Pedro Aceves Sanchez

Interests

Research

Networks and patterns in emergent systems, multiscale analysis, shallow water asymptotic, nonlinear nonlocal partial differential equations, fluid dynamics, computational mathematics, and mathematical biology.

Teaching

As an applied mathematician, I find fulfillment in instructing a variety of subjects within this realm. For instance, at the undergraduate level, I have experience teaching courses such as Calculus, Linear Algebra, Dynamical Systems, and Numerical Analysis. Moving forward, I aspire to extend my teaching portfolio at the graduate level to include subjects like Mathematical Analysis and Partial Differential Equations. In the past, I have taught a graduate-level course on Compressible Fluid Dynamics.

Courses

Scholarly Contributions

Journals/Publications

• Aceves-Sanchez, P., Bailo, R., Degond, P., & Mercier, Z. (2023). Pedestrian models with congestion effects.
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  We study the validity of the dissipative Aw-Rascle system as a macroscopicmodel for pedestrian dynamics. The model uses a congestion term (a singulardiffusion term) to enforce capacity constraints in the crowd density whileinducing a steering behaviour. Furthermore, we introduce a semi-implicit,structure-preserving, and asymptotic-preserving numerical scheme which canhandle the numerical solution of the model efficiently. We perform the firstnumerical simulations of the dissipative Aw-Rascle system in one and twodimensions. We demonstrate the efficiency of the scheme in solving an array ofnumerical experiments, and we validate the model, ultimately showing that itcorrectly captures the fundamental diagram of pedestrian flow.[Journal_ref: ]
• Aceves-Sanchez, P., Aymard, B., Peurichard, D., Kennel, P., Lorsignol, A., Plouraboué, F., Casteilla, L., & Degond, P. (2021). New model for the emergence of blood capillary networks. Networks and Heterogeneous Media, 16(Issue 1). doi:10.3934/nhm.2021001
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  We propose a new model for the emergence of blood capillary networks. We assimilate the tissue and extra cellular matrix as a porous medium, using Darcy’s law for describing both blood and interstitialuidows. Oxygen obeys a convection-diffusion-reaction equation describing advection by the blood, diffusion and consumption by the tissue. Discrete agents named capillary elements and modelling groups of endothelial cells are created or deleted according to different rules involving the oxygen concentration gradient, the blood velocity, the sheer stress or the capillary element density. Once created, a capillary element locally enhances the hydraulic conductivity matrix, contributing to a local increase of the blood velocity and oxygenow. No connectivity between the capillary elements is imposed. The coupling between blood, oxygenow and capillary elements provides a positive feedback mechanism which triggers the emergence of a network of channels of high hydraulic conductivity which we identify as new blood capillaries. We provide two different, biologically relevant geometrical settings and numerically analyze the in uence of each of the capillary creation mechanism in detail. All mechanisms seem to concur towards a harmonious network but the most important ones are those involving oxygen gradient and sheer stress. A detailed discussion of this model with respect to the literature and its potential future developments concludes the paper.
• Aceves-Sanchez, P., Degond, P., Keaveny, E. E., Manhart, A., Merino-Aceituno, S., & Peurichard, D. (2020). Large-Scale Dynamics of Self-propelled Particles Moving Through Obstacles: Model Derivation and Pattern Formation. Bulletin of Mathematical Biology, 82(Issue 10). doi:10.1007/s11538-020-00805-z
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  We model and study the patterns created through the interaction of collectively moving self-propelled particles (SPPs) and elastically tethered obstacles. Simulations of an individual-based model reveal at least three distinct large-scale patterns: travelling bands, trails and moving clusters. This motivates the derivation of a macroscopic partial differential equations model for the interactions between the self-propelled particles and the obstacles, for which we assume large tether stiffness. The result is a coupled system of nonlinear, non-local partial differential equations. Linear stability analysis shows that patterning is expected if the interactions are strong enough and allows for the predictions of pattern size from model parameters. The macroscopic equations reveal that the obstacle interactions induce short-ranged SPP aggregation, irrespective of whether obstacles and SPPs are attractive or repulsive.
• Aceves-Sanchez, P., & Cesbron, L. (2019). Fractional diffusion limit for a fractional Vlasov-Fokker-Planck equation. SIAM Journal on Mathematical Analysis, 51(Issue 1). doi:10.1137/17m1152073
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  This paper is devoted to the rigorous derivation of the macroscopic limit of a Vlasov- Fokker-Planck equation in which the Laplacian is replaced by a fractional Laplacian. The evolution of the density is governed by a fractional heat equation with the addition of a convective term coming from the external force. The analysis is performed by a modified test function method and by obtaining a priori estimates from quadratic entropy bounds. In addition, we give the proof of existence and uniqueness of solutions to the fractional Vlasov-Fokker-Planck equation.
• Aceves-Sánchez, P., Bostan, M., Carrillo, J. A., & Degond, P. (2019). Hydrodynamic limits for kinetic flocking models of Cucker-Smale type. Mathematical Biosciences and Engineering, 16(Issue 6). doi:10.3934/mbe.2019396
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  We analyse the asymptotic behavior for kinetic models describing the collective behavior of animal populations. We focus on models for self-propelled individuals, whose velocity relaxes toward the mean orientation of the neighbors. The self-propelling and friction forces together with the alignment and the noise are interpreted as a collision/interaction mechanism acting with equal strength. We show that the set of generalized collision invariants, introduced in [1], is equivalent in our setting to the more classical notion of collision invariants, i.e., the kernel of a suitably linearized collision operator. After identifying these collision invariants, we derive the fluid model, by appealing to the balances for the particle concentration and orientation. We investigate the main properties of the macroscopic model for a general potential with radial symmetry.
• Aceves-Sanchez, P., & Mellet, A. (2017). Anomalous diffusion limit for a linear Boltzmann equation with external force field. Mathematical Models and Methods in Applied Sciences, 27(Issue 5). doi:10.1142/s021820251750018x
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  This paper is devoted to the approximation of the linear Boltzmann equation by fractional diffusion equations. Most existing results address this question when there is no external acceleration field. The goal of this paper is to investigate the case where a given acceleration field is present. The main result of this paper shows that for an appropriate scaling of the acceleration field, the usual fractional diffusion equation is supplemented by an advection term. Both the critical and supercritical case are considered.
• Aceves-Sánchez, P., & Schmeiser, C. (2017). Fractional diffusion limit of a linear kinetic equation in a bounded domain. Kinetic and Related Models, 10(Issue 3). doi:10.3934/krm.2017021
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  A version of fractional diffusion on bounded domains, subject to 'homogeneous Dirichlet boundary conditions' is derived from a kinetic transport model with homogeneous inflow boundary conditions. For nonconvex domains, the result differs from standard formulations. It can be interpreted as the forward Kolmogorow equation of a stochastic process with jumps along straight lines, remaining inside the domain.
• Aceves-Sánchez, P., & Schmeiser, C. (2016). Fractional-diffusion-advection limit of a kinetic model. SIAM Journal on Mathematical Analysis, 48(Issue 4). doi:10.1137/15m1045387
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  A fractional diffusion equation with advection term is rigorously derived from a kinetic transport model with a linear turning operator, featuring a fat-tailed equilibrium distribution and a small directional bias due to a given vector field. The analysis is based on bounds derived by relative entropy inequalities and on two recently developed approaches for the macroscopic limit: a Fourier-Laplace transform method for spatially homogeneous data and the so called moment method, based on a modified test function.
• Aceves-Sánchez, P., Minzoni, A. A., & Panayotaros, P. (2013). Numerical study of a nonlocal model for water-waves with variable depth. Wave Motion, 50(Issue 1). doi:10.1016/j.wavemoti.2012.07.002
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  We study numerically the propagation of solitary waves in a Hamiltonian nonlocal shallow water model for bidirectional wave propagation in channels of variable depth. The derivation uses small wave amplitude and small depth variation expansions for the Dirichlet-Neumann operator in the fluid domain, and in the long wave regime we simplify the nonlinear and bottom topography terms, while keeping the exact linear dispersion. Solitons are seen to propagate robustly in channels with rapidly varying bottom topography, and their speed is predicted accurately by an effective equation obtained by the homogenization theory of Craig et al. (2005) [7]. We also study the evolution from peaked initial conditions and give evidence for solitary waves with limiting peakon profiles at an apparent threshold before blow-up. © 2012 Elsevier B.V.