressiveness of normalizing ow architectures, Wu, Kohler, and Noe introduced stochastic normalizing ows which combine deterministic, learnable ow transformations with stochastic sampling methods. In this paper, we consider stochastic normalizing ows from a Markov chain point of view. In particular, we replace transition densities by general Markov kernels and establish proofs via Radon-Nikodym derivatives, which allows us to incorporate distributions without densities in a sound way. Further, we generalize the results for sampling from posterior distributions as required in inverse problems. The performance of the proposed conditional stochastic normalizing ow is demonstrated by numerical examples.

(Keine Zusammenfassung in deutscher Sprache hinterlegt.) The modeling of physical properties often leads to the task of solving partial differential equations or integral equations. The results of some discretisation and linearisation process are matrix equations or linear systems of equations with special features. In the case of partial differential equations one exploits the local character of the differentiation by using some finite element method or finite difference scheme and gains a sparse system matrix. In the case of (nonlocal) integral operators low rank approximations seem to be the method of choice. These are either given explicitly by some multipole method or panel clustering technique or implicitly by rank revealing decompositions. Both types of matrices can be represented as so-called H-matrices. In this thesis we investigate algorithms that perform the addition, multiplication and inversion of H-matrices approximately. Under moderate assumptions the complexity of these new arithmetics is almost linear (linear up to logarithmic terms of order 1 to 3). The arithmetic operations can be performed adaptively, that is up to some given accuracy epsilon the relative error of the operations is zero. The question arises under which circumstances the inverse of an H-matrix can be approximated by an H-matrix. For the techniques used in this thesis we need very restrictive assumptions, but the numerical examples in the last part indicate that the approximability does not depend on these assumptions.

We introduce the concept of block sparse tensor trains to the mathematical community, which is known to physicists as sector decomposed matrix product states. We generalize it to a class of Laplace-like operators and show that it is well-behaved with the tangent space projection on the tensor train manifold. We discuss known tensor network methods in the context of this block sparsity. We give an explicit construction of the quantum mechanical Hamiltonian, as it appears in Quantum Chemistry, as a tensor train operator with quadratic rank bound. We introduce a novel method to solve the high dimensional Schr\"odinger equation, namely a preconditioned projected conjugate gradient method, where we use the inverse of the Fock operator as a preconditioner. We apply this method to the water and the nitrogen molecule. We discuss how tensor network methods can be used to efficiently learn equation systems based on data. For that the concept of the selection tensor is introduced, which enables us to reuse information for different equations of the equation system. We demonstrate this for a prototype system of equations, the Fermi-Pasta equations. We apply block sparsity to multi polynomial spaces which are used in polynomial regression. We show explicit bounds on the block sizes in the tensor train format for interesting multi polynomial spaces and discuss the expressivity of tensor trains for polynomial regression problems. Wir erläutern das Konzept von blocksparsen Tensortrains für die angewandte Mathematik. Dieses Konzept ist Physikern schon unter dem Namen sektorzerlegte Matrixproduktzustände bekannt. Wir verallgemeinern dieses Konzept für eine Klasse von Lapace-artigen Operatoren und zeigen, dass es sich gutartig mit der Tangentialraumprojektion verhält. Wir zeigen, welche Auswirkungen Blocksparsity auf bekannte Tensornetzwerkmethoden hat. Wir geben eine explizite Konstruktion des quantenmechanischen Hamiltonians als Tensortrainoperator mit quadratischem Rang an, wie er in der Quantumchemie auftaucht. Wir präsentieren eine neue Methode zum Lösen der hochdimensionalen Schrödingergleichung, nämlich ein vorkonditioniertes, projiziertes, konjugiertes Gradientenabstiegsverfahren, wo wir das Inverse des Fockoperators als Vorkonditionierer verwenden. Wir wenden dieses Verfahren auf das Wasser- und das Stickstoffmolekül an. Wir diskutieren, wie Tensornetzwerkmethoden benutzt werden können, um Gleichungssysteme effizient aus Daten zu lernen. Dafür erörtern wir das Konzept des Selectiontensors, welches uns erlaubt Informationen von verschiedenen Gleichungen des Gleichungssystems wiederzuverwenden. Wir zeigen, dass es möglich ist mit Hilfe dieses Ansatzes manche Gleichungssysteme effizienter zu lernen, zum Beispiel die Fermi-Pasta Gleichungen. Wir wenden Blocksparsity auf Multipolynomräume an, welche für polynomielle Regression verwendet werden. Wir zeigen explizite Schranken für die Blockgröße im Tensortrainformat für interessante Multipolynomräume und wir diskutieren die Darstellungsmöglichkeiten von Tensortrains im Kontext von polynomieller Regression.

Dissertation, RWTH Aachen University, 2020; Aachen 1 Online-Ressource (xvi, 205 Seiten) : Illustrationen, Diagramme (2020). = Dissertation, RWTH Aachen University, 2020, In this thesis, we develop an algebraic and graph theoretical reinterpretation of tensor networks and formats. We investigate properties of associated singular values and demonstrate their importance for high-dimensional approximation, in particular for model complexity adaption. This leads us to a concept of stability for iterative optimizations methods which we discuss at length for discrete matrix and tensor completion. We further generalize these ideas to the approximate interpolation of scattered data, and demonstrate the potential of the introduced algorithms on a data set that describes rolling press experiments. These largely algorithmic considerations are supplemented and supported by the theoretical examination of the interrelation between tensor singular values, and its relation to the quantum marginal problem. Tensor networks are essentially multilinear maps which reflect the connections between collections of tensors. In the first part, we discuss how two familiar concepts in mathematics yield an arithmetic that naturally describes such networks, and which formalizes the underlying, simple graph structures through universal assertions. The practicability of this calculus is reflected on by the straightforward implementations, which we provide also of well known algorithms. As a central theorem of this thesis serves the generalizing tree singular value decomposition, which, while not novel in its basic idea, incorporates various gauge conditions that stem from different, corresponding tensor formats. In the second part, we discuss details of high-dimensional, alternating least squares optimization in tree tensor networks, which are those families of tensors that form tree graphs. Due to the special properties of this class of formats, even high-dimensional problems can effectively be handled, in particular when the occurring, linear subproblems are solved via a conjugate gradient method. Subsequent to this introductory segment, we investigate the meaning of singular values in this context. As the model complexity is determined by the tensor ranks of the iterate, the proper calibration of such becomes essential in order to obtain reasonable solutions to recovery problems. Based on a specific definition of stability, we introduce and discuss modifications to standard alternating least squares as well as the relation to reweighted l1-minimization. We in particular demonstrate the use of these concepts for rank-adaptive algorithms. Such are further generalized from the discrete to the continuous setting, which we apply to the approximate interpolation of rolling press simulations. As the singular values associated to tensor networks stem from different matricizations of the same tensor, the question about the interrelation between such arises. In the third part we first show that the tensor feasibility problem is equivalent to a version of the quantum marginal problem. While the latter one has been well known in physics for multiple decades, the tensor version originating from mathematics has only recently been considered. We transfer several results into our setting and subsequently utilize the tree singular value decomposition in order to decouple high-dimensional feasibility problems into much simpler, smaller ones. Last but not least, we specifically consider this situation for the tensor train format, which leads us to cone theory, so-called honeycombs and the application of linear programming algorithms., Published by Aachen

In this thesis, we examine different approaches for efficient high dimensional data acquisition and reconstruction using low rank tensor decomposition techniques. High dimensional here refers to the order of the ambient tensor space in which the data is contained. Examples of such data include tomographic videos, solutions to parametric differential equations and quantum states of many particle systems. The major problem faced in any such high dimensional setting is the exponential scaling of the tensor space dimension with respect to the order, often referred to as the curse of dimensionality. A possible remedy are low rank tensor decomposition techniques, which allow for the storage and manipulation of a rich set of tensors in a data-sparse way, often avoiding the exponential scaling with respect to the order. Following the ideas of compressive sensing, we examine methods to exploit these very efficient representations to acquire high dimensional data from (incomplete) measurements. Our first approach uses randomized methods to construct a low rank representation using repeated random evaluations of the data. The second approach works non-intrusively and aims to reconstruct a complete low rank representation from different types of given incomplete measurements. The latter approach is known as the tensor recovery problem and contains as a special case the popular tensor completion problem. Finally, we employ tensor recovery techniques in a statistical learning setting to obtain low rank approximations of the complete solutions of parametric differential equations. This parametric representation in particular allows to predict the solution for any parameter combinations, even if those were not measured., Diese Arbeit untersucht unterschiedliche Methoden zur effizienten Erfassung und Rekonstruktion von hochdimensionalen Daten. Hochdimensional bezieht sich hierbei auf die Ordnung des Tensorraums, in dem die Daten dargestellt werden. Beispiele für hochdimensionale Daten in diesem Sinne sind tomografische Videos, Lösungen parametrischer partieller Differentialgleichungen, sowie Quantenzustände von Vielteilchensystemen. Das Hauptproblem in der Behandlung solch hochdimensionaler Systeme ist das exponentielle Skalieren der Dimension des Tensorraums bezüglich der Ordnung, der sogenannte Fluch der Dimensionen. Eine Möglichkeit solche Daten dennoch zu verwenden besteht in der Verwendung sogenannter Niedrigrang-Tensorzerlegungen. Diese Zerlegungen verallgemeinern den Matrixrang auf Tensoren höherer Ordnung und erlauben es sehr effizient mit Tensoren von niedrigem Rang zu arbeiten. Speziell erlauben sie teilweise eine linear in der Ordnung skalierende Darstellung entsprechender Niedrigrang-Tensoren. Den Ideen des compressive sensing folgend, untersuchen wir Methoden, die diese extrem effiziente Darstellung ausnutzen um hochdimensionale Daten aus Messungen zu bestimmen. Der erste Ansatz verwendet dazu wiederholte randomisierte Messungen der Daten, um eine Niedrigrang-Repräsentation zu konstruieren. Der zweite Ansatz verfolgt das Ziel eine Niedrigrang-Approximation aus vorgegebenen (unvollständigen) Messungen zu rekonstruieren. Dieser Ansatz ist als tensor recovery Problem bekannt und enthält als Spezialfall das bekannte tensor completion Problem. Im letzten Teil der Arbeit verwenden wir diese Rekonstruktionstechniken, um eine Niedrigrang-Darstellung für Lösungen von parametrischen partiellen Differentialgleichungen zu erlangen. Diese Niedrigrang-Darstellung erlaubt insbesondere, die Lösung einer solchen Differentialgleichung für beliebige Parameterkombinationen vorherzusagen.

Dissertation, RWTH Aachen University, 2019; Aachen 1 Online-Ressource (117 Seiten) : Illustrationen, Diagramme (2019). = Dissertation, RWTH Aachen University, 2019, Brain activity imaging using Electroencephalography (EEG) is a vital aspect of cognitive science and neurology. However, the reliability of brain activity localization based on EEG data has been a hard endeavor so far. In this dissertation, we propose a new, reliable mathematical algorithm for this purpose. First, we will provide an overview of the biological background of methodology in cognitive science. To localize brain activity from EEG data, we need to solve an elliptic partial differential equation (PDE) with a mathematical dipole as a right--hand side. For this, we use Finite Element Methods (FEM). However, FE cannot discretize a mathematical dipole. We solve this issue with regularization. Secondly, we list contemporary regularization approaches. The most successful one - the Venant Approach - has so far lacked a mathematical and theoretical foundation. This dissertation is expanding that approach both theoretically and practically. Thirdly, we formulate a new approach using the idea of the Venant Approach. It originates from the idea to substitute the mathematical dipole with a distribution of monopoles. Monopoles are placed exactly on FE nodes, using the corresponding monopole loads as FE values. The substitution of a mathematical dipole with a monopole distribution results in a new PDE. With assumptions for both monopole loads and electrical conductivity in the PDE, we prove that there is a unique solution for the PDE's ultra-weak formulation using the duality method. We can derive a more intuitive way to compute monopoles with the help of multipole expansion. This involves expanding the potential of the mathematical dipole and the potential of the monopole distribution and comparing the different terms. This new approach is called Multipole Approach. For the Multipole Approach, we can give an error bound with an error decay in the direction of the boundary for a small sphere of uniform conductivity around the dipole location. Fourthly and finally, we employ the most popular approaches, together with the new Multiple Approach in numerical computations. First, we test with unique conductivity in 2D. Second, we employ the approaches on spherical models with different layers of distinct conductivity in 3D. Finally, we compute a goal function scan on one 3D spherical model. We study the Multipole Approach in different configurations and find the optimal one. In this optimal case, the new approach outperforms the contemporary approaches effortlessly., Published by Aachen

Dissertation, RWTH Aachen University, 2017; Aachen, 181 Seiten : Illustrationen (2017). = Dissertation, RWTH Aachen University, 2017, To solve an elliptic PDE eigenvalue problem in practice, typically the finite element discretisation is used. From approximation theory it is known that only the smaller eigenvalues and their corresponding eigenfunctions can be well approximated by the finite element discretisation because the approximation error increases with increasing size of the eigenvalue. The number of well approximable eigenvalues or eigenfunctions, however, is unknown. In this work asymptotic estimates of these quantities are derived. For example, it is shown that for three-dimensional problems under certain smoothness assumptions on the data only the smallest $\Theta (N^{2/5})$ eigenvalues and only the eigenfunctions associated to the smallest $\Theta (N^{1/4})$ eigenvalues can be well approximated by the finite element discretisation, when the $N$-dimensional finite element spaces of piecewise affine functions with uniform mesh refinement are used. To solve the discretised elliptic PDE eigenvalue problem and to compute all well approximable eigenvalues and eigenfunctions, a new method is introduced which combines a recursive version of the automated multi-level substructuring (short AMLS) method with the concept of hierarchical matrices (short $\mathcal{H}$-matrices). AMLS is a domain decomposition technique for the solution of elliptic PDE eigenvalue problems where, after some transformation, a reduced eigenvalue problem is derived whose eigensolutions deliver approximations of the sought eigensolutions of the original problem. Whereas the classical AMLS method is very efficient for elliptic PDE eigenvalue problems posed in two dimensions, it is getting very expensive for three-dimensional problems, due to the fact that it computes the reduced eigenvalue problem via dense matrix operations. This problem is resolved by the use of hierarchical matrices. $\mathcal{H}$-matrices are a data-sparse approximation of dense matrices which, e.g., result from the inversion of the stiffness matrix that is associated to the finite element discretisation of an elliptic PDE operator. The big advantage of $\mathcal{H}$-matrices is that they provide matrix arithmetic with almost linear complexity. This fast $\mathcal{H}$-matrix arithmetic is used for the computation of the reduced eigenvalue problem. Beside this, the size of the reduced eigenvalue problem is bounded by a new recursive version of AMLS which further reduces the costs for the computation and the solution of this problem. Altogether this leads to a new method which is well-suited for three-dimensional problems and which allows us to compute a large amount of eigenpair approximations in optimal complexity., Published by Aachen

Tensors are obtained, e.g., by the consideration of a physical phenomena in a serie of measurements or by the discretisation of a multivariate function. Therein, the phenomena can be time-dependent, e.g., in the case of meteorological data or the evaluation of a functional value can be very expensive, e.g., if the solution of a partiell differential equation ist given by a huge dense linear system. These different situations lead to different restrictions for the generation of the tensor entries, which are grouped into three categories.In this work, we present three different strategies to approximate a high-dimensional tensor, which selects a set of approriate tensor entries for one of the three situations. Combined with our developed calculation routine, the tensor-cross-approximation, an approximation in the hierarchical format is available. The approximation is of significantly less storage complexity than the original tensor and can be used for further calculation instead of the orinigal one.We consider in detail how much the approximation accuracy ist influenced by changing the occuring variables of the tensor and the strategy. On this basis, we point their strengths and weaknesses out. We also looking beyond our calculation routine by comparing it with different optimization methods.All considerations and observations are joined together by given an advice in which case which method with which strategy is the most suitable one and how to use the strengths of the selected strategy in the best way.