arXiv subjects

Christian Seifert

Publications and source records attributed to Christian Seifert.

At least 19 recordsLinked to original sources

arXiv2026

Localisation of pseudospectra on discrete groups

In this paper we generalise two of the methods and corresponding results from our previous paper ``On spectral inclusion sets and computing the spectra and pseudospectra of bounded linear operators'' [J. Spectr. Theory 14 (2024), 719--804] from tridiagonal operators on $\ell^2(\Z)$ to band operators $A$ on $\ell^2(G,Y)$ with a countable Abelian group $G$ and a Hilbert space $Y$. Again, we cover the pseudospectra of $A$, with error-control, via a union of pseudospectra of finite and moderately sized ``local patches'' of $A$. While a major application is to understand the case $G=\Z^d$ that is immanent in many physical problems, our new approach to the so-called $\tau$ and $\tau_1$ methods immediately extends to countable Abelian groups $G$.

math.SP↗

arXiv2026

Strongly continuous and locally equicontinuous families of operators and their relation to bi-continuity

We study strongly continuous and locally equicontinuous families of operators on sequentially complete Hausdorff locally convex spaces. In case of Saks spaces, we relate the general notions to bi-continuity as well as equitightness. In this way, we recover and also generalise known results for special classes of operator families such as bi-continuous ($C$-)semigroups and ($C$-)cosine families by well-known results for the corresponding families in Hausdorff locally convex spaces.

math.FA↗

arXiv2026

On controllability, observability and stabilizability of the heat equation on discrete graphs

We consider linear control problems for the heat equation of the form $\dot f (t) = -Hf (t) + \mathbf{1}_D u (t)$, $f (0) \in \ell_2 (X,m)$, where $H$ is the weighted Laplacian on a discrete graph $(X,b,m)$, and where $D \subseteq X$ is relatively dense. We show cost-uniform $\alpha$-controllability by means of a weak observability estimate for the corresponding dual observation problem. We discuss optimality of our result as well as consequences on stabilizability properties.

math.OC↗

arXiv2025

Spatial Approximation for Evolutionary Equations

We consider evolutionary equations as introduced by R.\ Picard in 2009 and develop a general theory for approximation which can be seen as a theoretical foundation for numerical analysis for evolutionary equations. To demonstrate the approximation result, we apply it to a spatial discretisation of the heat equation using spectral methods.

math.FA↗

arXiv2024

Duality for Evolutionary Equations with Applications to Null Controllability

We study evolutionary equations in exponentially weighted $\mathrm{L}^{2}$-spaces as introduced by Picard in 2009. First, for a given evolutionary equation, we explicitly describe the $\nu$-adjoint system, which turns out to describe a system backwards in time. We prove well-posedness for the $\nu$-adjoint system. We then apply the thus obtained duality to introduce and study notions of null-controllability for evolutionary equations.

math.AP↗

arXiv2024

PyRIT: A Framework for Security Risk Identification and Red Teaming in Generative AI System

Generative Artificial Intelligence (GenAI) is becoming ubiquitous in our daily lives. The increase in computational power and data availability has led to a proliferation of both single- and multi-modal models. As the GenAI ecosystem matures, the need for extensible and model-agnostic risk identification frameworks is growing. To meet this need, we introduce the Python Risk Identification Toolkit (PyRIT), an open-source framework designed to enhance red teaming efforts in GenAI systems. PyRIT is a model- and platform-agnostic tool that enables red teamers to probe for and identify novel harms, risks, and jailbreaks in multimodal generative AI models. Its composable architecture facilitates the reuse of core building blocks and allows for extensibility to future models and modalities. This paper details the challenges specific to red teaming generative AI systems, the development and features of PyRIT, and its practical applications in real-world scenarios.

cs.CR↗

arXiv2024

Phi-3 Safety Post-Training: Aligning Language Models with a "Break-Fix" Cycle

Recent innovations in language model training have demonstrated that it is possible to create highly performant models that are small enough to run on a smartphone. As these models are deployed in an increasing number of domains, it is critical to ensure that they are aligned with human preferences and safety considerations. In this report, we present our methodology for safety aligning the Phi-3 series of language models. We utilized a "break-fix" cycle, performing multiple rounds of dataset curation, safety post-training, benchmarking, red teaming, and vulnerability identification to cover a variety of harm areas in both single and multi-turn scenarios. Our results indicate that this approach iteratively improved the performance of the Phi-3 models across a wide range of responsible AI benchmarks. Finally, we include additional red teaming strategies and evaluations that were used to test the safety behavior of Phi-3.5-mini and Phi-3.5-MoE, which were optimized for multilingual capabilities.

cs.CL↗

arXiv2024

Approximation of Random Evolution Equations of Parabolic type

In this paper, we present an abstract framework to obtain convergence rates for the approximation of random evolution equations corresponding to a random family of forms determined by finite-dimensional noise. The full discretization error in space, time, and randomness is considered, where polynomial chaos expansion (PCE) is used for the semi-discretization in randomness. The main result are regularity conditions on the random forms under which convergence of polynomial order in randomness is obtained depending on the smoothness of the coefficients and the Sobolev regularity of the initial value. In space and time, the same convergence rates as in the deterministic setting are achieved. To this end, we derive error estimates for vector-valued PCE as well as a quantified version of the Trotter--Kato theorem for form-induced semigroups. We apply the abstract framework to an anisotropic diffusion model with random diffusion coefficients.

math.FA↗

arXiv2024

On solitary waves for the Korteweg--de Vries equation on metric star graphs

We study the Korteweg--de Vries equation on a metric star graph and investigate existence of solitary waves on the metric graph in terms of the coefficients of the equation on each edge, the coupling condition at the central vertex of the star and the speeds of the travelling wave. We show that, with a continuity condition at the vertex, solitary waves can occur exactly when the parameters are chosen in a fairly special manner. We also consider coupling conditions beyond continuity.

math.AP↗

arXiv2023

Trust your BMS: Designing a Lightweight Authentication Architecture for Industrial Networks

With the advent of clean energy awareness and systems that rely on extensive battery usage, the community has seen an increased interest in the development of more complex and secure Battery Management Systems (BMS). In particular, the inclusion of BMS in modern complex systems like electric vehicles and power grids has presented a new set of security-related challenges. A concern is shown when BMS are intended to extend their communication with external system networks, as their interaction can leave many backdoors open that potential attackers could exploit. Hence, it is highly desirable to find a general design that can be used for BMS and its system inclusion. In this work, a security architecture solution is proposed intended for the communication between BMS and other system devices. The aim of the proposed architecture is to be easily applicable in different industrial settings and systems, while at the same time keeping the design lightweight in nature.

cs.CR↗

arXiv2023

Spectral Theory for Schrödinger operators on compact metric graphs with $δ$ and $δ'$ couplings: a survey

Spectral properties of Schrödinger operators on compact metric graphs are studied and special emphasis is put on differences in the spectral behavior between different classes of vertex conditions. We survey recent results especially for $δ$ and $δ'$ couplings and demonstrate the spectral properties on many examples. Amongst other things, properties of the ground state eigenvalue and eigenfunction and the spectral behavior under various perturbations of the metric graph or the vertex conditions are considered.

math.SP↗

arXiv2023

Final state observability estimates and cost-uniform approximate null-controllability for bi-continuous semigroups

We consider final state observability estimates for bi-continuous semigroups on Banach spaces, i.e. for every initial value, estimating the state at a final time $T>0$ by taking into account the orbit of the initial value under the semigroup for $t\in [0,T]$, measured in a suitable norm. We state a sufficient criterion based on an uncertainty relation and a dissipation estimate and provide two examples of bi-continuous semigroups which share a final state observability estimate, namely the Gauss-Weierstrass semigroup and the Ornstein-Uhlenbeck semigroup on the space of bounded continuous functions. Moreover, we generalise the duality between cost-uniform approximate null-controllability and final state observability estimates to the setting of locally convex spaces for the case of bounded and continuous control functions, which seems to be new even for the Banach spaces case.

math.OC↗

arXiv2023

Observability for Non-autonomous Systems

We study non-autonomous observation systems \begin{align*} \dot{x}(t) = A(t) x(t),\quad y(t) = C(t) x(t),\quad x(0) = x_0\in X, \end{align*} where $(A(t))$ is a strongly measurable family of closed operators on a Banach space $X$ and $(C(t))$ is a family of bounded observation operators from $X$ to a Banach space $Y$. Based on an abstract uncertainty principle and a dissipation estimate, we prove that the observation system satisfies a final-state observability estimate in $\mathrm{L}^r(E; Y)$ for measurable subsets $E \subseteq [0,T], T > 0$. We present applications of the above result to families $(A(t))$ of uniformly strongly elliptic differential operators as well as non-autonomous Ornstein-Uhlenbeck operators $P(t)$ on $\mathrm{L}^p(\mathbb{R}^d)$ with observation operators $C(t)u = u|_{Ω(t)}$. In the setting of non-autonomous strongly elliptic operators, we derive necessary and sufficient geometric conditions on the family of sets $(Ω(t))$ such that the corresponding observation system satisfies a final-state observability estimate.

math.FA↗

arXiv2023

Perturbations of non-autonomous second-order abstract Cauchy problems

In this paper we present time-dependent perturbations of second-order non-autonomous abstract Cauchy problems associated to a family of operators with constant domain. We make use of the equivalence to a first-order non-autonomous abstract Cauchy problem in a product space, which we elaborate in full detail. As an application we provide a perturbed non-autonomous wave equation.

math.FA↗

arXiv2023

Final State Observability in Banach spaces with applications to Subordination and Semigroups induced by L{é}vy processes

This paper generalizes the abstract method of proving an observability estimate by combining an uncertainty principle and a dissipation estimate. In these estimates we allow for a large class of growth/decay rates satisfying an integrability condition. In contrast to previous results, we use an iterative argument which enables us to give an asymptotically sharp estimate for the observation constant and which is explicit in the model parameters. We give two types of applications where the extension of the growth/decay rates naturally appear. By exploiting subordination techniques we show how the dissipation estimate of a semigroup transfers to subordinated semigroups. Furthermore, we apply our results to semigroups related to L{é}vy processes.

math.FA↗

arXiv2022

A note on the Lumer--Phillips theorem for bi-continuous semigroups

Given a Banach space $X$ and an additional coarser Hausdorff locally convex topology $τ$ on $X$ we characterise the generators of $τ$-bi-continuous semigroups in the spirit of the Lumer--Phillips theorem, i.e. by means of dissipativity w.r.t.~a directed system of seminorms and a range condition.

math.FA↗

arXiv2022

Observability and null-controllability for parabolic equations in $L_p$-spaces

We study (approximate) null-controllability of parabolic equations in $L_p(\mathbb{R}^d)$ and provide explicit bounds on the control cost. In particular we consider systems of the form $\dot{x}(t) = -A_p x(t) + \mathbf{1}_E u(t)$, $x(0) = x_0\in L_p (\mathbb{R}^d)$, with interior control on a so-called thick set $E \subset \mathbb{R}^d$, where $p\in [1,\infty)$, and where $A$ is an elliptic operator of order $m \in \mathbb{N}$ in $L_p(\mathbb{R}^d)$. We prove null-controllability of this system via duality and a sufficient condition for observability. This condition is given by an uncertainty principle and a dissipation estimate. Our result unifies and generalizes earlier results obtained in the context of Hilbert and Banach spaces. In particular, our result applies to the case $p=1$.

math.FA↗

arXiv2022

Non-autonomous Desch-Schappacher perturbations

We consider time-dependent Desch-Schappacher perturbations of non-autonomous abstract Cauchy problems and apply our result to non-autonomous uniformly strongly elliptic differential operators on $\mathrm{L}^p$-spaces.

math.FA↗