In this paper we provide a general framework to study a general class of linear and nonlinear kinetic equations with random uncertainties from the initial data or collision kernels, and their stochastic Galerkin (SG) approximations, in both incompressible Navier--Stokes and Euler (acoustic) regimes. First, we show that the general framework put forth in [C. Mouhot and L. Neumann, Nonlinearity, 19 (2006), pp. 969--998; M. Briant, J. Differential Equations, 259 (2005), pp. 6072--6141] based on hypocoercivity for the deterministic kinetic equations can be easily adopted for sensitivity analysis for random kinetic equations, which gives rise to an exponential convergence of the random solution toward the (deterministic) global equilibrium, under suitable conditions on the collision kernel. Then we use such theory to study the SG methods for the equations, establish hypocoercivity of the SG system and regularity of its solution, and study spectral accuracy and exponential decay in time of the numerical error of the method in a weighted Sobolev norm.

We introduce a model of multiagent dynamics for self-organized motion; individuals travel at a constant speed while trying to adopt the averaged body attitude of their neighbors. The body attitudes are represented through unitary quaternions. We prove the correspondence with the model presented in [P. Degond, A. Frouvelle, and S. Merino-Aceituno, Math. Models Methods Appl. Sci., 27 (2017), pp. 1005--1049], where the body attitudes are represented by rotation matrices. Differently from this previous work, the individual-based model introduced here is based on nematic (rather than polar) alignment. From the individual-based model, the kinetic and macroscopic equations are derived. The benefit of this approach, in contrast to that of the previous one, is twofold: first, it allows for a better understanding of the macroscopic equations obtained and, second, these equations are prone to numerical studies, which is key for applications.

In this paper, we suggest a new heterogeneous multiscale method (HMM) for the Helmholtz equation with high contrast. The method is constructed for a setting as in Bouchitte and Felbacq [C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 377-382], where the high contrast in the parameter leads to unusual effective parameters in the homogenized equation. We revisit existing homogenization approaches for this special setting and analyze the stability of the two-scale solution with respect to the wavenumber and the data. This includes a new stability result for solutions to the Helmholtz equation with discontinuous diffusion matrix. The HMM is defined as direct discretization of the two-scale limit equation. With this approach we are able to show quasi-optimality and an a priori error estimate under a resolution condition that inherits its dependence on the wavenumber from the stability constant for the analytical problem. Numerical experiments confirm our theoretical convergence results and examine the resolution condition. Moreover, the numerical simulation gives a good insight and explanation of the physical phenomenon of frequency band gaps.

Starting from Hadamard's method, we extend Babich's ansatz to the frequency-domain point-source (FDPS) Maxwell's equations in an inhomogeneous medium in the high-frequency regime. First, we develop...

The present work aims to justify on a formal level the obtention of the electron Boltzmann relation from a fully kinetic description of magnetically confined fusion plasmas, performing a suitable a...

Sea ice hosts a rich ecosystem of flora and fauna, from microscale to macroscale. Algae living in its porous brine microstructure, such as the diatom Melosira arctica, secrete gelatinous exopolymeric substances (EPS) which are thought to protect these communities from their cold and highly saline environment. Recent experimental work has shown significant changes in the structure and properties of young sea ice with entrained Melosira EPS, such as increased brine volume fraction, salt retention, pore tortuosity, and decreased fluid permeability. In particular, we find that the cross-sectional areas of the brine inclusions are described by a bimodal--lognormal distribution, which generalizes the classic lognormal distribution of Perovich and Gow. We propose a model for the effective fluid permeability of young, EPS-laden sea ice, consisting of a random network of pipes with cross-sectional areas chosen from this bimodal distribution. We consider an equilibrium model posed on a square lattice, incorporating...

In the paper, we present a strategy for accelerating posterior inference for unknown inputs in time fractional diffusion models. In many inference problems, the posterior may be concentrated in a small portion of the entire prior support. It will be much more efficient if we build and simulate a surrogate only over the significant region of the posterior. To this end, we construct a coarse model using the generalized multiscale finite element method (GMsFEM) and solve a least-squares problem for the coarse model with a regularizing Levenberg--Marquart algorithm. An intermediate distribution is built based on the approximate sampling distribution. For Bayesian inference, we use GMsFEM and the least-squares stochastic collocation method to obtain a reduced coarse model based on the intermediate distribution. To increase the sampling speed of Markov chain Monte Carlo, the DREAM$_{ZS}$ algorithm is used to explore the surrogate posterior density, which is based on the surrogate likelihood and the intermediate distribution. The proposed method with lower generalized polynomial chaos order gives the approximate posterior as accurately as the surrogate model directly based on the original prior. A few numerical examples for time fractional diffusion equations are carried out to demonstrate the performance of the proposed method with applications of the Bayesian inversion.

Digital Gabor filters are indispensable tools of local time-frequency analysis in signal processing. With strong orientation selectivity, discrete (tight) Gabor frames generated by two-dimensional ...

This paper introduces a novel framework for model adaptivity in the context of heterogeneous multiscale problems. The framework is based on the idea to interpret model adaptivity as a minimization problem of local error indicators that are derived in the general context of the dual weighted residual (DWR) method. Based on the optimization approach a postprocessing strategy is formulated that lifts the requirement of strict a priori knowledge about applicability and quality of effective models. This allows for the systematic, “goal-oriented” tuning of effective models} with respect to a quantity of interest. The framework is tested numerically on elliptic diffusion problems with different types of heterogeneous, random coefficients, as well as an advection-diffusion problem with a strong microscopic, random advection field.

Multi-scale/level computer models are efficient for simulating solutions of partial differential equations on nontrivial geometries using a large number of modes. This is because fine-scale structures in the quantities of interest cannot, in practice, be resolved using a single-level finite element method (FEM) applied directly to the full model, or standard spectral basis functions are not appropriate for the nontrivial geometry and related constraints. The multi-scale/level FEM (MFEM) is based on first applying a FEM for a reduced model problem, obtained by restriction to either (i) several local subdomains or (ii) a dominant linear differential operator in the model. The resulting structure-aware precomputed solutions are then used as basis functions to simulate the full model in the MFEM framework. If these are eigenfunctions of a reduced model eigenvalue problem, such basis functions provide spectrally accurate approximations to the solution of the full model and the resulting method is called the spectral MFEM (SMFEM). In this article we develop, analyze, and implement time-space fully discrete implicit SMFEM (ISMFEM) algorithms for efficiently simulating a Ginzburg--Landau (GL) system modeling superconductivity on a class of superconducting surfaces $S$. The spatial linear differential operator in the time-dependent nonlinear GL model is the Schrödinger operator $( i \nabla + {A}_0)^2$ on $S$, with magnetic vector potential ${A}_0$. For our ISMFEM, the spectrally accurate basis functions are eigenfunctions of a reduced stationary linear model governed by $(i \nabla + {A}_0)^2$. In a recent work we developed, analyzed, and implemented an algorithm to compute these functions using an analytic-numeric hybrid method, in conjunction with a high-order FEM. Using these precomputed solutions as basis functions, the spectrally accurate algorithms developed and analyzed for our new GL computer models on nontrivial geometries are validated using both nonlinear and linearized implicit time discretizations.