Authors:Manuel Friedrich, Leonard Kreutz Pages: 1 - 48 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We consider finite discrete systems consisting of two different atomic types and investigate ground-state configurations for configurational energies featuring two-body short-ranged particle interactions. The atomic potentials favor some reference distance between different atomic types and include repulsive terms for atoms of the same type, which are typical assumptions in models for ionic dimers. Our goal is to show a two-dimensional crystallization result. More precisely, we give conditions in order to prove that energy minimizers are connected subsets of the hexagonal lattice where the two atomic types are alternately arranged in the crystal lattice. We also provide explicit formulas for the ground-state energy. Finally, we characterize the net charge, i.e. the difference of the number of the two atomic types. Analyzing the deviation of configurations from the hexagonal Wulff shape, we prove that for ground states consisting of [math] particles the net charge is at most of order [math] where the scaling is sharp. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2019-06-06T10:38:08Z DOI: 10.1142/S0218202519500362
Authors:Jȩdrzej Jabłoński, Dariusz Wrzosek Pages: 1 - 33 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. Radon-measure-valued solutions to a size structured population model of the McKendrick–von Foerster-type are analytically studied under general assumptions on individuals’ growth, birth and mortality rates. The model is used to describe changes in size structure of zooplankton when prey size-dependent mortality rate is a consequence of a planktivorous fish foraging in low prey-density environment (commonly found in predator-controlled populations). The model of foraging is based on the optimization of the rate of net energy intake as a function of predator speed. Mortality is defined as an operator on a metric space of nonnegative Radon measures equipped with the bounded Lipschitz distance. The solutions to the size structured model of zooplankton population are studied analytically and numerically. Numerical solutions (derived using the Escalator Boxcar Train (EBT)-like schema), in particular those starting from Dirac deltas corresponding to distinct cohorts, exhibit regularization in time and convergence to the same stationary state. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2019-06-06T10:38:07Z DOI: 10.1142/S0218202519500313
Authors:G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. Poyato, J. Soler Pages: 1 - 105 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. This paper presents a review and critical analysis on the modeling of the dynamics of vehicular traffic, human crowds and swarms seen as living and, hence, complex systems. It contains a survey of the kinetic models developed in the last 10 years on the aforementioned topics so that overlapping with previous reviews can be avoided. Although the main focus of this paper lies on the mesoscopic models for collective dynamics, we provide a brief overview on the corresponding micro and macroscopic models, and discuss intermediate role of mesoscopic model between them. Moreover, we provide a number of selected challenging research perspectives for readers’ attention. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2019-06-06T10:38:06Z DOI: 10.1142/S0218202519500374
Authors:Tao Luo, Shu Wang, Yan-Lin Wang Pages: 1 - 19 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. The singular limit from compressible Euler–Poisson equation in nonthermal plasma to incompressible Euler equation with an ill-prepared initial data is investigated in this paper by constructing approximate solutions of the appropriate order via an asymptotic expansion. Nonlinear asymptotic stability of initial layer approximation is established with the convergence rate. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2019-06-06T10:38:06Z DOI: 10.1142/S0218202519500337
Authors:K. Disser, J. Rehberg Pages: 1 - 33 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We establish the well-posedness of the transient van Roosbroeck system in three space dimensions under realistic assumptions on the data: non-smooth domains, discontinuous coefficient functions and mixed boundary conditions. Moreover, within this analysis, recombination terms may be concentrated on surfaces and interfaces and may not only depend on charge-carrier densities, but also on the electric field and currents. In particular, this includes Avalanche recombination. The proofs are based on recent abstract results on maximal parabolic and optimal elliptic regularity of divergence-form operators. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2019-06-06T10:38:05Z DOI: 10.1142/S0218202519500350
Authors:Lorenzo Mascotto, Ilaria Perugia, Alexander Pichler Pages: 1 - 38 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We introduce a novel virtual element method (VEM) for the two-dimensional Helmholtz problem endowed with impedance boundary conditions. Local approximation spaces consist of Trefftz functions, i.e. functions belonging to the kernel of the Helmholtz operator. The global trial and test spaces are not fully discontinuous, but rather interelement continuity is imposed in a nonconforming fashion. Although their functions are only implicitly defined, as typical of the VEM framework, they contain discontinuous subspaces made of functions known in closed form and with good approximation properties (plane-waves, in our case). We carry out an abstract error analysis of the method, and derive [math]-version error estimates. Moreover, we initiate its numerical investigation by presenting a first test, which demonstrates the theoretical convergence rates. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2019-06-06T10:38:04Z DOI: 10.1142/S0218202519500301
Authors:Jakob Zech, Dinh Dũng, Christoph Schwab Pages: 1 - 65 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We analyze the complexity of the sparse-grid interpolation and sparse-grid quadrature of countably-parametric functions which take values in separable Banach spaces with unconditional bases. Assuming a suitably quantified holomorphic dependence on the parameters, we establish dimension-independent convergence rate bounds for sparse-grid approximation schemes. Analogous results are shown in the case that the parametric families are obtained as approximate solutions of corresponding parametric-holomorphic, nonlinear operator equations as considered in [A. Cohen and A. Chkifa and Ch. Schwab: Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs, J. Math. Pures Appl. 103 (2015) 400–428], for example by means of stable, finite-dimensional approximations. We discuss in detail nonlinear Petrov–Galerkin projections. Error and convergence rate bounds for constructive and explicit multilevel, sparse tensor approximation schemes combining sparse-grid interpolation in the parameter space and general, multilevel discretization schemes in the physical domain are proved. The present results unify and generalize earlier works in terms of the admissible multilevel approximations in the physical domain (comprising general stable Petrov–Galerkin and discrete Petrov–Galerkin schemes, collocation and stable domain approximations) and in terms of the admissible operator equations (comprising smooth, nonlinear locally well-posed operator equations). Additionally, a novel computational strategy to localize sequences of nested index sets for the anisotropic Smolyak interpolation in parameter space is developed which realizes best [math]-term benchmark convergence rates. We also consider Smolyak-type quadratures in this general setting, for which we establish improved convergence rates based on cancellations in the integrands’ gpc expansions by symmetries of quadratures and the probability measure [J. Zech and Ch. Schwab: Convergence rates of high dimensional Smolyak quadrature, Report 2017-27, SAM ETH Zürich (2017)]. Several examples illustrating the abstract theory include domain uncertainty quantification (UQ) for general, linear, second-order, elliptic advection–reaction–diffusion equations on polygonal domains, where optimal convergence rates of FEM are known to require local mesh refinement near corners. Further applications of the presently developed theory comprise evaluations of posterior expectations in Bayesian inverse problems. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2019-06-06T10:38:02Z DOI: 10.1142/S0218202519500349
Authors:Marvin Fritz, Ernesto A. B. F. Lima, J. Tinsley Oden, Barbara Wohlmuth Pages: 1 - 41 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. A mathematical analysis of local and nonlocal phase-field models of tumor growth is presented that includes time-dependent Darcy–Forchheimer–Brinkman models of convective velocity fields and models of long-range cell interactions. A complete existence analysis is provided. In addition, a parameter-sensitivity analysis is described that quantifies the sensitivity of key quantities of interest to changes in parameter values. Two sensitivity analyses are examined; one employing statistical variances of model outputs and another employing the notion of active subspaces based on existing observational data. Remarkably, the two approaches yield very similar conclusions on sensitivity for certain quantities of interest. The work concludes with the presentation of numerical approximations of solutions of the governing equations and results of numerical experiments on tumor growth produced using finite element discretizations of the full tumor model for representative cases. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2019-06-06T10:38:00Z DOI: 10.1142/S0218202519500325
Authors:Francesca Romana Guarguaglini, Marco Papi, Flavia Smarrazzo Pages: 1 - 45 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. In this paper, we study a hyperbolic–elliptic system on a network which arises in biological models involving chemotaxis. We also consider suitable transmission conditions at internal points of the graph which on one hand allow discontinuous density functions at nodes, and on the other guarantee the continuity of the fluxes at each node. Finally, we prove local and global existence of non-negative solutions — the latter in the case of small (in the [math]-norm) initial data — as well as their uniqueness. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2019-04-10T09:19:56Z DOI: 10.1142/S021820251950026X
Authors:Jean-David Benamou, Guillaume Carlier, Simone Di Marino, Luca Nenna Pages: 1 - 31 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We propose an entropy minimization viewpoint on variational mean-field games with diffusion and quadratic Hamiltonian. We carefully analyze the time discretization of such problems, establish [math]-convergence results as the time step vanishes and propose an efficient algorithm relying on this entropic interpretation as well as on the Sinkhorn scaling algorithm. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2019-04-10T09:19:56Z DOI: 10.1142/S0218202519500283
Authors:Guilherme Mazanti, Filippo Santambrogio Pages: 1 - 52 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. This paper considers a mean field game model inspired by crowd motion where agents want to leave a given bounded domain through a part of its boundary in minimal time. Each agent is free to move in any direction, but their maximal speed is bounded in terms of the average density of agents around their position in order to take into account congestion phenomena. After a preliminary study of the corresponding minimal-time optimal control problem, we formulate the mean field game in a Lagrangian setting and prove existence of Lagrangian equilibria using a fixed point strategy. We provide a further study of equilibria under the assumption that agents may leave the domain through the whole boundary, in which case equilibria are described through a system of a continuity equation on the distribution of agents coupled with a Hamilton–Jacobi equation on the value function of the optimal control problem solved by each agent. This is possible thanks to the semiconcavity of the value function, which follows from some further regularity properties of optimal trajectories obtained through Pontryagin Maximum Principle. Simulations illustrate the behavior of equilibria in some particular situations. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2019-04-10T09:19:55Z DOI: 10.1142/S0218202519500258
Authors:Hoai-Minh Nguyen, Loc Tran Pages: 1 - 42 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We study the approximate cloaking via transformation optics for electromagnetic waves in the time harmonic regime in which the cloaking device only consists of a layer constructed by the mapping technique. Due to the fact that no-lossy layer is required, resonance might appear and the analysis is delicate. We analyze both non-resonant and resonant cases. In particular, we show that the energy can blow up inside the cloaked region in the resonant case and/whereas cloaking is achieved in both cases. Moreover, the degree of visibility depends on the compatibility of the source inside the cloaked region and the system. These facts are new and distinct from known mathematical results in the literature. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2019-04-10T09:19:54Z DOI: 10.1142/S0218202519500271
Authors:Yvon Maday, Carlo Marcati Pages: 1 - 33 Abstract: Mathematical Models and Methods in Applied Sciences, Ahead of Print. We study the regularity in weighted Sobolev spaces of Schrödinger-type eigenvalue problems, and we analyze their approximation via a discontinuous Galerkin (dG) [math] finite element method. In particular, we show that, for a class of singular potentials, the eigenfunctions of the operator belong to analytic-type non-homogeneous weighted Sobolev spaces. Using this result, we prove that an isotropically graded [math] dG method is spectrally accurate, and that the numerical approximation converges with exponential rate to the exact solution. Numerical tests in two and three dimensions confirm the theoretical results and provide an insight into the behavior of the method for varying discretization parameters. Citation: Mathematical Models and Methods in Applied Sciences PubDate: 2019-04-10T09:19:54Z DOI: 10.1142/S0218202519500295
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