arXiv subjects

Felix Meyer

Publications and source records attributed to Felix Meyer.

5 recordsLinked to original sources

arXiv2026

Existence of large-data solutions to a thermo-piezoelectric system and forward operator analysis for associated inverse problems

We consider an inverse problem governed by the initial-boundary value problem for the thermo-piezoelectric Kelvin-Voigt dynamical system \[ \left\{ \begin{aligned} \rho(z,t) u_{tt} &= \frac{d}{dz} \big( \Gamma(\Theta) u_{zt} +p_1 u_z +p_2(z,t)\phi_z^0 +p_2(z,t)\chi_z -\beta \Theta \big), \\[1ex] 0 &=-\frac{d}{dz} \big( p_2(z,t)u_z -p_3(z,t)\phi_z^0 -p_3(z,t)\chi_z\big), \\ b(z,t)\Theta_t &= \frac{d}{dz}(k(z,t)\Theta_z) +\Gamma(\Theta)u_{zt}^2 -\beta \Theta u_{zt}. \end{aligned} \right. \] in an open bounded interval $\Omega\subset\mathbb{R}$, for the evolution of the displacement variable $u$, the electric potential $\phi^0$ and the temperature $\Theta\geq 0$, where $\chi$ is a given Dirichlet lift function. Assuming that the coefficients $\beta, p_1 \in \mathbb{R}^+$ and the the parameter functions $\rho$, $\Gamma$, $p_2$, $p_3$, $b$ and $k$ are strictly positive and bounded, a global-in-time existence result is established for weak solutions. We show that this can be achieved under energy- and entropy-minimal assumptions, in the sense that global weak solutions are shown to exist for any initial data $$u_0\in W^{1,2}(\Omega)\mbox{ with }u_0|_{\partial\Omega}\in\mathbb{R},\quad u_{0t}\in L^2(\Omega)\mbox{ with }u_{0t}|_{\partial\Omega}\in\mathbb{R}\quad\mbox{and}\quad 0\leq\Theta_0\in L^2(\Omega).$$ The qualitative analysis of the evolution problem then allows to model and analyze the structural properties of the corresponding forward operator arising in inverse parameter identification. Therein, two modeling approaches of the observation operator as approximations of the electrical surface charge are presented and results on their well-definedness and boundedness are established. Building on these results, we prove well-definedness, boundedness, and continuous Fr'{e}chet differentiability of the forward operator.

math.AP↗

arXiv2026

Global smooth solutions in a one-dimensional thermoviscoelastic model with temperature-dependent paramaters

This manuscript is concerned with the system \begin{align*} \left\{ \begin{array}{l} u_{tt} = (\gamma(\Theta) u_{xt})_x + (a(x,t) u_x)_x +(f(\Theta))_x, \\[1mm] \Theta_t = D\Theta_{xx} + \gamma(\Theta) u_{xt}^2 + f(\Theta) u_{xt}, \end{array} \right. \end{align*} which is used to describe thermoviscoelastic developments in one-dimensional Kelvin-Voigt materials. \abs It is assumed that $a,\gamma$ and $f$ are sufficiently smooth functions that satisfy $$c_\gamma<\gamma(\zeta) 0$ and $\alpha \in (0,5/6)$. Under these conditions, this study then establishes a result on the existence of global classical solutions for sufficiently smooth but arbitrarily large initial data.

math.AP↗

arXiv2026

Large time existence in a thermoviscoelastic evolution problem with mildly temperature-dependent parameters

We consider \begin{align*} \label{HS} \left\{ \begin{array}{l} u_{tt} = (\gamma(\Theta) u_{xt})_x + a (\gamma(\Theta) u_x)_x +(f(\Theta))_x, \\[1mm] \Theta_t = D\Theta_{xx} + \Gamma(\Theta) u_{xt}^2 + F(\Theta) u_{xt}, \end{array}\right. \qquad \qquad (\star) \end{align*} under Neumann boundary conditions for $u$ and Dirichlet boundary conditions for $\Theta$ in a bounded interval $\Omega\subset\mathbb{R}$. \abs This model is a generalization of the classical system for the description of strain and temperature evolution in a thermo-viscoelastic material following a Kelvin-Voigt material law, in which $\gamma\equiv \Gamma$ and $f\equiv F$. Different variations of this model have already been analyzed in the past and the present study draws upon a known result concerning the existence of classical solutions, which are local in time, for suitably smooth initial data, arbitrary $a>0$, $D>0$ and $\gamma,f\in C^2([0,\infty))$ as well as $\Gamma,F\in C^1([0,\infty))$ with $\gamma>0,\Gamma\ge0$ and $F(0)=0$. Our work focuses on proving that existence times for classical solutions can be arbitrarily large, assuming sublinear temperature dependencies of $\gamma$ and $f$, and further $|F(s)|\le C_F(1+s)^\alpha$ for some $C_F>0$ and $\alpha\in(0,1)$. In particular, for any given $T_\star$, initial mass $M$ and $0<\underline\gamma<\overline\gamma$, there exists a constant $\delta_\star(M,T_\star,a,D, \Omega, \underline\gamma, \overline\gamma,C_F,\alpha)>0$, such that if $$\underline\gamma \le\gamma\le \overline\gamma\quad\mbox{ and }\quad 0\le \Gamma\le \overline\gamma \quad \mbox{ as well as } \quad\|\gamma'\|_{L^\infty([0,\infty))}\le \delta_\star \quad \mbox{ and }\quad \|f'\|_{L^\infty([0,\infty))}\le \delta_\star $$ hold, the maximal existence time of the classical solution to $(\star)$ surpasses $T_\star$.

math.AP↗

arXiv2026

Large-data global solutions to a quasilinear model for viscuos acoustic wave propagation in a non-isothermal setting

The manuscript considers the model for conversion of mechanical energy into heat during acoustic wave propagation in the presence of temperature-dependent elastic parameters, as given by \[ \left\{ \begin{array}{l} u_{tt} = (\gamma(\Theta) u_{xt})_x + a (\gamma(\Theta) u_x)_x, \\[1mm] \Theta_t = D\Theta_{xx} + \gamma(\Theta) u_{xt}^2. \end{array} \right. \qquad \qquad (\star) \] It is firstly shown that when considered along with no-flux boundary conditions in an open bounded real interval $\Omega$, under the assumption that $\gamma\in C^2([0,\infty))$ is such that $\gamma>0$ and $\gamma'\ge 0$ on $[0,\infty)$ as well as \[ D\cdot (\gamma+D) \cdot \gamma'' + 2\gamma \gamma'^2 \le 0 \qquad \mbox{on } [0,\infty), \] for all suitably regular initial data this problem admits a globally defined classical solution. This complements recent findings in the literature, according to which ($\star$) may admit solutions blowing up in finite time whenever $\gamma$ is positive and nondecreasing on $[0,\infty)$ with $\int_0^\infty \frac{d\xi}{\gamma(\xi)} < \infty$. Apart from that, it is found that if the additional assumption \[ a|\Omega|^2 \le \frac{\pi^2 \gamma(0)}{1+\sqrt{1+\frac{\gamma(0)}{D}}} \] is satisfied, the all these solutions stabilize toward some spatially homogeneous equilibrium in the large time limit.

math.AP↗

arXiv2021

Automatic Learning of Subword Dependent Model Scales

To improve the performance of state-of-the-art automatic speech recognition systems it is common practice to include external knowledge sources such as language models or prior corrections. This is usually done via log-linear model combination using separate scaling parameters for each model. Typically these parameters are manually optimized on some held-out data. In this work we propose to optimize these scaling parameters via automatic differentiation and stochastic gradient decent similar to the neural network model parameters. We show on the LibriSpeech (LBS) and Switchboard (SWB) corpora that the model scales for a combination of attentionbased encoder-decoder acoustic model and language model can be learned as effectively as with manual tuning. We further extend this approach to subword dependent model scales which could not be tuned manually which leads to 7% improvement on LBS and 3% on SWB. We also show that joint training of scales and model parameters is possible and gives additional 6% improvement on LBS.

cs.CL↗