Authors:Massimo Fornasier, Hui Huang, Lorenzo Pareschi, Philippe Sünnen Pages: 2725 - 2751 Abstract: Mathematical Models and Methods in Applied Sciences, Volume 30, Issue 14, Page 2725-2751, 30 December 2020. We introduce a new stochastic differential model for global optimization of nonconvex functions on compact hypersurfaces. The model is inspired by the stochastic Kuramoto–Vicsek system and belongs to the class of Consensus-Based Optimization methods. In fact, particles move on the hypersurface driven by a drift towards an instantaneous consensus point, computed as a convex combination of the particle locations weighted by the cost function according to Laplace’s principle. The consensus point represents an approximation to a global minimizer. The dynamics is further perturbed by a random vector field to favor exploration, whose variance is a function of the distance of the particles to the consensus point. In particular, as soon as the consensus is reached, then the stochastic component vanishes. In this paper, we study the well-posedness of the model and we derive rigorously its mean-field approximation for large particle limit. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-01-18T08:00:00Z DOI: 10.1142/S0218202520500530 Issue No:Vol. 30, No. 14 (2021)
Authors:Christoph Ortner, Jack Thomas Pages: 2753 - 2797 Abstract: Mathematical Models and Methods in Applied Sciences, Volume 30, Issue 14, Page 2753-2797, 30 December 2020. We consider atomistic geometry relaxation in the context of linear tight binding models for point defects. A limiting model as Fermi-temperature is sent to zero is formulated, and an exponential rate of convergence for the nuclei configuration is established. We also formulate the thermodynamic limit model at zero Fermi-temperature, extending the results of [H. Chen, J. Lu and C. Ortner, Thermodynamic limit of crystal defects with finite temperature tight binding, Arch. Ration. Mech. Anal. 230 (2018) 701–733]. We discuss the non-trivial relationship between taking zero temperature and thermodynamic limits in the finite Fermi-temperature models. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-01-06T08:00:00Z DOI: 10.1142/S0218202520500542 Issue No:Vol. 30, No. 14 (2021)
Authors:Andrea Bonito, Ricardo H. Nochetto, Dimitrios Ntogkas Pages: 1 - 43 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We propose a new discontinuous Galerkin (dG) method for a geometrically nonlinear Kirchhoff plate model for large isometric bending deformations. The minimization problem is nonconvex due to the isometry constraint. We present a practical discrete gradient flow that decreases the energy and computes discrete minimizers that satisfy a prescribed discrete isometry defect. We prove [math]-convergence of the discrete energies and discrete global minimizers. We document the flexibility and accuracy of the dG method with several numerical experiments. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2021-01-06T08:00:00Z DOI: 10.1142/S0218202521500044
Authors:Félix del Teso, Jørgen Endal, Juan Luis Vázquez Pages: 1 - 49 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We study the existence and properties of solutions and free boundaries of the one-phase Stefan problem with fractional diffusion posed in [math]. In terms of the enthalpy [math], the evolution equation reads [math], while the temperature is defined as [math] for some constant [math] called the latent heat, and [math] stands for the fractional Laplacian with exponent [math]. We prove the existence of a continuous and bounded selfsimilar solution of the form [math] which exhibits a free boundary at the change-of-phase level [math]. This level is located at the line (called the free boundary) [math] for some [math]. The construction is done in 1D, and its extension to [math]-dimensional space is shown. We also provide well-posedness and basic properties of very weak solutions for general bounded data [math] in several dimensions. The temperatures [math] of these solutions are continuous functions that have finite speed of propagation, with possible free boundaries. We obtain estimates on the growth in time of the support of [math] for solutions with compactly supported initial temperatures. Besides, we show the property of conservation of positivity for [math] so that the support never recedes. On the contrary, the enthalpy [math] has infinite speed of propagation and we obtain precise estimates on the tail. The limits [math], [math], [math] and [math] are also explored, and we find interesting connections with well-studied diffusion problems. Finally, we propose convergent monotone finite-difference schemes and include numerical experiments aimed at illustrating some of the obtained theoretical results, as well as other interesting phenomena. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-12-28T08:00:00Z DOI: 10.1142/S0218202521500032
Authors:Gregor Corbin, Axel Klar, Christina Surulescu, Christian Engwer, Michael Wenske, Juanjo Nieto, Juan Soler Pages: 1 - 46 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We deduce a model for glioma invasion that accounts for the dynamics of brain tissue being actively degraded by tumor cells via excessive acidity production, but also according to the local orientation of tissue fibers. Our approach has a multiscale character: we start with a microscopic description of single cell dynamics including biochemical and/or biophysical effects of the tumor microenvironment, translated on the one hand into cell stress and corresponding forces and on the other hand into receptor binding dynamics. These lead on the mesoscopic level to kinetic equations involving transport terms with respect to all considered kinetic variables and eventually, by appropriate upscaling, to a macroscopic reaction–diffusion equation for glioma density with multiple taxis, coupled to (integro-)differential equations characterizing the evolution of acidity and macro- and mesoscopic tissue. Our approach also allows for a switch between fast and slower moving regimes, according to the local tissue anisotropy. We perform numerical simulations to assess the importance of each tactic term and investigate the influence of two models for tissue dynamics on the tumor shape. We also suggest how the model can be used to perform a numerical necrosis-based tumor grading or support radiotherapy planning by dose painting. Finally, we discuss alternative ways of including cell level environmental influences in such a multiscale modeling approach, ultimately leading in the macroscopic limit to (multiple) taxis. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-12-28T08:00:00Z DOI: 10.1142/S0218202521500056
Authors:Stefano Almi, Sandro Belz, Stefano Micheletti, Simona Perotto Pages: 1 - 45 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. In this paper, we derive a new 2D brittle fracture model for thin shells via dimension reduction, where the admissible displacements are only normal to the shell surface. The main steps include to endow the shell with a small thickness, to express the three-dimensional energy in terms of the variational model of brittle fracture in linear elasticity, and to study the [math]-limit of the functional as the thickness tends to zero. The numerical discretization is tackled by first approximating the fracture through a phase field, following an Ambrosio–Tortorelli like approach, and then resorting to an alternating minimization procedure, where the irreversibility of the crack propagation is rigorously imposed via an inequality constraint. The minimization is enriched with an anisotropic mesh adaptation driven by an a posteriori error estimator, which allows us to sharply track the whole crack path by optimizing the shape, the size, and the orientation of the mesh elements. Finally, the overall algorithm is successfully assessed on two Riemannian settings and proves not to bias the crack propagation. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-12-22T08:00:00Z DOI: 10.1142/S0218202521500020
Authors:Christian Rohde, Lars von Wolff Pages: 1 - 35 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We consider the incompressible flow of two immiscible fluids in the presence of a solid phase that undergoes changes in time due to precipitation and dissolution effects. Based on a seminal sharp interface model a phase-field approach is suggested that couples the Navier–Stokes equations and the solid’s ion concentration transport equation with the Cahn–Hilliard evolution for the phase fields. The model is shown to preserve the fundamental conservation constraints and to obey the second law of thermodynamics for a novel free energy formulation. An extended analysis for vanishing interfacial width reveals that in this limit the sharp interface model is recovered, including all relevant transmission conditions. Notably, the new phase-field model is able to realize Navier-slip conditions for solid–fluid interfaces in the limit. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-12-17T08:00:00Z DOI: 10.1142/S0218202521500019
Authors:Yifeng Xu, Irwin Yousept, Jun Zou Pages: 1 - 28 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. An adaptive edge element method is designed to approximate a quasilinear [math]-elliptic problem in magnetism, based on a residual-type a posteriori error estimator and general marking strategies. The error estimator is shown to be both reliable and efficient, and its resulting sequence of adaptively generated solutions converges strongly to the exact solution of the original quasilinear system. Numerical experiments are provided to verify the validity of the theoretical results. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-12-10T08:00:00Z DOI: 10.1142/S0218202520500554
Authors:Sylvain Billiard, Maxime Derex, Ludovic Maisonneuve, Thomas Rey Pages: 1 - 33 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. Understanding how knowledge emerges and propagates within groups is crucial to explain the evolution of human populations. In this work, we introduce a mathematically oriented model that draws on individual-based approaches, inhomogeneous Markov chains and learning algorithms, such as those introduced in [F. Cucker and S. Smale, On the mathematical foundations of learning, Bull. Amer. Math. Soc. 39 (2002) 1–49; F. Cucker, S. Smale and D. X. Zhou, Modeling language evolution, Found. Comput. Math. 4 (2004) 315–343]. After deriving the model, we study some of its mathematical properties, and establish theoretical and quantitative results in a simplified case. Finally, we run numerical simulations to illustrate some properties of the model. Our main result is that, as time goes to infinity, individuals’ knowledge can converge to a common shared knowledge that was not present in the convex combination of initial individuals’ knowledge. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-12-07T08:00:00Z DOI: 10.1142/S0218202520500529
Authors:Dongfen Bian, Yan Guo, Ian Tice Pages: 1 - 82 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. The [math]-pinch is a classical steady state for the MHD model, where a confined plasma fluid is separated by vacuum, in the presence of a magnetic field which is generated by a prescribed current along the [math]-direction. We develop a scaled variational framework to study its stability in the presence of viscosity effect, and demonstrate that any such [math]-pinch is always unstable. We also establish the existence of a largest growing mode, which dominates the linear growth of the linear MHD system. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2020-12-05T08:00:00Z DOI: 10.1142/S0218202520500566
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