arXiv subjects

Gonzalo Galiano

Publications and source records attributed to Gonzalo Galiano.

At least 19 recordsLinked to original sources

arXiv2025

Bifurcation in dynamic problems with seasonal succession

We investigate the bifurcation structure of equilibria in a class of non-autonomous ordinary differential equations governed by a season length parameter, $τ$, which determines the alternation between growth and decline dynamics. This structure models biological systems exhibiting seasonal variation, such as insect population dynamics or infectious disease transmission. Using the Crandall-Rabinowitz bifurcation theorem, we establish the existence of a critical threshold $τ^*$ at which a bifurcation from the extinction equilibrium occurs. We also explore the emergence of secondary bifurcations from, in general, explicitly unknown non-trivial equilibria which can only be treated numerically. Our results are illustrated with a two-species competitive Lotka-Volterra model for the growth season and a Malthusian model for the decline season for which primary and secondary bifurcations may be computed analytically, allowing the validation of numerical approximations. Our analysis shows how seasonality drives transitions between extinction of both populations, of only one, and coexistence of both populations.

math.DS↗

arXiv2024

On a cross-diffusion system arising in image denosing

We study a generalization of a cross-diffusion problem deduced from a nonlinear complex-variable diffusion model for signal and image denoising. We prove the existence of weak solutions of the time-independent problem with fidelity terms under mild conditions on the data problem. Then, we show that this translates on the well-posedness of a quasi-steady state approximation of the evolution problem, and also prove the existence of weak solutions of the latter under more restrictive hypothesis. We finally perform some numerical simulations for image denoising, comparing the performance of the cross-diffusion model and its corresponding scalar Perona-Malik equation.

math.AP↗

arXiv2024

An economic cross-diffusion mutualistic model for cities emergence

We study an evolution cross-diffusion problem with mutualistic Lotka-Volterra reaction term to modelize the long-term spatial distribution of labor and capital. The mutualistic behavior is deduced from the gradient flow associated to profits maximization. We perform a linear and weakly nonlinear stability analysis and find conditions under which the uniform optimum of profits becomes unstable, leading to pattern formation. The patterns alternate regions of high and low concentrations of both labor and capital, which may be interpreted as cities. Finally, numerical simulations based on the weakly nonlinear analysis, as well as in a finite element approximation, are provided.

math.AP↗

arXiv2024

Error analysis of some nonlocal diffusion discretization schemes

We study two numerical approximations of solutions of nonlocal diffusion evolution problems which are inspired in algorithms for computing the bilateral denoising filtering of an image, and which are based on functional rearrangements and on the Fourier transform. Apart from the usual time-space discretization, these algorithms also use the discretization of the range of the solution (quantization). We show that the discrete approximations converge to the continuous solution in suitable functional spaces, and provide error estimates. Finally, we present some numerical experiments illustrating the performance of the algorithms, specially focusing in the execution time.

math.NA↗

arXiv2024

Well-posedness of an evolution problem with nonlocal diffusion

We prove the well-posedness of a general evolution reaction-nonlocal diffusion problem under two sets of assumptions. In the first set, the main hypothesis is the Lipschitz continuity of the range kernel and the bounded variation of the spatial kernel and the initial datum. In the second set of assumptions, we relax the Lipschitz continuity of the range kernel to Hölder continuity, and assume monotonic behavior. In this case, the spatial kernel and the initial data can be just integrable functions. The main applications of this model are related to the fields of Image Processing and Population Dynamics.

math.AP↗

arXiv2024

Well-posedness of a cross-diffusion population model with nonlocal diffusion

We prove the existence and uniqueness of solution of a nonlocal cross-diffusion competitive population model for two species. The model may be considered as a version, or even an approximation, of the paradigmatic Shigesada-Kawasaki-Teramoto cross-diffusion model, in which the usual diffusion differential operator is replaced by an integral diffusion operator. The proof of existence of solutions is based on a compactness argument, while the uniqueness of solution is achieved through a duality technique.

math.AP↗

arXiv2022

Symmetrization in nonlocal diffusion problems

We study Talenti's type symmetrization properties for solutions of linear stationary and evolution problems. Our main result establishes the comparison in norm between the solution of a problem and its symmetric version when nonlocal diffusion defined through integrable kernels is replacing the usual local diffusion defined by a second order differential operator. Using an approximation argument, we recover, as a corollary of our results, the classical Talenti's theorem. A novelty of our approach is that we replace the measure geometric tools employed in Talenti's proof by the use of the Riesz's rearrangement inequality, giving thus an alternative and somehow simpler proof than Talenti's one.

math.AP↗

arXiv2021

Convergence of solutions of a rescaled evolution nonlocal cross-diffusion problem to its local diffusion counterpart

We prove that, under a suitable rescaling of the integrable kernel defining the nonlocal diffusion terms, the corresponding sequence of solutions of the Shigesada-Kawasaki-Teramoto nonlocal cross-diffusion problem converges to a solution of the usual problem with local diffusion. In particular, the result may be regarded as a new proof of existence of solutions for the local diffusion problem.

math.AP↗

arXiv2020

Turing instability analysis of a singular cross-diffusion problem

The population model of Busenberg and Travis is a paradigmatic model in ecology and tumour modelling due to its ability to capture interesting phenomena like the segregation of populations. Its singular mathematical structure enforces the consideration of regularized problems to deduce properties as fundamental as the existence of solutions. In this article we perform a weakly nonlinear stability analisys of a general class of regularized problems to study the convergence of the instability modes in the limit of the regularization parameter. We demonstrate with some specific examples that the pattern formation observed in the regularized problems, with unbounded wave numbers, is not present in the limit problem due to the amplitude decay of the oscillations. We also check the results of the stability analysis with direct finite element simulations of the problem.

math.AP↗

arXiv2018

Non-convex non-local flows for saliency detection

We propose and numerically solve a new variational model for automatic saliency detection in digital images. Using a non-local framework we consider a family of edge preserving functions combined with a new quadratic saliency detection term. Such term defines a constrained bilateral obstacle problem for image classification driven by p-Laplacian operators, including the so-called hyper-Laplacian case (0 < p < 1). The related non-convex non-local reactive flows are then considered and applied for glioblastoma segmentation in magnetic resonance fluid-attenuated inversion recovery (MRI-Flair) images. A fast convolutional kernel based approximated solution is computed. The numerical experiments show how the non-convexity related to the hyperLaplacian operators provides monotonically better results in terms of the standard metrics.

cs.CV↗

arXiv2016

Well-posedness of a nonlinear integro-differential problem and its rearranged formulation

We study the existence and uniqueness of solutions of a nonlinear integro-differential problem which we reformulate introducing the notion of the decreasing rearrangement of the solution. A dimensional reduction of the problem is obtained and a detailed analysis of the properties of the solutions of the model is provided. Finally, a fast numerical method is devised and implemented to show the performance of the model when typical image processing tasks such as filtering and segmentation are performed.

cs.CV↗

arXiv2015

Analysis of a splitting-differentiation population model leading to cross-diffusion

Starting from the dynamical system model capturing the splitting-differentiation process of populations, we extend this notion to show how the speciation mechanism from a single species leads to the consideration of several well known evolution cross-diffusion partial differential equations. Among the different alternatives for the diffusion terms, we study the model introduced by Busenberg and Travis, for which we prove the existence of solutions in the one-dimensional spatial case. Using a direct parabolic regularization technique, we show that the problem is well posed in the space of bounded variation functions, and demonstrate with a simple example that this is the best regularity expected for solutions. We numerically compare our approach to other alternative regularizations previously introduced in the literature, for the particular case of the contact inhibition problem. Simulation experiments indicate that the numerical scheme arising from the approximation introduced in this article outperforms those of the existent models from the stability point of view.

math.AP↗

arXiv2015

On a fast bilateral filtering formulation using functional rearrangements

We introduce an exact reformulation of a broad class of neighborhood filters, among which the bilateral filters, in terms of two functional rearrangements: the decreasing and the relative rearrangements. Independently of the image spatial dimension (one-dimensional signal, image, volume of images, etc.), we reformulate these filters as integral operators defined in a one-dimensional space corresponding to the level sets measures. We prove the equivalence between the usual pixel-based version and the rearranged version of the filter. When restricted to the discrete setting, our reformulation of bilateral filters extends previous results for the so-called fast bilateral filtering. We, in addition, prove that the solution of the discrete setting, understood as constant-wise interpolators, converges to the solution of the continuous setting. Finally, we numerically illustrate computational aspects concerning quality approximation and execution time provided by the rearranged formulation.

cs.CV↗

arXiv2014

On a new formulation of nonlocal image filters involving the relative rearrangement

Nonlocal filters are simple and powerful techniques for image denoising. In this paper we study the reformulation of a broad class of nonlocal filters in terms of two functional rearrangements: the decreasing and the relative rearrangements. Independently of the dimension of the image, we reformulate these filters as integral operators defined in a one-dimensional space corresponding to the level sets measures. We prove the equivalency between the original and the rearranged versions of the filters and propose a discretization in terms of constant-wise interpolators, which we prove to be convergent to the solution of the continuous setting. For some particular cases, this new formulation allows us to perform a detailed analysis of the filtering properties. Among others, we prove that the filtered image is a contrast change of the original image, and that the filtering procedure behaves asymptotically as a shock filter combined with a border diffusive term, responsible for the staircaising effect and the loss of contrast.

cs.CV↗

arXiv2014

Neighborhood filters and the decreasing rearrangement

Nonlocal filters are simple and powerful techniques for image denoising. In this paper, we give new insights into the analysis of one kind of them, the Neighborhood filter, by using a classical although not very common transformation: the decreasing rearrangement of a function (the image). Independently of the dimension of the image, we reformulate the Neighborhood filter and its iterative variants as an integral operator defined in a one-dimensional space. The simplicity of this formulation allows to perform a detailed analysis of its properties. Among others, we prove that the filter behaves asymptotically as a shock filter combined with a border diffusive term, responsible for the staircaising effect and the loss of contrast.

cs.CV↗

arXiv2013

On a cross-diffusion segregation problem arising from a model of interacting particles

We prove the existence of solutions of a cross-diffusion parabolic population problem. The system of partial differential equations is deduced as the limit equations satisfied by the densities corresponding to an interacting particles system modeled by stochastic differential equations. According to the values of the diffusion parameters related to the intra and inter-population repulsion intensities, the system may be classified in terms of an associated matrix. For proving the existence of solutions when the matrix is positive definite, we use a fully discrete finite element approximation in a general functional setting. If the matrix is only positive semi-definite, we use a regularization technique based on a related cross-diffusion model under more restrictive functional assumptions. We provide some numerical experiments demonstrating the weak and strong segregation effects corresponding to both types of matrices.

math.AP↗

arXiv2013

Existence and nonuniqueness of segregated solutions to a class of cross-diffusion systems

We study the the Dirichlet problem for the cross-diffusion system \[ \partial_tu_i=\operatorname{div}\left(a_iu_i\nabla (u_1+u_2)\right)+f_i(u_1,u_2),\quad i=1,2,\quad a_i=const>0, \] in the cylinder $Q=Ω\times (0,T]$. The functions $f_i$ are assumed to satisfy the conditions $f_1(0,r)=0$, $f_2(s,0)=0$, $f_1(0,r)$, $f_2(s,0)$ are locally Lipschitz-continuous. It is proved that for suitable initial data $u_0$, $v_0$ the system admits segregated solutions $(u_1,u_2)$ such that $u_i\in L^{\infty}(Q)$, $u_1+u_2\in C^{0}(\overline{Q})$, $u_1+u_2>0$ and $u_1\cdot u_2=0$ everywhere in $Q$. We show that the segregated solution is not unique and derive the equation of motion of the surface $Γ$ which separates the parts of $Q$ where $u_1>0$, or $u_2>0$. The equation of motion of $Γ$ is a modification of the Darcy law in filtration theory. Results of numerical simulation are presented.

math.AP↗

arXiv2013

On a non-local spectrogram for denoising one-dimensional signals

In previous works, we investigated the use of local filters based on partial differential equations (PDE) to denoise one-dimensional signals through the image processing of time-frequency representations, such as the spectrogram. In this image denoising algorithms, the particularity of the image was hardly taken into account. We turn, in this paper, to study the performance of non-local filters, like Neighborhood or Yaroslavsky filters, in the same problem. We show that, for certain iterative schemes involving the Neighborhood filter, the computational time is drastically reduced with respect to Yaroslavsky or nonlinear PDE based filters, while the outputs of the filtering processes are similar. This is heuristically justified by the connection between the (fast) Neighborhood filter applied to a spectrogram and the corresponding Nonlocal Means filter (accurate) applied to the Wigner-Ville distribution of the signal. This correspondence holds only for time-frequency representations of one-dimensional signals, not to usual images, and in this sense the particularity of the image is exploited. We compare though a series of experiments on synthetic and biomedical signals the performance of local and non-local filters.

cs.CV↗