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Hendrik Vogt

Publications and source records attributed to Hendrik Vogt.

At least 19 recordsLinked to original sources
arXiv2024

Sharp Gaussian upper bounds for Schrödinger semigroups on the half-line

In 1998, V. Liskevich and Y. Semenov showed sharp Gaussian upper bounds for Schrödinger semigroups on $\mathbb R^3$ with potentials satisfying a global Kato class condition. Using similar basic ideas we show sharp Gaussian upper bounds for Schrödinger semigroups on the half-line, also assuming a suitable global Kato class condition. Our proof strategy includes a new technique of weighted ultracontractivity estimates.

math.FA↗
arXiv2023

Defensive Perception: Estimation and Monitoring of Neural Network Performance under Deployment

In this paper, we propose a method for addressing the issue of unnoticed catastrophic deployment and domain shift in neural networks for semantic segmentation in autonomous driving. Our approach is based on the idea that deep learning-based perception for autonomous driving is uncertain and best represented as a probability distribution. As autonomous vehicles' safety is paramount, it is crucial for perception systems to recognize when the vehicle is leaving its operational design domain, anticipate hazardous uncertainty, and reduce the performance of the perception system. To address this, we propose to encapsulate the neural network under deployment within an uncertainty estimation envelope that is based on the epistemic uncertainty estimation through the Monte Carlo Dropout approach. This approach does not require modification of the deployed neural network and guarantees expected model performance. Our defensive perception envelope has the capability to estimate a neural network's performance, enabling monitoring and notification of entering domains of reduced neural network performance under deployment. Furthermore, our envelope is extended by novel methods to improve the application in deployment settings, including reducing compute expenses and confining estimation noise. Finally, we demonstrate the applicability of our method for multiple different potential deployment shifts relevant to autonomous driving, such as transitions into the night, rainy, or snowy domain. Overall, our approach shows great potential for application in deployment settings and enables operational design domain recognition via uncertainty, which allows for defensive perception, safe state triggers, warning notifications, and feedback for testing or development and adaptation of the perception stack.

cs.CV↗
arXiv2023

On sequences of sectorial forms converging `from above'

We present a form convergence theorem for sequences of sectorial forms and their associated semigroups in a complex Hilbert space. Roughly speaking, the approximating forms $a_n$ are all `bounded below' by the limiting form $a$, but in contrast to the previous literature there is no monotonicity hypothesis on the sequence. Moreover, the forms are not supposed to be closed or densely defined. For a sectorial form one obtains an associated linear relation, whose negative generates a degenerate strongly continuous semigroup of linear operators. Our hypotheses on the sequence of forms imply strong resolvent convergence of the associated linear relations, which in turn implies convergence of the corresponding semigroups. The result is illustrated by two examples, one of them closely related to the Galerkin method of numerical analysis.

math.FA↗
arXiv2021

On Hausdorff measure and an inequality due to Maz'ya

We give an "elementary" proof of an inequality due to Maz'ya. As a prerequisite we prove an approximation property for the Hausdorff measure. We also comment on the relations between Maz'ya's inequality, the isoperimetric inequality and the Sobolev inequality.

math.FA↗
arXiv2019

Secrecy Rate Region of SWIPT Wiretap Interference Channels

The secrecy rate region of wiretap interference channels with a multi-antenna passive eavesdropper is studied under receiver energy harvesting constraints. To stay operational in the network, the legitimate receivers demand energy alongside information, which is fulfilled by power transmission and exploiting a power splitting (PS) receiver. By simultaneous wireless information and power transfer (SWIPT), the amount of leakage to the eavesdropper increases, which in turn reduces the secrecy rates. For this setup, lower-bounds for secure communication rate are derived without imposing any limitation at the eavesdropper processing. These lower-bounds are then compared with the rates achieved by assuming the worst-case linear eavesdropper processing. We show that in certain special cases the worst-case eavesdropper does not enlarge the achievable secure rate region in comparison to the unconstrained eavesdropper case. It turns out that in order to achieve the Pareto boundary of the secrecy rate region, smart tuning of the transmit power and receiver PS coefficient is required. Hence, we propose an efficient algorithm to optimize these parameters jointly in polynomial-time. The secrecy rate region characterization is formulated as a weighted max-min optimization problem. This problem turns out to be a non-convex problem due to the non-convex constrained set. This set is replaced by a convex subset that in consequence leads to an achievable suboptimal solution which is improved iteratively. By solving the problem efficiently, we obtain the amount of rate loss for providing secrecy, meanwhile satisfying the energy demands.

cs.IT↗
arXiv2018

State-Space Adaptive Nonlinear Self-Interference Cancellation for Full-Duplex Communication

Full-duplex transmission comprises the ability to transmit and receive at the same time on the same frequency band. It allows for more efficient utilization of spectral resources, but raises the challenge of strong self-interference (SI). Cancellation of SI is generally implemented as a multi-stage approach. This work proposes a novel adaptive SI cancellation algorithm in the digital domain and a comprehensive analysis of state-of-the-art adaptive cancellation techniques. Inspired by recent progress in acoustic echo control, we introduce a composite state-space model of the nonlinear SI channel in cascade structure. We derive a SI cancellation algorithm that decouples the identification of linear and nonlinear elements of the composite state. They are estimated separately and consecutively in each adaptation cycle by a Kalman filter in DFT domain. We show that this adaptation can be supported by a-priori signal orthogonalization and decoding of the signal-of-interest (SoI). In our simulation results, we analyze the performance by evaluating residual interference, system identification accuracy and communication rate. Based on the results, we provide recommendations for system design. In case of input orthogonalization, our Kalman filter solution in cascade structure delivers best performance with low computational complexity. In this configuration, the performance lines up with that of the monolithic (parallel) Kalman filter or the recursive-least squares (RLS) algorithms. We show that the Kalman-based algorithm is superior over the RLS under time-variant conditions if the SoI is decoded and in this way the covariance information required by the Kalman filter can be provided to it.

cs.IT↗
arXiv2018

$L_\infty$-estimates for the torsion function and $L_\infty$-growth of semigroups satisfying Gaussian bounds

We investigate selfadjoint $C_0$-semigroups on Euclidean domains satisfying Gaussian upper bounds. Major examples are semigroups generated by second order uniformly elliptic operators with Kato potentials and magnetic fields. We study the long time behaviour of the $L_\infty$ operator norm of the semigroup. As an application we prove a new $L_\infty$-bound for the torsion function of a Euclidean domain that is close to optimal.

math.AP↗
arXiv2018

A note on perturbations of $C_0$-semigroups

This article deals with a variation of constants type inequality for semigroups acting consistently on a scale of Banach spaces. This inequality can be characterized by a corresponding (easy to verify) inequality for their generators. The results have applications to heat kernel estimates and provide a unified perspective to estimates of these type. Moreover, bi-continuous semigroups can be treated as well.

math.FA↗
arXiv2017

On the $L^p$-theory of $C_0$-semigroups associated with second-order elliptic operators with complex singular coefficients

We study $L^p$-theory of second-order elliptic divergence type operators with complex measurable coefficients. The major aspect is that we allow complex coefficients in the main part of the operator, too. We investigate generation of analytic $C_0$-semigroups under very general conditions on the coefficients, related to the notion of form-boundedness. We determine an interval $J$ in the $L^p$-scale, not necessarily containing $p=2$, in which one obtains a consistent family of quasi-contractive semigroups. This interval is close to optimal, as shown by several examples. In the case of uniform ellipticity we construct a family of semigroups in an extended range of $L^p$-spaces, and we prove $p$-independence of the analyticity sector and of the spectrum of the generators.

math.AP↗
arXiv2017

Holomorphic families of forms, operators and $C_0$-semigroups

If $z\mapsto a_z$ is a holomorphic function with values in the sectorial forms in a Hilbert space, then the associated operator valued function $z\mapsto A_z$ is resolvent holomorphic. We give a proof of this result of Kato, on the basis of the Lax-Milgram lemma. We also show that the $C_0$-semigroups $T_z$ generated by $-A_z$ depend holomorphically on $z$.

math.FA↗
arXiv2017

Numerical Range and Quadratic Numerical Range for Damped Systems

We prove new enclosures for the spectrum of non-selfadjoint operator matrices associated with second order linear differential equations $\ddot{z}(t) + D \dot{z} (t) + A_0 z(t) = 0$ in a Hilbert space. Our main tool is the quadratic numerical range for which we establish the spectral inclusion property under weak assumptions on the operators involved; in particular, the damping operator only needs to be accretive and may have the same strength as $A_0$. By means of the quadratic numerical range, we establish tight spectral estimates in terms of the unbounded operator coefficients $A_0$ and $D$ which improve earlier results for sectorial and selfadjoint $D$; in contrast to numerical range bounds, our enclosures may even provide bounded imaginary part of the spectrum or a spectral free vertical strip. An application to small transverse oscillations of a horizontal pipe carrying a steady-state flow of an ideal incompressible fluid illustrates that our new bounds are explicit.

math.SP↗
arXiv2017

The Passive Eavesdropper Affects my Channel: Secret-Key Rates under Real-World Conditions (Extended Version)

Channel-reciprocity based key generation (CRKG) has gained significant importance as it has recently been proposed as a potential lightweight security solution for IoT devices. However, the impact of the attacker's position in close range has only rarely been evaluated in practice, posing an open research problem about the security of real-world realizations. Furthermore, this would further bridge the gap between theoretical channel models and their practice-oriented realizations. For security metrics, we utilize cross-correlation, mutual information, and a lower bound on secret-key capacity. We design a practical setup of three parties such that the channel statistics, although based on joint randomness, are always reproducible. We run experiments to obtain channel states and evaluate the aforementioned metrics for the impact of an attacker depending on his position. It turns out the attacker himself affects the outcome, which has not been adequately regarded yet in standard channel models.

cs.IT↗
arXiv2016

Bands in $L_p$-spaces

For a general measure space $(Ω,μ)$, it is shown that for every band $M$ in $L_p(μ)$ there exists a decomposition $μ=μ'+μ^{\prime\prime}$ such that $M=L_p(μ')=\{f\in L_p(μ);f=0\ μ^{\prime\prime}\text{-a.e.}\}$. The theory is illustrated by an example, with an application to absorption semigroups.

math.FA↗
arXiv2016

Full-Duplex vs. Half-Duplex Secret-Key Generation

Full-duplex (FD) communication is regarded as a key technology in future 5G and Internet of Things (IoT) systems. In addition to high data rate constraints, the success of these systems depends on the ability to allow for confidentiality and security. Secret-key agreement from reciprocal wireless channels can be regarded as a valuable supplement for security at the physical layer. In this work, we study the role of FD communication in conjunction with secret-key agreement. We first introduce two complementary key generation models for FD and half-duplex (HD) settings and compare the performance by introducing the key-reconciliation function. Furthermore, we study the impact of the so called probing-reconciliation trade-off, the role of a strong eavesdropper and analyze the system in the high SNR regime. We show that under certain conditions, the FD mode enforces a deteriorating impact on the capabilities of the eavesdropper and offers several advantages in terms of secret-key rate over the conventional HD setups. Our analysis reveals as an interesting insight that perfect self-interference cancellation is not necessary in order to obtain performance gains over the HD mode.

cs.IT↗