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Matthias Langer

Publications and source records attributed to Matthias Langer.

36 records · Page 2Linked to original sources
arXiv2020

Distributed Training of Deep Learning Models: A Taxonomic Perspective

Distributed deep learning systems (DDLS) train deep neural network models by utilizing the distributed resources of a cluster. Developers of DDLS are required to make many decisions to process their particular workloads in their chosen environment efficiently. The advent of GPU-based deep learning, the ever-increasing size of datasets and deep neural network models, in combination with the bandwidth constraints that exist in cluster environments require developers of DDLS to be innovative in order to train high quality models quickly. Comparing DDLS side-by-side is difficult due to their extensive feature lists and architectural deviations. We aim to shine some light on the fundamental principles that are at work when training deep neural networks in a cluster of independent machines by analyzing the general properties associated with training deep learning models and how such workloads can be distributed in a cluster to achieve collaborative model training. Thereby we provide an overview of the different techniques that are used by contemporary DDLS and discuss their influence and implications on the training process. To conceptualize and compare DDLS, we group different techniques into categories, thus establishing a taxonomy of distributed deep learning systems.

cs.DC↗
arXiv2020

Spectral enclosures for a class of block operator matrices

We prove new spectral enclosures for the non-real spectrum of a class of $2\times2$ block operator matrices with self-adjoint operators $A$ and $D$ on the diagonal and operators $B$ and $-B^*$ as off-diagonal entries. One of our main results resembles Gershgorin's circle theorem. The enclosures are applied to $J$-frame operators.

math.SP↗
arXiv2020

Discrete fragmentation systems in weighted $\ell^1$ spaces

We investigate an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters. We assume that each cluster is composed of identical units (monomers) and we allow mass to be lost, gained or conserved during each fragmentation event. By formulating the initial-value problem for the system as an abstract Cauchy problem (ACP), posed in an appropriate weighted $\ell^1$ space, and then applying perturbation results from the theory of operator semigroups, we prove the existence and uniqueness of physically relevant, classical solutions for a wide class of initial cluster distributions. Additionally, we establish that it is always possible to identify a weighted $\ell^1$ space on which the fragmentation semigroup is analytic, which immediately implies that the corresponding ACP is well posed for any initial distribution belonging to this particular space. We also investigate the asymptotic behaviour of solutions, and show that, under appropriate restrictions on the fragmentation coefficients, solutions display the expected long-term behaviour of converging to a purely monomeric steady state. Moreover, when the fragmentation semigroup is analytic, solutions are shown to decay to this steady state at an explicitly defined exponential rate.

math.FA↗
arXiv2018

Spectral enclosures for non-self-adjoint extensions of symmetric operators

The spectral properties of non-self-adjoint extensions $A_{[B]}$ of a symmetric operator in a Hilbert space are studied with the help of ordinary and quasi boundary triples and the corresponding Weyl functions. These extensions are given in terms of abstract boundary conditions involving an (in general non-symmetric) boundary operator $B$. In the abstract part of this paper, sufficient conditions for sectoriality and m-sectoriality as well as sufficient conditions for $A_{[B]}$ to have a non-empty resolvent set are provided in terms of the parameter $B$ and the Weyl function. Special attention is paid to Weyl functions that decay along the negative real line or inside some sector in the complex plane, and spectral enclosures for $A_{[B]}$ are proved in this situation. The abstract results are applied to elliptic differential operators with local and non-local Robin boundary conditions on unbounded domains, to Schrödinger operators with $δ$-potentials of complex strengths supported on unbounded hypersurfaces or infinitely many points on the real line, and to quantum graphs with non-self-adjoint vertex couplings.

math.SP↗
arXiv2017

Quasi boundary triples and semibounded self-adjoint extensions

In this note semibounded self-adjoint extensions of symmetric operators are investigated with the help of the abstract notion of quasi boundary triples and their Weyl functions. The main purpose is to provide new sufficient conditions on the parameters in the boundary space to induce self-adjoint realizations, and to relate the decay of the Weyl function to estimates on the lower bound of the spectrum. The abstract results are illustrated with uniformly elliptic second order PDEs on domains with non-compact boundaries.

math.SP↗
arXiv2017

Path Laplacian operators and superdiffusive processes on graphs. I. One-dimensional case

We consider a generalization of the diffusion equation on graphs. This generalized diffusion equation gives rise to both normal and superdiffusive processes on infinite one-dimensional graphs. The generalization is based on the $k$-path Laplacian operators $L_{k}$, which account for the hop of a diffusive particle to non-nearest neighbours in a graph. We first prove that the $k$-path Laplacian operators are self-adjoint. Then, we study the transformed $k$-path Laplacian operators using Laplace, factorial and Mellin transforms. We prove that the generalized diffusion equation using the Laplace- and factorial-transformed operators always produce normal diffusive processes independently of the parameters of the transforms. More importantly, the generalized diffusion equation using the Mellin-transformed $k$-path Laplacians $\sum_{k=1}^{\infty}k^{-s}L_{k}$ produces superdiffusive processes when $1<s<3$.

math.FA↗
arXiv2017

Spectrum of $J$-frame operators

A $J$-frame is a frame $\mathcal{F}$ for a Krein space $(\mathcal{H}, [\, , \,])$ which is compatible with the indefinite inner product $[\, , \, ]$ in the sense that it induces an indefinite reconstruction formula that resembles those produced by orthonormal bases in $\mathcal{H}$. With every $J$-frame the so-called $J$-frame operator is associated, which is a self-adjoint operator in the Krein space $\mathcal{H}$. The $J$-frame operator plays an essential role in the indefinite reconstruction formula. In this paper we characterize the class of $J$-frame operators in a Krein space by a $2\times 2$ block operator representation. The $J$-frame bounds of $\mathcal{F}$ are then recovered as the suprema and infima of the numerical ranges of some uniformly positive operators which are build from the entries of the $2\times 2$ block representation. Moreover, this $2\times 2$ block representation is utilized to obtain enclosures for the spectrum of $J$-frame operators, which finally leads to the construction of a square root. This square root allows a complete description of all $J$-frames associated with a given $J$-frame operator.

math.FA↗
arXiv2016

Variational principles for self-adjoint operator functions arising from second-order systems

Variational principles are proved for self-adjoint operator functions arising from variational evolution equations of the form \[ \langle\ddot{z}(t),y \rangle + \mathfrak{d}[\dot{z} (t), y] + \mathfrak{a}_0 [z(t),y] = 0. \] Here $\mathfrak{a}_0$ and $\mathfrak{d}$ are densely defined, symmetric and positive sesquilinear forms on a Hilbert space $H$. We associate with the variational evolution equation an equivalent Cauchy problem corresponding to a block operator matrix $\mathcal{A}$, the forms \[ \mathfrak{t}(λ)[x,y] := λ^2\langle x,y\rangle + λ\mathfrak{d}[x,y] + \mathfrak{a}_0[x,y], \] where $λ\in \mathbb C$ and $x,y$ are in the domain of the form $\mathfrak{a}_0$, and a corresponding operator family $T(λ)$. Using form methods we define a generalized Rayleigh functional and characterize the eigenvalues above the essential spectrum of $\mathcal{A}$ by a min-max and a max-min variational principle. The obtained results are illustrated with a damped beam equation.

math.FA↗
arXiv2015

Trace formulae for Schrödinger operators with singular interactions

Let $Σ\subset\mathbb{R}^d$ be a $C^\infty$-smooth closed compact hypersurface, which splits the Euclidean space $\mathbb{R}^d$ into two domains $Ω_\pm$. In this note self-adjoint Schrödinger operators with $δ$ and $δ'$-interactions supported on $Σ$ are studied. For large enough $m\in\mathbb{N}$ the difference of $m$th powers of resolvents of such a Schrödinger operator and the free Laplacian is known to belong to the trace class. We prove trace formulae, in which the trace of the resolvent power difference in $L^2(\mathbb{R}^d)$ is written in terms of Neumann-to-Dirichlet maps on the boundary space $L^2(Σ)$.

math.SP↗
arXiv2015

Spectral properties of unbounded J-self-adjoint matrices

We study the spectrum of unbounded J-self-adjoint block operator matrices. In particular, we prove enclosures for the spectrum, provide a sufficient condition for the spectrum being real and derive variational principles for certain real eigenvalues even in the presence of non-real spectrum. The latter lead to lower and upper bounds and asymptotic estimates for eigenvalues.

math.SP↗
arXiv2015

Triple Variational Principles for Self-Adjoint Operator Functions

For a very general class of unbounded self-adjoint operator function we prove upper bounds for eigenvalues which lie within arbitrary gaps of the essential spectrum. These upper bounds are given by triple variations. Furthermore, we find conditions which imply that a point is in the resolvent set. For norm resolvent continuous operator functions we show that the variational inequality becomes an equality.

math.SP↗
arXiv2015

Spectral asymptotics for resolvent differences of elliptic operators with $δ$ and $δ^\prime$-interactions on hypersurfaces

We consider self-adjoint realizations of a second-order elliptic differential expression on ${\mathbb R}^n$ with singular interactions of $δ$ and $δ^\prime$-type supported on a compact closed smooth hypersurface in ${\mathbb R}^n$. In our main results we prove spectral asymptotics formulae with refined remainder estimates for the singular values of the resolvent difference between the standard self-adjoint realizations and the operators with a $δ$ and $δ^\prime$-interaction, respectively. Our technique makes use of general pseudodifferential methods, classical results on spectral asymptotics of $ψ$do's on closed manifolds and Krein-type resolvent formulae.

math.SP↗
arXiv2013

Trace formulae and singular values of resolvent power differences of self-adjoint elliptic operators

In this note self-adjoint realizations of second order elliptic differential expressions with non-local Robin boundary conditions on a domain $Ω\subset\dR^n$ with smooth compact boundary are studied. A Schatten--von Neumann type estimate for the singular values of the difference of the $m$th powers of the resolvents of two Robin realizations is obtained, and for $m>\tfrac{n}{2}-1$ it is shown that the resolvent power difference is a trace class operator. The estimates are slightly stronger than the classical singular value estimates by M. Sh. Birman where one of the Robin realizations is replaced by the Dirichlet operator. In both cases trace formulae are proved, in which the trace of the resolvent power differences in $L^2(Ω)$ is written in terms of the trace of derivatives of Neumann-to-Dirichlet and Robin-to-Neumann maps on the boundary space $L^2(\partialΩ)$.

math.SP↗
arXiv2012

Schrödinger operators with delta and delta'-potentials supported on hypersurfaces

Self-adjoint Schrödinger operators with $δ$ and $δ'$-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman--Schwinger principle and a variant of Krein's formula are shown. Furthermore, Schatten--von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with $δ$ and $δ'$-potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity.

math.SP↗
arXiv2010

Spectral estimates for resolvent differences of self-adjoint elliptic operators

The notion of quasi boundary triples and their Weyl functions is an abstract concept to treat spectral and boundary value problems for elliptic partial differential equations. In the present paper the abstract notion is further developed, and general theorems on resolvent differences belonging to operator ideals are proved. The results are applied to second order elliptic differential operators on bounded and exterior domains, and to partial differential operators with $δ$ and $δ'$-potentials supported on hypersurfaces.

math.SP↗
arXiv2009

A remark on Schatten-von Neumann properties of resolvent differences of generalized Robin Laplacians on bounded domains

In this note we investigate the asymptotic behaviour of the $s$-numbers of the resolvent difference of two generalized self-adjoint, maximal dissipative or maximal accumulative Robin Laplacians on a bounded domain $Ω$ with smooth boundary $\partialΩ$. For this we apply the recently introduced abstract notion of quasi boundary triples and Weyl functions from extension theory of symmetric operators together with Krein type resolvent formulae and well-known eigenvalue asymptotics of the Laplace-Beltrami operator on $\partialΩ$. It will be shown that the resolvent difference of two generalized Robin Laplacians belongs to the Schatten-von Neumann class of any order $p$ for which $p>(dimΩ-1)/3$. Moreover, we also give a simple sufficient condition for the resolvent difference of two generalized Robin Laplacians to belong to a Schatten-von Neumann class of arbitrary small order. Our results extend and complement classical theorems due to M.Sh.Birman on Schatten-von Neumann properties of the resolvent differences of Dirichlet, Neumann and self-adjoint Robin Laplacians.

math.SP↗