arXiv subjects

Bertrand Morel

Publications and source records attributed to Bertrand Morel.

7 recordsLinked to original sources

arXiv2021

Evaluating uncertainties in electrochemical impedance spectra of solid oxide fuel cells

Electrochemical impedance spectroscopy (EIS) is a widely used tool for characterization of fuel cells and other electrochemical conversion systems. When applied to the on-line monitoring in the context of in-field applications, the disturbances, drifts and sensor noise may cause severe distortions in the evaluated spectra, especially in the low-frequency part. Failure to ignore the random effects can result in misinterpreted spectra and, consequently, in misleading diagnostic reasoning. This fact has not been often addressed in the research so far. In this paper, we propose an approach to the quantification of the spectral uncertainty, which relies on evaluating the uncertainty of the equivalent circuit model (ECM). We apply the computationally efficient variational Bayes (VB) method and compare the quality of the results with those obtained with the Markov chain Monte Carlo (MCMC) algorithm. Namely, MCMC algorithm returns accurate distributions of the estimated model parameters, while VB approach provides the approximate distributions. By using simulated and real data we show that approximate results provided by VB approach, although slightly over-optimistic, are still close to the more realistic MCMC estimates. A great advantage of the VB method for online monitoring is low computational load, which is several orders of magnitude lower compared to MCMC. The performance of VB algorithm is demonstrated on a case of ECM parameters estimation in a 6 cell solid oxide fuel cell (SOFC) stack. The complete numerical implementation for recreating the results can be found at https://repo.ijs.si/lznidaric/variational-bayes-supplementary-material.

stat.CO↗

arXiv2007

Un problème de type Yamabe sur les variétés compactes spinorielles compactes

Let $(M,g,\si)$ be a compact spin manifold of dimension $n \geq 2$. Let $λ_1^+(\tilde{g})$ be the smallest positive eigenvalue of the Dirac operator in the metric $\tilde{g} \in [g]$ conformal to $g$. We then define $\lamin(M,[g],\si) = \inf_{\tilde{g} \in [g]} λ_1^+(\tilde{g}) \Vol(M,\tilde{g})^{1/n} $. We show that $0< \lamin(M,[g],\si) \leq \lamin(\mS^n)$. %=\frac{n}{2} \om_n^{1 \over n}$ . We find sufficient conditions for which we obtain strict inequality $\lamin(M,[g],\si) < \lamin(\mS^n)$. This strict inequality has applications to conformal spin geometry. ----- Soit $(M,g,\si)$ une variété spinorielle compacte de dimension $n \geq 2$. %Si $\tilde{g} \in [g]$ est une métrique conforme à $g$, On note $λ_1^+(\tilde{g})$ la plus petite valeur propre $>0$ de l'opérateur de Dirac dans la métrique $\tilde{g} \in [g]$ conforme à $g$. On définit $\lamin(M,[g],\si) = \inf_{\tilde{g} \in [g]} λ_1^+(\tilde{g}) \Vol(M,\tilde{g})^{1/n} $. On montre que $0< \lamin(M,[g],\si) \leq \lamin(\mS^n)$. %= \frac{n}{2} \om_n^{1 \over n}$ On trouve des conditions suffisantes pour lesquelles on obtient l'inégalité stricte $\lamin(M,[g],\si) < \lamin(\mS^n)$. Cette inégalité stricte a des applications en géométrie spinorielle conforme.

math.DG↗

arXiv2007

A spinorial analogue of Aubin's inequality

Let $(M,g,\si)$ be a compact Riemannian spin manifold of dimension $\geq 2$. For any metric $\tilde g$ conformal to $g$, we denote by $\tildeλ$ the first positive eigenvalue of the Dirac operator on $(M,\tilde g,\si)$. We show that $$\inf_{\tilde{g} \in [g]} \tildeλ\Vol(M,\tilde g)^{1/n} \leq (n/2) \Vol(S^n)^{1/n}.$$ This inequality is a spinorial analogue of Aubin's inequality, an important inequality in the solution of the Yamabe problem. The inequality is already known in the case $n \geq 3$ and in the case $n = 2$, $\ker D=\{0\}$. Our proof also works in the remaining case $n=2$, $\ker D\neq \{0\}$. With the same method we also prove that any conformal class on a Riemann surface contains a metric with $2\tildeλ^2\leq \tildeμ$, where $\tildeμ$ denotes the first positive eigenvalue of the Laplace operator.

math.DG↗

arXiv2006

Mass endomorphism and spinorial Yamabe type problems on conformally flat manifolds

Let M be a compact manifold equipped with a Riemannian metric g and a spin structure \si. We let $λ(M,[g],\si)= \inf_{\tilde{g} \in [g]} λ_1^+(\tilde{g}) Vol(M,\tilde{g})^{1/n}$ where $λ_1^+(\tilde{g})$ is the smallest positive eigenvalue of the Dirac operator D in the metric $\tilde{g}$. A previous result stated that $λ(M,[g],\si) \leq λ(\mS^n) =\frac{n}{2} \om_n^{1/n}$ where \om_n stands for the volume of the standard n-sphere. In this paper, we study this problem for conformally flat manifolds of dimension n \geq 2 such that D is invertible. E.g. we show that strict inequality holds in dimension $n\equiv 0,1,2\mod 4$ if a certain endomorphism does not vanish. Because of its tight relations to the ADM mass in General Relativity, the endomorphism will be called mass endomorphism. We apply the strict inequality to spin-conformal spectral theory and show that the smallest positive Dirac eigenvalue attains its infimum inside the enlarged volume-1-conformal class of g.

math.DG↗

arXiv2003

Surfaces in S^3 and H^3 via Spinors

We generalize the spinorial characterization of isometric immersions of surfaces in R^3 given by T. Friedrich (On the spinor representation of surfaces in Euclidean 3-space, J. Geom. Phys. 28 (1998)) to surfaces in S^3 and H^3. The main argument is the interpretation of the energy-momentum tensor associated with a special spinor field as a second fundamental form. It turns out that such a characterization of isometric immersions in terms of a special section of the spinor bundle also holds in the case of hypersurfaces in the Euclidean 4-space.

math.DG↗

arXiv2003

The energy-momentum tensor as a second fundamental form

We show that it is natural to consider the energy-momentum tensor associated with a spinor field as the second fundamental form of an isommetric immersion. In particular we give a generalization of the warped product construction over a Riemannian manifold leading to this interpretation. Special sections of the spinor bundle, generalizing the notion of Killing spinor, are studied. First applications of such a construction are then given.

math.DG↗

arXiv2001

Eigenvalue estimates for the Dirac-Schrödinger operators

We give new estimates for the eigenvalues of the hypersurface Dirac operator in terms of the intrinsic energy-momentum tensor, the mean curvature and the scalar curvature. We also discuss their limiting cases as well as the limiting cases of the estimates obtained by X. Zhang and O. Hijazi in [13] and [10]. We compare these limiting cases with those corresponding to the Friedrich and Hijazi inequalities. We conclude by comparing these results to intrinsic estimates for the Dirac-Schrödinger operator D_f = D - f/2.

math.DG↗