arXiv subjects

Axel Osses

Publications and source records attributed to Axel Osses.

At least 19 recordsLinked to original sources

arXiv2026

Observability of wave and plate equations with rough coefficients and interfaces: a multiplier approach

The goal of this article is to derive observability properties for wave and plate equations with rough coefficients in the principal part, including the case of interfaces, through a regularization process. Specifically, we demonstrate that if a singular coefficient can be approximated by a sequence of smooth coefficients corresponding to uniformly observable systems, then the observability inequalities can be passed to the limit. This allows us to derive observability properties for the limit system. While this strategy is natural, we show that it can be used to establish new observability results in several settings, particularly in the presence of interfaces meeting the boundary, or multiple interfaces intersecting at a point, under a suitable multiplier-type condition that imposes sign constraints on the jumps of the coefficients at the interfaces. A key advantage of this approach is that it avoids the need for a detailed analysis of the regularity of solutions near the sets where the coefficients are singular.

math.AP↗

arXiv2025

Koopman Kalman Filter (KKF): An asymptotically optimal nonlinear filtering algorithm with error bounds and its application to parameter estimation

In this article, we propose a new filtering algorithm based in the Koopman operator, showing that a nonlinear filtering problem can be seen as an equivalent problem where the dynamics is infinite dimensional, but linear. Using Extended Dynamic Mode Decomposition (EDMD), we create a finite dimensional approximation of the filtering problem of dimension $N$, in state and error covariance matrix, that accomplishes an error bound of order $O(N^{-1/2})$ in both where $N$ denotes the number of points used in the Koopman approximation. The algorithm is denominated Koopman Kalman Filter (KKF), and has computational complexity $O(T\cdot N^3)$ in time, and $O(T \cdot N^2)$ in space, where $T$ is the number of iterations of the filtering problem. We test the algorithm in linear and nonlinear dynamics cases, showing and effective error bound with respect to the Kalman filter, that corresponds to the optimal solution in the linear case, and equals the error performance of other methods in the state of the art, but with a much lower execution time. Also, we propose a parameter estimation algorithm based in KKF, comparing it with Markov Chain Monte Carlo techniques, showing similar performance with lower execution time.

math.DS↗

arXiv2025

Novel reaction-diffusion PDE model for fingerprint-like pattern emergence via the Schnakenberg mechanism

Fingerprint analysis and fingerprint identification have been the most widely used tools for human identification. To this day, various models have been proposed to explain how fingerprints are formed, ranging from the fibroblast model, which focuses on cell-collagen interactions, to the buckling of thin layers model, both yielding significant results. In this work, we present a reaction-diffusion model of Schnakenberg type, featuring an anisotropic diffusion matrix that follows the ridge orientations supplied by other traditional fingerprint-generation models, and notably yet allows minutiae -- i.e. characteristic microstructures embedded in fingerprints -- to emerge. The statistical analysis of the minutiae distribution in a randomly generated fingerprint collection is consistent with observations in real fingerprints. The model can numerically generate fingerprint-like patterns corresponding to the four basic classifications -- arches, ulnar loops, radial loops, and whorls -- as well as a variety of derived forms. The generated patterns emerge on a convex domain that mimics the geometry of a fingertip, exhibiting the diverse types of minutiae typically analyzed in fingerprint identification and showing strong agreement with those observed in human fingerprints. This model also provides insight into how levels of certainty in human identification can be achieved when based on minutiae positions. All the algorithms are implemented in an open source software named GenCHSin.

q-bio.QM↗

arXiv2025

Improved error bounds for Koopman operator and reconstructed trajectories approximations with kernel-based methods

In this article, we propose a new error bound for Koopman operator approximation using Kernel Extended Dynamic Mode Decomposition. The new estimate is $O(N^{-1/2})$, with a constant related to the probability of success of the bound, given by Hoeffding's inequality, similar to other methodologies, such as Philipp et al. Furthermore, we propose a \textit{lifting back} operator to obtain trajectories generated by embedding the initial state and iterating a linear system in a higher dimension. This naturally yields an $O(N^{-1/2})$ error bound for mean trajectories. Finally, we show numerical results including an example of nonlinear system, exhibiting successful approximation with exponential decay faster than $-1/2$, as suggested by the theoretical results.

math.NA↗

arXiv2024

Methods based on Radon transform for non-affine deformable image registration of noisy images

Deformable image registration is a standard engineering problem used to determine the distortion experienced by a body by comparing two images of it in different states. This study introduces two new DIR methods designed to capture non-affine deformations using Radon transform-based similarity measures and a classical regularizer based on linear elastic deformation energy. It establishes conditions for the existence and uniqueness of solutions for both methods and presents synthetic experimental results comparing them with a standard method based on the sum of squared differences similarity measure. These methods have been tested to capture various non-affine deformations in images, both with and without noise, and their convergence rates have been analyzed. Furthermore, the effectiveness of these methods was also evaluated in a lung image registration scenario.

cs.CV↗

arXiv2023

An improved spectral inequality for sums of eigenfunctions

We establish a new spectral inequality for the quantified estimation of the $H^s$-norm, $s\ge 0$ of a finite linear combination of eigenfunctions in a domain in terms of its $H^s$-norm in a strictly open subset of the whole domain. The corresponding upper bound depends exponentially on the square root of the frequency number associated to the linear combination.

math.AP↗

arXiv2023

An Open Waveguide with a Thin High Contrast Core Layer: Asymptotic Analysis and Inverse Detection Problem

We investigate the Helmholtz equation in a two dimensional open waveguide with a thin and high contrast core layer. We develop an asymptotic analysis of the Green function of the problem, and through it we identify and characterize the appearance of resonant frequencies. For waves originating outside of the core, the waveguide response at these resonant frequencies is vastly different than the response at non-resonant frequencies. Using this phenomenon and multifrequency measurements containing the first resonance, we propose, theoretically analyze, and numerically validate a reconstruction algorithm to identify the location, thickness and index of refraction of the core layer.

math.AP↗

arXiv2023

Unsupervised reconstruction of accelerated cardiac cine MRI using Neural Fields

Cardiac cine MRI is the gold standard for cardiac functional assessment, but the inherently slow acquisition process creates the necessity of reconstruction approaches for accelerated undersampled acquisitions. Several regularization approaches that exploit spatial-temporal redundancy have been proposed to reconstruct undersampled cardiac cine MRI. More recently, methods based on supervised deep learning have been also proposed to further accelerate acquisition and reconstruction. However, these techniques rely on usually large dataset for training, which are not always available. In this work, we propose an unsupervised approach based on implicit neural field representations for cardiac cine MRI (so called NF-cMRI). The proposed method was evaluated in in-vivo undersampled golden-angle radial multi-coil acquisitions for undersampling factors of 26x and 52x, achieving good image quality, and comparable spatial and improved temporal depiction than a state-of-the-art reconstruction technique.

eess.IV↗

arXiv2023

Modeling of cardiac fibers as oriented liquid crystals

In this work we propose a mathematical model that describes the orientation of ventricular cardiac fibers. These fibers are commonly computed as the normalized gradient of certain harmonic potentials, so our work consisted in finding the equations that such a vector field satisfies, considering the unitary norm constraint. The resulting equations belong to the Frank-Oseen theory of nematic liquid crystals, which yield a bulk of mathematical properties to the cardiac fibers, such as the characterization of singularities. The numerical methods available in literature are computationally expensive and not sufficiently robust for the complex geometries obtained from the human heart, so we also propose a preconditioned projected gradient descent scheme that circumvents these difficulties in the tested scenarios. The resulting model further confirms recent experimental observations of liquid crystal behavior of soft tissue, and provides an accurate mathematical description of such behavior.

math.NA↗

arXiv2020

Lipschitz stability for Backward Heat Equation with application to Fluorescence Microscopy

This second version of the manuscript includes, in the appendices, an erratum that points out an error on the published version and offers alternative results for the Lipschitz stability analysis of the backward heat propagation problem and its applications to light sheet fluorescence microscopy. Abstract of the first version: In this work, we study a Lipschitz stability result in the reconstruction of a compactly supported initial temperature for the heat equation in $\mathbb{R}^n$, from measurements along a positive time interval and over an open set containing its support. We take advantage of the explicit dependency of solutions to the heat equation with respect to the initial condition. By means of Carleman estimates we obtain an analogous result for the case when the observation is made along an exterior region $\omega\times(\tau,T)$, such that the unobserved part $\mathbb{R}^n\backslash\omega$ is bounded. In the latter setting, the method of Carleman estimates gives a general conditional logarithmic stability result when initial temperatures belong to a certain admissible set, and without the assumption of compactness of support. Furthermore, we apply these results to deduce a similar result for the heat equation in $\mathbb{R}$ for measurements available on a curve contained in $\mathbb{R}\times [0,\infty)$, from where a stability estimate for an inverse problem arising in 2D Fluorescence Microscopy is deduced as well. In order to further understand this Lipschitz stability, in particular, the magnitude of its stability constant with respect to the noise level of the measurements, a numerical reconstruction is presented based on the construction of a linear system for the inverse problem in Fluorescence Microscopy. We investigate the stability constant with the condition number of the corresponding matrix.

math.AP↗

arXiv2019

Mathematical Modeling for 2D Light-Sheet Fluorescence Microscopy image reconstruction

We study an inverse problem for Light Sheet Fluorescence Microscopy (LSFM), where the density of fluorescent molecules needs to be reconstructed. Our first step is to present a mathematical model to describe the measurements obtained by an optic camera during an LSFM experiment. Two meaningful stages are considered: excitation and fluorescence. We propose a paraxial model to describe the excitation process which is directly related with the Fermi pencil-beam equation. For the fluorescence stage, we use the transport equation to describe the transport of photons towards the detection camera. For the mathematical inverse problem that we obtain after the modeling, we present a uniqueness result, recasting the problem as the recovery of the initial condition for the heat equation in $\mathbb{R}\times(0,\infty)$ from measurements in a space-time curve. Additionally, we present numerical experiments to recover the density of the fluorescent molecules by discretizing the proposed model and facing this problem as the solution of a large and sparse linear system. Some iterative and regularized methods are used to achieve this objective. The results show that solving the inverse problem achieves better reconstructions than the direct acquisition method that is currently used.

math.AP↗

arXiv2019

Cooling in the shade of warped transition disks

The mass of the gaseous reservoir in young circumstellar disks is a crucial initial condition for the formation of planetary systems, but estimates vary by orders of magnitude. In some disks with resolvable cavities, sharp inner disk warps cast two-sided shadows on the outer rings; can the cooling of the gas as it crosses the shadows bring constraints on its mass? The finite cooling timescale should result in dust temperature decrements shifted ahead of the optical/IR shadows in the direction of rotation. However, some systems show temperature drops, while others do not. The depth of the drops and the amplitude of the shift depend on the outer disk surface density Sigma through the extent of cooling during the shadow crossing time, and also on the efficiency of radiative diffusion. These phenomena may bear observational counterparts, which we describe with a simple one-dimensional model. An application to the HD142527 disk suggests an asymmetry in its shadows, and predicts a >~10deg shift for a massive gaseous disk, with peak Sigma > 8.3 g/cm2. Another application to the DoAr44 disk limits the peak surface density to Sigma < 13g/cm2

astro-ph.EP↗

arXiv2018

An inner warp in the DoAr 44 T Tauri transition disk

Optical/IR images of transition disks (TDs) have revealed deep intensity decrements in the rings of HAeBes HD142527 and HD100453, that can be interpreted as shadowing from sharply tilted inner disks, such that the outer disks are directly exposed to stellar light. Here we report similar dips in SPHERE+IRDIS differential polarized imaging (DPI) of TTauri DoAr44. With a fairly axially symmetric ring in the submm radio continuum, DoAr44 is likely also a warped system. We constrain the warp geometry by comparing radiative transfer predictions with the DPI data in H band (Q_ϕ(H)) and with a re-processing of archival 336GHz ALMA observations. The observed DPI shadows have coincident radio counterparts, but the intensity drops are much deeper in Q_ϕ(H) (~88%), compared to the shallow drops at 336GHz (~24%). Radiative transfer predictions with an inner disk tilt of ~30+-5deg approximately account for the observations. ALMA long-baseline observations should allow the observation of the warped gas kinematics inside the cavity of DoAr44.

astro-ph.SR↗

arXiv2017

ALMA observations of Elias 2-24: a protoplanetary disk with multiple gaps in the Ophiuchus Molecular Cloud

We present ALMA 1.3 mm continuum observations at 0.2" (25 au) resolution of Elias 2-24, one of the largest and brightest protoplanetary disks in the Ophiuchus Molecular Cloud, and report the presence of three partially resolved concentric gaps located at ~20, 52, and 87 au from the star. We perform radiative transfer modeling of the disk to constrain its surface density and temperature radial profile and place the disk structure in the context of mechanisms capable of forming narrow gaps such as condensation fronts and dynamical clearing by actively forming planets. In particular, we estimate the disk temperature at the locations of the gaps to be 23, 15, and 12 K (at 20, 52, and 87 au respectively), very close to the expected snow-lines of CO (23-28 K) and N2 (12-15 K). Similarly, by assuming that the widths of the gaps correspond to 4-8 x the Hill radii of forming planets (as suggested by numerical simulations), we estimate planet masses in the range of 0.2-1.5 M_Jup, 1.0-8.0 M_Jup, and 0.02-0.15 M_Jup for the inner, middle, and outer gap, respectively. Given the surface density profile of the disk, the amount of "missing mass" at the location of each one of these gaps (between 4 and 20 M_Jup) is more than sufficient to account for the formation of such planets.

astro-ph.EP↗

arXiv2015

Simultaneous source and attenuation reconstruction in SPECT using ballistic and single scattering data

In medical SPECT imaging, we seek to simultaneously obtain the internal radioactive sources and the attenuation map using not only ballistic measurements but also first order scattering measurements. The problem is modeled using the radiative transfer equation by means of an explicit nonlinear operator that gives the ballistic and scattering measurements as a function of the radioactive source and attenuation distributions. First, by differentiating this nonlinear operator we obtain a linearized inverse problem. Then, under regularity hypothesis for the source distribution and attenuation map and considering small attenuations, we rigorously prove that the linear operator is invertible and we compute its inverse explicitly. This allows to prove local uniqueness for the nonlinear inverse problem. Finally, using the previous inversion result for the linear operator, we propose a new type of iterative algorithm for simultaneous source and attenuation recovery for SPECT based on Neumann series and a Newton-Raphson algorithm.

math.AP↗

arXiv2014

Stability of an inverse problem for the discrete wave equation and convergence results

Using uniform global Carleman estimates for discrete elliptic and semi-discrete hyperbolic equations, we study Lipschitz and logarithmic stability for the inverse problem of recovering a potential in a semi-discrete wave equation, discretized by finite differences in a 2-d uniform mesh, from boundary or internal measurements. The discrete stability results, when compared with their continuous counterparts, include new terms depending on the discretization parameter h. From these stability results, we design a numerical method to compute convergent approximations of the continuous potential.

math.AP↗

arXiv2012

Fixed-points in Random Boolean Networks: The impact of parallelism in the scale-free topology case

Fixed points are fundamental states in any dynamical system. In the case of gene regulatory networks (GRNs) they correspond to stable genes profiles associated to the various cell types. We use Kauffman's approach to model GRNs with random Boolean networks (RBNs). We start this paper by proving that, if we fix the values of the source nodes (nodes with in-degree 0), the expected number of fixed points of any RBN is one (independently of the topology we choose). For finding such fixed points we use the α-asynchronous dynamics (where every node is updated independently with probability 0 < α < 1). In fact, it is well-known that asynchrony avoids the cycle attractors into which parallel dynamics tends to fall. We perform simulations and we show the remarkable property that, if for a given RBN with scale-free topology and α-asynchronous dynamics an initial configuration reaches a fixed point, then every configuration also reaches a fixed point. By contrast, in the parallel regime, the percentage of initial configurations reaching a fixed point (for the same networks) is dramatically smaller. We contrast the results of the simulations on scale-free networks with the classical Erdos-Renyi model of random networks. Everything indicates that scale-free networks are extremely robust. Finally, we study the mean and maximum time/work needed to reach a fixed point when starting from randomly chosen initial configurations.

q-bio.CB↗

arXiv2011

Nonlinear Neumann boundary stabilization of the wave equation using rotated multipliers

The rotated multipliers method is performed in the case of the boundary stabilization by means of a(linear or non-linear) Neumann feedback. this method leads to new geometrical cases concerning the "active" part of the boundary where the feedback is apllied. Due to mixed boundary conditions, these cases generate singularities. Under a simple geometrical conditon concerning the orientation of boundary, we obtain a stabilization result in both cases.

math.OC↗