CP decomposition and low-rank approximation of antisymmetric tensors

For the antisymmetric tensors the paper examines a low-rank approximation which is represented via only three vectors. We describe a suitable low-rank format and propose an alternating least squares structure-preserving algorithm for finding such approximation. The case of partial antisymmetry is also discussed. The algorithms are implemented in Julia programming language and their numerical performance is discussed.Comment: 16 pages, 4 table

Hypermatrix factors for string and membrane junctions

The adjoint representations of the Lie algebras of the classical groups SU(n), SO(n), and Sp(n) are, respectively, tensor, antisymmetric, and symmetric products of two vector spaces, and hence are matrix representations. We consider the analogous products of three vector spaces and study when they appear as summands in Lie algebra decompositions. The Z3-grading of the exceptional Lie algebras provide such summands and provides representations of classical groups on hypermatrices. The main natural application is a formal study of three-junctions of strings and membranes. Generalizations are also considered.Comment: 25 pages, 4 figures, presentation improved, minor correction

n-ary algebras: a review with applications

This paper reviews the properties and applications of certain n-ary generalizations of Lie algebras in a self-contained and unified way. These generalizations are algebraic structures in which the two entries Lie bracket has been replaced by a bracket with n entries. Each type of n-ary bracket satisfies a specific characteristic identity which plays the r\^ole of the Jacobi identity for Lie algebras. Particular attention will be paid to generalized Lie algebras, which are defined by even multibrackets obtained by antisymmetrizing the associative products of its n components and that satisfy the generalized Jacobi identity (GJI), and to Filippov (or n-Lie) algebras, which are defined by fully antisymmetric n-brackets that satisfy the Filippov identity (FI). Three-Lie algebras have surfaced recently in multi-brane theory in the context of the Bagger-Lambert-Gustavsson model. Because of this, Filippov algebras will be discussed at length, including the cohomology complexes that govern their central extensions and their deformations (Whitehead's lemma extends to all semisimple n-Lie algebras). When the skewsymmetry of the n-Lie algebra is relaxed, one is led the n-Leibniz algebras. These will be discussed as well, since they underlie the cohomological properties of n-Lie algebras. The standard Poisson structure may also be extended to the n-ary case. We shall review here the even generalized Poisson structures, whose GJI reproduces the pattern of the generalized Lie algebras, and the Nambu-Poisson structures, which satisfy the FI and determine Filippov algebras. Finally, the recent work of Bagger-Lambert and Gustavsson on superconformal Chern-Simons theory will be briefly discussed. Emphasis will be made on the appearance of the 3-Lie algebra structure and on why the A_4 model may be formulated in terms of an ordinary Lie algebra, and on its Nambu bracket generalization.Comment: Invited topical review for JPA Math.Theor. v2: minor changes, references added. 120 pages, 318 reference

Unconventional contributions to dynamical low-rank approximation of tree tensor networks

Die vorliegende Arbeit befasst sich mit der numerischen Zeitintegration von hoch-dimensionalen, zeit-abhängigen, gewöhnlichen Differentialgleichungen, welche beispielsweise bei der Diskretisierung von partiellen Differentialgleichungen auftreten. Treten in den Problemen sehr hohe Dimensionen auf, sind Standardtechniken nicht mehr durchführbar. Die Anzahl und Größe der Objekte, welche gespeichert und verarbeitet werden müssen, übersteigt bei Weitem die Kapazitäten eines Standard-Computers. Dieser Umstand wird häufig als der Fluch der Dimension bezeichnet. Deshalb müssen andere Ansätze zur Behandlung solcher Probleme herangezogen werden, wie zum Beispiel dynamische Approximationen von niedrigem Rang. Dynamische Approximationen von niedrigem Rang beruhen auf der Projektion der Zeitableitung einer zeitabhängigen Lösung eines gegebenen Problems auf den Tangentialraum eines Lösungsraums, welcher eine niedrigere Komplexität aufweist. Die Mannigfaltigkeit der Matrizen von festem Rang r sind ein Beispiel für einen solchen Lösungsraums. Auf dieser speziellen Mannigfaltigkeit wurde mit Hilfe einer Zerlegung der Projektion (=Projektions-Splitting) ein Integrator vorgestellt. Dieser erhält die spezielle Struktur der Objekte bei, was ein effizientes Speichern ermöglicht. Weiter weist der Integrator Robustheit bezüglich kleiner Singulärwerte auf, was essentiell ist um eine ausreichende Genauigkeit der Approximation zu gewährleisten. Für hoch-dimensionale Probleme existieren diverse Mannigfaltigkeiten von niedrigem Rang, auf welchen analog Integratoren vorgestellt wurden, die ebenfalls auf einer Zerlegung der Projektion beruhen. Die Literatur ist reich an Arbeiten zu dynamischen Approximationen von niedrigem Rang für hoch-dimensionale, niedrig-rangige und schleifen-freie Darstellungen, wie zum Beispiel Tucker-Tensoren oder Tensoren in hierarchischer Tucker Darstellung. Wir präsentieren hier die Erweiterung des Projektions-Splitting Integrators für Matrizen auf die Klasse der Tensor-Netzwerke, welche nach Konstruktion alle oben genannten Darstellungen verallgemeinert. Weiter wird eine nicht-triviale Modifikation des Projektions-Splitting Integrators vorgestellt, welche die (Schief-)Symmetrie von Matrizen und Tucker-Tensoren erhält. Diese Modifikation wurde auf Matrizen von niedrigem Rang und Tensoren in Tucker Darstellung verallgemeinert. Aufgrund der ungewöhnlichen Herleitung und Konstruktion des neuen Integrators, bezeichnen wir diesen als unkonventionellen Integrator. Im Unterschied zum Projektions-Splitting Integrator für Matrizen und Tucker-Tensoren, ermöglicht der unkonventionelle Integrator paralleles Rechnen. Er erhält die ursprünglichen vorteilhaften Eigenschaften des Projektions-Splitting Integrators für Matrizen und Tucker-Tensoren und bietet mehr Stabilität bei stark dissipativen Problemen.The present thesis deals with the numerical time-integration of high-dimensional time-dependent ordinary differential equations arising from, e.g. the discretization of high-dimensional time-dependent partial differential equations. Because the size of such problems is assumed to be extremely large, standard discretization techniques are not feasible. The number of quantities needed to be stored and treated exceeds standard capacities of common computational devices; a circumstance usually referred to as curse of dimensionality. A different ansatz is required to be used, i.e. the dynamical low-rank approximation. Dynamical low-rank approximation consists of projecting the time-derivative of the time-dependent solution of the given problem onto the tangent space of a search space of lower complexity, e.g. the manifold of low-rank matrices of fixed rank r. In the manifold of low-rank matrices, an efficient numerical integrator based on a projector-splitting approach has been recently proposed. This numerical integrator has been proven to retain a low-memory footprint and to be robust with respect to the presence of small singular values, desirable property which is needed for achieving sufficient precision in the final approximation. Different low-rank manifolds exist in the high-dimensional setting, and analogous projector-splitting integrators have been there proposed. In the literature, exhaustive and complete research has been carried out for dynamical low-rank approximation of high-dimensional low-rank loop-free formats, such as tensors in Tucker format and tensors in hierarchical Tucker format. We present here the extension of the projector-splitting integrator for matrices to the most general class of tree tensor networks, which by construction includes all mentioned low-rank loop-free formats. Furthermore, a non-trivial modification of the projector-splitting integrator, preserving (skew-)symmetry for matrices and tensors in Tucker format, is presented. The latter results are generalized to low-rank matrices and tensors in Tucker format. Due to its unusual derivation and construction, the new derived numerical integrator is referred to as the unconventional integrator. In contrast to the original projector-splitting integrator for matrices and tensors in Tucker format, the new unconventional integrator introduces more parallelism. It preserves the original excellent properties of the projector-splitting integrator for matrices and tensors in Tucker format and provides more stability when strong dissipative problems are considered

Tensor Networks and Hierarchical Tensors for the Solution of High-dimensional Partial Differential Equations

Hierarchical tensors can be regarded as a generalisation, preserving many crucial features, of the singular value decomposition to higher-order tensors. For a given tensor product space, a recursive decomposition of the set of coordinates into a dimension tree gives a hierarchy of nested subspaces and corresponding nested bases. The dimensions of these subspaces yield a notion of multilinear rank. This rank tuple, as well as quasi-optimal low-rank approximations by rank truncation, can be obtained by a hierarchical singular value decomposition. For fixed multilinear ranks, the storage and operation complexity of these hierarchical representations scale only linearly in the order of the tensor. As in the matrix case, the set of hierarchical tensors of a given multilinear rank is not a convex set, but forms an open smooth manifold. A number of techniques for the computation of low-rank approximations have been developed, including local optimisation techniques on Riemannian manifolds as well as truncated iteration methods, which can be applied for solving high-dimensional partial differential equations. In a number of important cases, quasi-optimality of approximation ranks and computational complexity have been analysed. This article gives a survey of these developments. We also discuss applications to problems in uncertainty quantification, to the solution of the electronic Schrödinger equation in the strongly correlated regime, and to the computation of metastable states in molecular dynamics

HOT–Lines: Tracking Lines in Higher Order Tensor Fields

Tensors occur in many areas of science and engineering. Especially, they are used to describe charge, mass and energy transport (i.e. electrical conductivity tensor, diffusion tensor, thermal conduction tensor resp.) If the locale transport pattern is complicated, usual second order tensor representation is not sufficient. So far, there are no appropriate visualization methods for this case. We point out similarities of symmetric higher order tensors and spherical harmonics. A spherical harmonic representation is used to improve tensor glyphs. This paper unites the definition of streamlines and tensor lines and generalizes tensor lines to those applications where second order tensors representations fail. The algorithm is tested on the tractography problem in diffusion tensor magnetic resonance imaging (DT-MRI) and improved for this special application

MOND via Matrix Gravity

MOND theory has arisen as a promising alternative to dark matter in explaining the collection of discrepancies that constitute the so-called missing mass problem. The MOND paradigm is briefly reviewed. It is shown that MOND theory can be incorporated in the framework of the recently proposed Matrix Gravity. In particular, we demonstrate that Matrix Gravity contains MOND as a particular case, which adds to the validity of Matrix Gravity and proves it is deserving of further inquiry.Comment: 31 pages, minor correction

Matematické metody výpočtů elektronové struktury velkých systémů

This thesis focuses on mathematical methods of the quantum chemistry. It consists of several thematic parts. The first part focuses on tensor numerical methods which serve as a~tool for storing and numerical treatment of large multidimensional data. We focus on an efficient numerical representation of several types of basis functions that can be used in electronic structure calculations. In the second part we present our development of an~optimization algorithm intended for solving Kohn-Sham equations. It can be understood as an alternative to standard iterative methods, where problems with the convergence to the ground state energy occur. Finally, our parallel software for electronic structure calculations based on the Hartree-Fock approximation and the density functional theory together with achieved results is presented.Tato disertační práce se zaměřuje na matematické metody kvantové chemie a skládá se z několika tematických okruhů. V první části se věnuje tenzorovým numerickým metodám jakožto efektivnímu nástroji pro práci s rozsáhlýmí vícedimenzionálními numerickými daty. V rámci našeho výzkumu se zabýváme numerickou reprezentací bázových funkcí využívaných ve výpočtech elektronových struktur. Ve druhé části se věnujeme vývoji optimalizačního algoritmu, který je určen pro řešení Kohnovy-Shamovy rovnice. Algoritmus je primárně určen jako alternativa ke standardním metodám, u nichž jsou známy problémy s konvergencí k energii základního stavu. Na závěr se věnujeme námi vyvíjenému paralelnímu softwaru pro výpočty elektronových struktur založených na Hartreeho-Fockově aproximaci a teorii funkcionálu hustoty. V této části jsou rovněž prezentovány vybrané dosažené výsledky.470 - Katedra aplikované matematikyvyhově