arXiv subjects

Ana Carpio

Publications and source records attributed to Ana Carpio.

At least 19 recordsLinked to original sources

arXiv2026

Approximation of solutions of parameter-dependent problems by residual neural networks

We develop a convergent scheme to train neural networks involving analytic activation functions based on gradient flows. Convergence properties are guaranteed by Lojasiewicz theory. The main advantage of this approach is its simplicity of implementation. The coefficients of the network are approximated by solving a system of ordinary differential equations. We test the method by constructing residual neural network approximations of solutions of parametric problems. The dependence of the solutions of simple ordinary differential equations on a few parameters is correctly reproduced. The solutions of inverse problems involving wave constraints which depend on a few parameters can be reasonably approximated, even in regions in which the problem is severely ill posed.

math.NA↗

arXiv2026

Simultaneous Estimation of Seabed and Its Roughness With Longitudinal Waves

This paper introduces an infinite-dimensional Bayesian framework for acoustic seabed tomography, leveraging wave scattering to simultaneously estimate the seabed and its roughness. Tomography is considered an ill-posed problem where multiple seabed configurations can result in similar measurement patterns. We propose a novel approach focusing on the statistical isotropy of the seabed. Utilizing fractional differentiability to identify seabed roughness, the paper presents a robust numerical algorithm to estimate the seabed and quantify uncertainties. Extensive numerical experiments validate the effectiveness of this method, offering a promising avenue for large-scale seabed exploration.

stat.AP↗

arXiv2025

Hierarchical topological clustering

Topological methods have the potential of exploring data clouds without making assumptions on their the structure. Here we propose a hierarchical topological clustering algorithm that can be implemented with any distance choice. The persistence of outliers and clusters of arbitrary shape is inferred from the resulting hierarchy. We demonstrate the potential of the algorithm on selected datasets in which outliers play relevant roles, consisting of images, medical and economic data. These methods can provide meaningful clusters in situations in which other techniques fail to do so.

cs.LG↗

arXiv2025

Asymptotic models for the evolution of a circular biofilm

We study a class of lubrication-type equations modeling the spread of thin poroelastic biofilms at air/agar interfaces. Starting from a biofilm slab model, we perform a formal multi-scale asymptotic expansion to derive a reduced nonlinear evolution equation with time-dependent coefficients. First, we establish a local-in-time existence result as well as a continuation criteria. Moreover, under suitable assumptions on the radius of the biofilm, we show the existence of global in time solutions. We conclude with some numerical simulations illustrating the emergence of instabilities and potential singularities in finite time.

math.AP↗

arXiv2025

Quantifying uncertainty in inverse scattering problems set in layered environments

The attempt to solve inverse scattering problems often leads to optimization and sampling problems that require handling moderate to large amounts of partial differential equations acting as constraints. We focus here on determining inclusions in a layered medium from the measurement of wave fields on the surface, while quantifying uncertainty and addressing the effect of wave solver quality. Inclusions are characterized by a few parameters describing their material properties and shapes. We devise algorithms to estimate the most likely configurations by optimizing cost functionals with Bayesian regularizations and wave constraints. In particular, we design an automatic Levenberg-Marquardt-Fletcher type scheme based on the use of algorithmic differentiation and adaptive finite element meshes for time dependent wave equation constraints with changing inclusions. In synthetic tests with a single frequency, this scheme converges in few iterations for increasing noise levels. To attain a global view of other possible high probability configurations and asymmetry effects we resort to parallelizable affine invariant Markov Chain Monte Carlo methods, at the cost of solving a few million wave problems. This forces the use of prefixed meshes. While the optimal configurations remain similar, we encounter additional high probability inclusions influenced by the prior information, the noise level and the layered structure, effect that can be reduced by considering more frequencies. We analyze the effect on the calculations of working with adaptive and fixed meshes, under a simple choice of non-reflecting boundary conditions in truncated layered domains for which we establish wellposedness and convergence results.

math.NA↗

arXiv2024

Analysis of a two phase flow model of biofilm spread

Free boundaries of biofilms advancing on surfaces evolve according to conservation laws coupled with systems of partial differential equations for velocities, pressures and chemicals affecting cell behavior. Thin film approximations lead to complicated quasi-stationary systems coupling stationary transport equations and compressible Stokes systems with convection-reaction-diffusion equations.We establish existence, uniqueness and stability of solutions of the different submodels involved and then obtain well posedness results for the full system. Our analysis relies on the construction of weak solutions for the steady transport equations under sign assumptions and the reformulation of the compressible Stokes problem as an elliptic system with enhanced regularity properties on the pressure. We need to consider velocity fields whose divergence and normal boundary components satisfy sign conditions, instead of vanishing as classical results require. Applications include the study of cells, biofilms and tissues, where one phase is a liquid solution, whereas the other one is assorted biomass.

math.AP↗

arXiv2024

Dynamics of bacterial aggregates in microflows

Biofilms are bacterial aggregates that grow on moist surfaces. Thin homogeneous biofilms naturally formed on the walls of conducts may serve as biosensors, providing information on the status of microsystems (MEMS) without disrupting them. However, uncontrolled biofilm growth may largely disturb the environment they develop in, increasing the drag and clogging the tubes. To ensure controlled biofilm expansion we need to understand the effect of external variables on their structure. We formulate a hybrid model for the computational study of biofilms growing in laminar microflows. Biomass evolves according to stochastic rules for adhesion, erosion and motion, informed by numerical approximations of the flow fields at each stage. The model is tested studying the formation of streamers in three dimensional corner flows, gaining some insight on the effect of external variables on their structure.

cond-mat.soft↗

arXiv2024

Biofilms as poroelastic materials

Biofilms are bacterial aggregates encased in a self-produced polymeric matrix which attach to moist surfaces and are extremely resistant to chemicals and antibiotics. Recent experiments show that their structure is defined by the interplay of elastic deformations and liquid transport within the biofilm, in response to the cellular activity and the interaction with the surrounding environment. We propose a poroelastic model for elastic deformation and liquid transport in three dimensional biofilms spreading on agar surfaces. The motion of the boundaries can be described by the combined use of Von Karman type approximations for the agar/biofilm interface and thin film approximations for the biofilm/air interface. Bacterial activity informs the macroscopic continuous model through source terms and residual stresses, either phenomenological or derived from microscopic models. We present a procedure to estimate the structure of such residual stresses, based on a simple cellular automata description of bacterial activity. Inspired by image processing, we show that a filtering strategy effectively smooths out the rough tensors provided by the stochastic cellular automata rules, allowing us to insert them in the macroscopic model without numerical instability.

cond-mat.soft↗

arXiv2024

Biofilm growth on rugose surfaces

A stochastic model is used to assess the effect of external parameters on the development of submerged biofilms on smooth and rough surfaces. The model includes basic cellular mechanisms, such as division and spreading, together with an elementary description of the interaction with the surrounding flow and probabilistic rules for EPS matrix generation, cell decay and adhesion. Insight on the interplay of competing mechanisms as the flow or the nutrient concentration change is gained. Erosion and growth processes combined produce biofilm structures moving downstream. A rich variety of patterns are generated: shrinking biofilms, patches, ripple-like structures traveling downstream, fingers, mounds, streamer-like patterns, flat layers, porous and dendritic structures. The observed regimes depend on the carbon source and the type of bacteria.

cond-mat.soft↗

arXiv2024

Incorporating Cellular Stochasticity in Solid--Fluid Mixture Biofilm Models

The dynamics of cellular aggregates is driven by the interplay of mechanochemical processes and cellular activity. Although deterministic models may capture mechanical features, local chemical fluctuations trigger random cell responses, which determine the overall evolution. Incorporating stochastic cellular behavior in macroscopic models of biological media is a challenging task. Herein, we propose hybrid models for bacterial biofilm growth, which couple a two phase solid/fluid mixture description of mechanical and chemical fields with a dynamic energy budget-based cellular automata treatment of bacterial activity. Thin film and plate approximations for the relevant interfaces allow us to obtain numerical solutions exhibiting behaviors observed in experiments, such as accelerated spread due to water intake from the environment, wrinkle formation, undulated contour development, and the appearance of inhomogeneous distributions of differentiated bacteria performing varied tasks.

cond-mat.soft↗

arXiv2024

Biofilm mechanics and patterns

From multicellular tissues to bacterial colonies, three dimensional cellular structures arise through the interaction of cellular activities and mechanical forces. Simple bacterial communities provide model systems for analyzing such interaction. Biofilms are bacterial aggregates attached to wet surfaces and encased in a self-produced polymeric matrix. Biofilms in flows form filamentary structures that contrast with the wrinkled layers observed on air/solid interfaces. We are able to reproduce both types of shapes through elastic rod and plate models that incorporate information from the biomass production and differentiations process, such as growth rates, growth tensors or inner stresses, as well as constraints imposed by the interaction with environment. A more precise study of biofilm dynamics requires tackling water absorption from its surroundings and fluid transport within the biological system. This process alters the material properties of the biofilm and the overall stresses. We analyze whether poroelastic approaches can provide a suitable combined description of fluid-like and solid-like biofilm behavior.

cond-mat.soft↗

arXiv2023

Well posedness of fluid/solid mixture models for biofilm spread

Two phase solid-fluid mixture models are ubiquitous in biological applications. For instance, models for growth of tissues and biofilms combine time dependent and quasi-stationary boundary value problems set in domains whose boundary moves in response to variations in the mechano-chemical variables. For a model of biofilm spread, we show how to obtain better posed models by characterizing the time derivatives of relevant quasi-stationary magnitudes in terms of additional boundary value problems. We also give conditions for well posedness of time dependent submodels set in moving domains depending on the motion of the boundary. After constructing solutions for transport, diffusion and elliptic submodels for volume fractions, displacements, velocities, pressures and concentrations with the required regularity, we are able to handle the full model of biofilm spread in moving domains assuming we know the dynamics of the boundary. These techniques are general and can be applied in models with a similar structure arising in biological and chemical engineering applications.

math.AP↗

arXiv2023

Object based Bayesian full-waveform inversion for shear elastography

We develop a computational framework to quantify uncertainty in shear elastography imaging of anomalies in tissues. We adopt a Bayesian inference formulation. Given the observed data, a forward model and their uncertainties, we find the posterior probability of parameter fields representing the geometry of the anomalies and their shear moduli. To construct a prior probability, we exploit the topological energies of associated objective functions. We demonstrate the approach on synthetic two dimensional tests with smooth and irregular shapes. Sampling the posterior distribution by Markov Chain Monte Carlo (MCMC) techniques we obtain statistical information on the shear moduli and the geometrical properties of the anomalies. General affine-invariant ensemble MCMC samplers are adequate for shapes characterized by parameter sets of low to moderate dimension. However, MCMC methods are computationally expensive. For simple shapes, we devise a fast optimization scheme to calculate the maximum a posteriori (MAP) estimate representing the most likely parameter values. Then, we approximate the posterior distribution by a Gaussian distribution found by linearization about the MAP point to capture the main mode at a low computational cost.

math.NA↗

arXiv2022

Immersed boundary approach to biofilm spread on surfaces

We propose a computational framework to study the growth and spread of bacterial biofilms on interfaces, as well as the action of antibiotics on them. Bacterial membranes are represented by boundaries immersed in a fluid matrix and subject to interaction forces. Growth, division and death of bacterial cells follow dynamic energy budget rules, in response to variations in environmental concentrations of nutrients, toxicants and substances released by the cells. In this way, we create, destroy and enlarge boundaries, either spherical or rod-like. Appropriate forces represent details of the interaction between cells, and the interaction with the environment. Numerical simulations illustrate the evolution of top views and diametral slices of small biofilm seeds, as well as the action of antibiotics. We show that cocktails of antibiotics targeting active and dormant cells can entirely eradicate a biofilm.

q-bio.CB↗

arXiv2020

Bayesian approach to inverse scattering with topological priors

We propose a Bayesian inference framework to estimate uncertainties in inverse scattering problems. Given the observed data, the forward model and their uncertainties, we find the posterior distribution over a finite parameter field representing the objects. To construct the prior distribution we use a topological sensitivity analysis. We demonstrate the approach on the Bayesian solution of 2D inverse problems in light and acoustic holography with synthetic data. Statistical information on objects such as their center location, diameter size, orientation, as well as material properties, are extracted by sampling the posterior distribution. Assuming the number of objects known, comparison of the results obtained by Markov Chain Monte Carlo sampling and by sampling a Gaussian distribution found by linearization about the maximum a posteriori estimate show reasonable agreement. The latter procedure has low computational cost, which makes it an interesting tool for uncertainty studies in 3D. However, MCMC sampling provides a more complete picture of the posterior distribution and yields multi-modal posterior distributions for problems with larger measurement noise. When the number of objects is unknown, we devise a stochastic model selection framework.

math.NA↗

arXiv2017

Dynamic energy budget approach to evaluate antibiotic effects on biofilms

Quantifying the action of antibiotics on biofilms is essential to devise therapies against chronic infections. Biofilms are bacterial communities attached to moist surfaces, sheltered from external aggressions by a polymeric matrix. Coupling a dynamic energy budget based description of cell metabolism to surrounding concentration fields, we are able to approximate survival curves measured for different antibiotics. We reproduce numerically stratified distributions of cell types within the biofilm and introduce ways to incorporate different resistance mechanisms. Qualitative predictions follow that are in agreement with experimental observations, such as higher survival rates of cells close to the substratum when employing antibiotics targeting active cells or enhanced polymer production when antibiotics are administered. The current computational model enables validation and hypothesis testing when developing therapies.

physics.bio-ph↗

arXiv2016

Well posedness of an integrodifferential kinetic model of Fokker-Planck type for angiogenesis

Tumor induced angiogenesis processes including the effect of stochastic motion and branching of blood vessels can be described coupling a (nonlocal in time) integrodifferential kinetic equation of Fokker-Planck type with a diffusion equation for the tumor induced angiogenic factor. The chemotactic force field depends on the flux of blood vessels through the angiogenic factor. We develop an existence and uniqueness theory for this system under natural assumptions on the initial data. The proof combines the construction of fundamental solutions for associated linearized problems with comparison principles, sharp estimates of the velocity integrals and compactness results for this type of kinetic and parabolic operators.

math.AP↗

arXiv2016

Well posedness of an angiogenesis related integrodifferential diffusion model

We prove existence and uniqueness of nonnegative solutions for a nonlocal in time integrodifferential diffusion system related to angiogenesis descriptions. Fundamental solutions of appropriately chosen parabolic operators with bounded coefficients allow us to generate sequences of approximate solutions. Comparison principles and integral equations provide uniform bounds ensuring some convergence properties for iterative schemes and providing stability bounds. Uniqueness follows from chained integral inequalities.

math.AP↗