The continuous time random walk (CTRW) underlies many fundamental processes in non-equilibrium statistical physics. When the jump length of CTRW obeys a power-law distribution, its corresponding Fokker-Planck equation has space fractional derivative, which characterizes L\'{e}vy flights. Sometimes the infinite variance of L\'{e}vy flight discourages it as a physical approach; exponentially tempering the power-law jump length of CTRW makes it more `physical' and the tempered space fractional diffusion equation appears. This paper provides the basic strategy of deriving the high order quasi-compact discretizations for space fractional derivative and tempered space fractional derivative. The fourth order quasi-compact discretization for space fractional derivative is applied to solve space fractional diffusion equation and the unconditional stability and convergence of the scheme are theoretically proved and numerically verified. Furthermore, the tempered space fractional diffusion equation is effectively solved by its counterpart of the fourth order quasi-compact scheme; and the convergence orders are verified numerically.

We compare three types of mathematical models of growth factor reaction and diffusion in angiogenesis: one describes the reaction on the blood capillary surface, one in the capillary volume, and one on the capillary centerline. Firstly, we explore the analytical properties of these models including solution regularity and positivity. We prove that the surface-reaction models have smooth and positive solutions, and the volume-reaction models have continuous and positive solutions. The line-reaction models utilize distributions on the capillary centerline to represent the reaction line source. The line-reaction model-I employs the Dirac delta function and the mean value of the growth factor around the centerline, which gives a valid model. The line-reaction model-II and III use the local value of the growth factor, which either create singulaity of decouple the reaction from diffusion, thus invalid. Secondly, we compare the programming complexity and computational cost of these models in numerical implementations: the surface-reaction model is the most complicated and suitable for small domains, while the volume-reaction and linear-reaction models are simpler and suitable for large domains with a large number of blood capillaries. Finally, we qauantitatively compare these models in the prediction of the growth factor dynamics. It turns out the volume-reaction and line-reaction model-I agree well with the surface-reaction model for most parameters used in literature, but may differ significantly when the diffusion constant is small.

We study boundary value problems of a quasi-one-dimensional steady-state Poisson-Nernst-Planck model with a local hard-sphere potential for ionic flows of two oppositely charged ion species through an ion channel, focusing on effects of ion sizes and ion valences. The flow properties of interest, individual fluxes and total flow rates of the mixture, depend on multiple physical parameters such as boundary conditions (boundary concentrations and boundary potentials) and diffusion coefficients, in addition to ion sizes and ion valences. For the relatively simple setting and assumptions of the model in this paper, we are able to characterize, almost completely, the distinct effects of the nonlinear interplay between these physical parameters. The boundaries of different parameter regions are identified through a number of critical values that are explicitly expressed in terms of the physical parameters. We believe our results will provide useful insights for numerical and even experimental studies of ionic flows through membrane channels.

We consider the Benard convection in a three-dimensional domain bounded below by a fixed flatten boundary and above by a free moving surface. The domain is horizontally periodic. The fluid dynamics are governed by the Boussinesq approximation and the effect of surface tension is neglected on the free surface. Here we develop a local well-posedness theory for the equations of general case in the framework of the nonlinear energy method.

In the current work we demonstrate the principal possibility of prediction of the response of the largest Lyapunov exponent of a chaotic dynamical system to a small constant forcing perturbation via a linearized relation, which is computed entirely from the unperturbed dynamics. We derive the formal representation of the corresponding linear response operator, which involves the (computationally infeasible) infinite time limit. We then compute suitable finite-time approximations of the corresponding linear response operator, and compare its response predictions with actual, directly perturbed and measured, responses of the largest Lyapunov exponent. The test dynamical system is a 20-variable Lorenz 96 model, run in weakly, moderately, and strongly chaotic regimes. We observe that the linearized response prediction is a good approximation for the moderately and strongly chaotic regimes, and less so in the weakly chaotic regime due to intrinsic nonlinearity in the response of the Lyapunov exponent, which the linearized approximation is incapable of following.

In this work, we study the quasineutral limit of the one-dimensional Vlasov-Poisson equation for ions with massless thermalized electrons. We prove new weak-strong stability estimates in the Wasserstein metric that allow us to extend and improve previously known convergence results. In particular, we show that given a possibly unstable analytic initial profile, the formal limit holds for sequences of measure initial data converging sufficiently fast in the Wasserstein metric to this profile. This is achieved without assuming uniform analytic regularity.

Semi-discrete Runge-Kutta schemes for nonlinear diffusion equations of parabolic type are analyzed. Conditions are determined under which the schemes dissipate the discrete entropy locally. The dissipation property is a consequence of the concavity of the difference of the entropies at two consecutive time steps. The concavity property is shown to be related to the Bakry-Emery approach and the geodesic convexity of the entropy. The abstract conditions are verified for quasilinear parabolic equations (including the porous-medium equation), a linear diffusion system, and the fourth-order quantum diffusion equation. Numerical experiments for various Runge-Kutta finite-difference discretizations of the one-dimensional porous-medium equation show that the entropy-dissipation property is in fact global.

We present a finite difference method to compute the principal eigenvalue and the corresponding eigenfunction for a large class of second order elliptic operators including notably linear operators in nondivergence form and fully nonlinear operators.

The principal eigenvalue is computed by solving a finite-dimensional nonlinear min-max optimization problem.
We prove the convergence of the method and we discuss its implementation. Some examples where the exact solution is explicitly known show the effectiveness of the method.

We show that, for the space of Borel probability measures on a Borel subset of a Polish metric space, the extreme points of the Prokhorov, Monge-Wasserstein and Kantorovich metric balls about a measure whose support has at most n points, consist of measures whose supports have at most n+2 points. Moreover, we use the Strassen and Kantorovich-Rubinstein duality theorems to develop representations of supersets of the extreme points based on linear programming, and then develop these representations towards the goal of their efficient computation.

This paper studies the non-autonomous micropolar fluid flows in two-dimensional bounded domains with external forces containing infinite delay effects. The authors first prove the global well-posedness of the weak solutions and then establish the existence of the pullback attractors for the associated process.

We formulate the large deviations for a class of two scale chemical kinetic processes motivated from biological applications. The result is successfully applied to treat a genetic switching model with positive feedbacks. The corresponding Hamiltonian is convex with respect to the momentum variable as a by-product of the large deviation theory. This property ensures its superiority in the rare event simulations compared with the result obtained by formal WKB asymptotics. The result is of general interest to understand the large deviations for multiscale problems.

The paper is concerned with the Burgers-Hilbert equation u_t + (u^2/2)_x = H[u], where the right hand side is a Hilbert transform. Unique entropy admissible solutions are constructed, locally in time, having a single shock. In a neighborhood of the shock curve, a detailed description of the solution is provided.

In liquid crystals, the existing experiments and simulations suggest that for various types of molecules, no homogeneous phase is found breaking the molecular symmetry. It has been proved for rod-like molecules. We conjecture that it holds for two types of two-fold symmetries, and prove it for some molecules with these symmetries.

The alternating minimization is an efficient method for solving the convex minimization whose objective function is a sum of differentiable function and a separable nonsmooth one. Variants and extensions of the alternating minimization method have been developed in recent years. In this paper, we consider the convergence rate of several existing alternating minimization schemes. We improve the proved big-O convergence rate of these algorithms to little-o under an error bound condition which is actually quite common in applications. We also investigate the convergence of a variant of alternating minimization proposed in this paper.

This paper deals with a Boltzmann-type kinetic model describing the interplay between vehicle dynamics and safety aspects in vehicular traffic. Sticking to the idea that the macroscopic characteristics of traffic flow, including the distribution of the driving risk along a road, are ultimately generated by one-to-one interactions among drivers, the model links the personal (i.e., individual) risk to the changes of speeds of single vehicles and implements a probabilistic description of such microscopic interactions in a Boltzmann-type collisional operator. By means of suitable statistical moments of the kinetic distribution function, it is finally possible to recover macroscopic relationships between the average risk and the road congestion, which show an interesting and reasonable correlation with the well-known free and congested phases of the flow of vehicles.

In this paper, we propose numerical methods for computing the boundary local time of reflecting Brownian motion (RBM) for a bounded domain in R^3 and its use in the probabilistic representation of the solution of the Laplace equation with the Neumann boundary condition. Approximations of RBM based on walk-on-spheres (WOS) and random walk on lattices are discussed and tested for sampling RBM paths and their applicability in finding accurate approximation of the local time and discretization of the probabilistic formula. Numerical tests for several domains (a cube, a sphere, an ellipsoid, and a non-convex nonsmooth domain) have shown the convergence of the numerical methods as the time length of RBM paths and number of paths sampled increase.

We prove existence and uniqueness of solutions to a transport equation modelling vehicular traffic in which the velocity field depends non-locally on the downstream traffic density via a discontinuous anisotropic kernel. The result is obtained recasting the problem in the space of probability measures equipped with the $\infty$-Wasserstein distance. We also show convergence of solutions of a finite dimensional system, which provide a particle method to approximate the solutions to the original problem.

In this note, we propose a discrete model to study one-dimensional transport equations with non-local drift and supercritical dissipation. The inspiration for our model is the equation $$ \theta_t + (H\theta) \theta_x +(-\Delta)^\alpha \theta =0 $$ where $H$ is the Hilbert transform. For our discrete model, we present blow-up results that are analogous to the known results for the above equation. In addition, we will prove regularity for our discrete model which suggests supercritical regularity in the range $1/4<\alpha<1/2$ in the continuous setting.

We consider the inverse problem of determining the highly oscillatory coefficient a in partial differential equations of the form -nabla \cdot (a \nabla u_ + bu = f from given measurements of the solutions. Here, indicates the smallest characteristic wavelength in the problem (0<\epsilon <<1). In addition to the general difficulty of finding an inverse is the challenge of multiscale modeling, which is hard even for forward computations. The inverse problem in its full generality is typically ill-posed, and one common approach is to reduce the dimension by seeking effective parameters. We will here include microscale features directly in the inverse problem and avoid ill-posedness by assuming that the microscale can be accurately represented by a low-dimensional parametrization. The basis for our inversion will be a coupling of the parametrization to analytic homogenization or a coupling to efficient multiscale numerical methods when analytic homogenization is not available. We will analyze the reduced problem, b=0, by proving uniqueness of the inverse in certain problem classes and by numerical examples and also include numerical model examples for medical imaging, b>0, and exploration seismology, b<0.

In this paper we study the non-relativistic and low Mach num-ber limits of two P1 approximation model arising in radiation hydrodynamics in T^3, i.e. the barotropic model and the Navier-Stokes-Fourier model. For the barotropic model, we consider the case that the initial data is a small perturbation of stable equilbria while for the Navier-Stokes-Fourier model, we consider the case that the initial data is large. For both models, we prove the convergence to the solution of the incompressible Navier-Stokes equations with/without stationary transport equations.

In this paper, a numerical scheme for a generalized planar Ginzburg-Landau energy in a circular geometry is studied. A spectral-Galerkin method is utilized, and a stability analysis and an error estimate for the scheme are presented. It is shown that the scheme is unconditionally stable. We present numerical simulation results that have been obtained by using the scheme with various sets of boundary data, including those the form u(\theta)=exp(id\theta), where the integer d denotes the topological degree of the solution. These numerical results are in good agreement with the experimental and analytical results. Results include the computation of bifurcations from pure bend or splay patterns to spiral patterns for d=1, energy decay curves for d=1, spectral accuracy plots for d=2 and computations of metastable or unstable higher-energy solutions as well as the lowest energy ground state solutions for values of d ranging from two to five.

We analyze the weak-coupling limit of the random Schrodinger equation with low frequency initial data and a slowly decorrelating random potential. For the probing signal with a sufficiently long wavelength, we prove a homogenization result, that is, the properly compensated wave field admits a deterministic limit in the "very low" frequency regime. The limit is "anomalous" in the sense that the solution behaves as exp(-Dt^s) with s>1 rather than the "usual" exp(-Dt) homogenized behavior when the random potential is rapidly decorrelating. Unlike in rapidly decorrelating potentials, as we decrease the wavelength of the probing signal, stochasticity appears in the asymptotic limit-- there exists a critical scale depending on the random potential which separates the deterministic and stochastic regimes.

In this work we extend a recent kinetic traffic model [G. Puppo, M. Semplice, A. Tosin and G. Visconti, submitted, 2015] to the case of more than one class of vehicles, each of which is characterized by few different microscopic features. We consider a Boltzmann-like framework with only binary interactions, which take place among vehicles belonging to the various classes. Our approach differs from the multi-population kinetic model proposed in [G. Puppo, M. Semplice, A. Tosin and G. Visconti, Comm. Math. Sci., 2015] because here we assume continuous velocity spaces and we introduce a parameter describing the physical velocity jump performed by a vehicle that increases its speed after an interaction. The model is discretized in order to investigate numerically the structure of the resulting fundamental diagrams and the system of equations is analyzed by studying well posedness. Moreover, we compute the equilibria of the discretized model and we show that the exact asymptotic kinetic distributions can be obtained with a small number of velocities in the grid. Finally, we introduce a new probability law in order to attenuate the sharp capacity drop occurring in the diagrams of traffic.

We prove weak-strong uniqueness results for the compressible Navier-Stokes system with degenerate viscosity coefficients and with vacuum in one dimension. In other words, we give conditions on the weak solution constructed in [Q.S. Jiu, Z.P. Xin, Kinet. Relat. Models, 1. 313-330, 2008] so that it is unique. The novelty consists in dealing with initial density \rho_0 which contains vacuum. To do this we use the notion of relative entropy developed recently by Germain, Feireisl et al and Mellet and Vasseur combined with our new formulation of the compressible system [10, 11]) (more precisely we introduce a new effective velocity v which makes the system parabolic on the density and hyperbolic on the velocity v).

Filtering is concerned with the sequential estimation of the state, and uncertainties, of a Markovian system, given noisy observations. It is particularly difficult to achieve accurate filtering in complex dynamical systems, such as those arising in turbulence, in which effective low-dimensional representation of the desired probability distribution is challenging. Nonetheless recent advances have shown considerable success in filtering based on certain carefully chosen simplifications of the underlying system, which allow closed form filters. This leads to filtering algorithms with significant, but judiciously chosen, model error. The purpose of this article is to analyze the effectiveness of these simplified filters, and to suggest modifications of them which lead to improved filtering in certain time-scale regimes. We employ a Markov switching process for the true signal underlying the data, rather than working with a fully resolved DNS PDE model. Such Markov switching models haven been demonstrated to provide an excellent surrogate test-bed for the turbulent bursting phenomena which make filtering of complex physical models, such as those arising in atmospheric sciences, so challenging.

Motivated by some models arising in quantum plasma dynamics, in this paper we study the Maxwell-Schrodinger system with a power-type nonlinearity. We show the local well-posedness in H^(R^3) \times H^{3/2}(R^3) and the global existence of finite energy weak solutions, these results are then applied to the analysis of finite energy weak solutions for Quantum Magnetohydrodynamic systems.

The multi-dimensional Euler-Poisson system describes the dynamic behavior of many important physical flows, yet as a hyperbolic system its solution can blow up for some initial configurations. This paper strives to advance our understanding on the critical threshold phenomena through the study of a two-dimensional weakly restricted Euler-Poisson (WREP) system. This system can be viewed as an improved model upon the restricted Euler-Poisson (REP) system introduced in [H. Liu and E. Tadmor, Comm. Math. Phys. 228 (2002), 435-466]. We identify upper-thresholds for finite time blow up of solutions for WREP equations with attractive/repulsive forcing. It is shown that the thresholds depend on the size of the initial density relative to the initial velocity gradient through both trace and a nonlinear quantity.

The transformed l_1 penalty (TL1) functions are a one parameter family of bilinear transformations composed with the absolute value function. When acting on vectors, the TL1 penalty interpolates l_0 and l_1 similar to l_p norm, where p is in (0,1). In our companion paper, we showed that TL1 is a robust sparsity promoting penalty in compressed sensing (CS) problems for a broad range of incoherent and coherent sensing matrices. Here we develop an explicit fixed point representation for the TL1 regularized minimization problem. The TL1 thresholding functions are in closed form for all parameter values. In contrast, the lp thresholding functions (p is in [0,1]) are in closed form only for p=0,1,1/2, 2/3, known as hard, soft, half, and 2/3 thresholding respectively. The TL1 threshold values differ in subcritical (supercritical) parameter regime where the TL1 threshold functions are continuous (discontinuous) similar to soft-thresholding (half-thresholding) functions. We propose TL1 iterative thresholding algorithms and compare them with hard and half thresholding algorithms in CS test problems. For both incoherent and coherent sensing matrices, a proposed TL1 iterative thresholding algorithm with adaptive subcritical and supercritical thresholds (TL1IT-s1 for short), consistently performs the best in sparse signal recovery with and without measurement noise.

This work studies the stability of a class of globally hyperbolic moment systems (GHMS) with the single relaxation-time collision model in the sense of hyperbolic relaxation systems. We prove the equilibrium stability of the GHMS in both one- and multi-dimensional space. For a five-moment system in one dimension, we prove its linear instability for some quiescent nonequilibrium states and demonstrate numerically the nonlinear instability of the nonequilibrium states.

In this paper we obtain a weak solution to a quantum energy-transport model for semiconductors. The model is formally derived from the quantum hydrodynamic model in the large-time and small-velocity regime by J\"{u}ngel and Mili\v{s}i\'{c} (Nonlinear Anal.: Real World Appl., 12(2011), pp. 1033-1046). It consists of a fourth-order nonlinear parabolic equation for the electron density, an elliptic equation for the electron temperature, and the Poisson equation for the electric potential. Our solution is global in the time variable, while the N space variables ($N<6$) lie in a bounded Lipschitz domain with a mixed boundary condition. The existence proof is based upon a carefully-constructed approximation scheme which generates a sequence of positive approximate solutions. These solutions are so regular that they can be used to form a variety of test functions to produce a priori estimates. Then these estimates are shown to be enough to justify passing to the limit in the approximate problems.

We study a class of nonlinear stochastic partial differential equations with dissipative nonlinear drift, driven by L\'evy noise. We define a Hilbert-Banach setting in which we prove existence and uniqueness of solutions under general assumptions on the drift and the L\'evy noise. We then prove a decomposition of the solution process into a stationary component, the law of which is identified with the unique invariant probability measure $\mu$ of the process, and a component which vanishes asymptotically for large times in the $L^p(\mu)$-sense, for all $1 \leq p < +\infty$.

The maximum entropy principle is widely used in diverse fields. We address the issue of why the second order maximum entropy model, by using only firing rates and second order correlations of neurons as constraints, can well capture the observed distribution of neuronal firing patterns in many neuronal networks, thus, conferring its great advantage in that the degree of complexity in the analysis of neuronal activity data reduces drastically from O(2^n) to O(n^2), where n is the number of neurons under consideration. We first derive an expression for the effective interactions of the n-th order maximum entropy model using all orders of correlations of neurons as constraints and show that there exists a recursive relation among the effective interactions in the model. Then, via a perturbative analysis, we explore a possible dynamical state in which this recursive relation gives rise to the strengths of higher order interactions always smaller than the lower orders. Finally, we invoke this hierarchy of effective interactions to provide a possible mechanism underlying the success of the second order maximum entropy model and to predict whether such a model can successfully capture the observed distribution of neuronal firing patterns.

The charge motion in vacuum and the induced currents on the electrodes can be related through the Shockley-Ramo (SR) theorem. In this paper, we develop a generalized Shockley-Ramo (GSR) theorem, which could be used to study the motion of macro charged particles in electrolytes. It could be widely applied to biological and physical environments, such as the voltage-gated ion channels. With the procedure of renormalizing of charge and dipole, the generalized theorem provides a direct relationship between the induced currents and the macro charge velocity. Compared with the original Shockley-Ramo theorem, the generalized Shockley-Ramo theorem avoids integrating all the ionic flux, which could reduce the computational cost significantly.

In this paper we consider a model of thermal explosion in porous media. The model consists of two reaction-diffusion equations in a bounded domain with Dirichlet boundary conditions and describes the initial stage of evolution of pressure and temperature fields. Under certain conditions, the classical solution of these equations exists only on finite time interval after which it forms a singularity and becomes unbounded (blows up). This behavior raises a natural question whether this solution can be extended, in a weak sense, after blow up time. We prove that the answer to this question is no, that is, the solution becomes unbounded in entire domain immediately after the singularity is formed. From a physical perspective our results imply that autoignition in porous materials occurs simultaneously in entire domain.

The stabilized semi-implicit time-stepping method is an efficient algorithm to simulate phased field problems with fourth order dissipation. We consider the 3D Cahn-Hilliard equation and prove unconditional energy stability of the corresponding stabilized semi-implicit Fourier spectral scheme independent of the time step. We do not impose any Lipschitz-type assumption on the nonlinearity. It is shown that the size of the stabilization term depends only on the initial data and the diffusion coefficient. Unconditional Sobolev bounds of the numerical solution are obtained and the corresponding error analysis under nearly optimal regularity assumptions is established.

The chemical reaction rate from reactant to product depends on the geometry of potential energy surface (PES) as well as the temperature. We consider a design problem of how to choose the best PES from a given family of smooth potential functions in order to maximize (or minimize) the reaction rate for a given chemical reaction. By utilizing the transition-path theory, we relate reaction rate to committor functions which solves boundary-value elliptic problems, and perform the sensitivity analysis of the underlying elliptic equations via adjoint approach. We derive the derivative of the reaction rate with respect to the potential function. The shape derivative with respect to the domains defining reactant and product is also investigated. The numerical optimization method based on the gradient is applied for two simple numerical examples to demonstrate the feasibility of our approach.

Explicit energy-transport equations for the spinorial carrier transport in ferromagnetic semiconductors are calculated from a general spin energy-transport system that was derived by Ben Abdallah and El Hajj from a spinorial Boltzmann equation. The novelty of our approach are the simplifying assumptions leading to explicit models which extend both spin drift-diffusion and semiclassical energy-transport equations. The explicit models allow us to examine the interplay between the spin and charge degrees of freedom. In particular, the dissipation of the entropy (or free energy) is quantified, and the existence of weak solutions to a time-discrete version of one of the models is proved, using novel truncation arguments. Numerical experiments in one-dimensional multilayer structures using a finite-volume discretization illustrate the effect of the temperature and the polarization parameter.

The Hodge projection of a vector field is the divergence-free component of its Helmholtz decomposition. In a bounded domain, a boundary condition needs to be supplied to the decomposition. The decomposition with the non-penetration boundary condition is equivalent to solving the Poisson equation with the Neumann boundary condition. The Gibou-Min method is an application of the Poisson solver by Purvis and Burkhalter to the decomposition. In the decomposition by the Gibou-Min method, an important $L^{2}$-orthogonality holds between the gradient field and the solenoidal field, which is similar to the continuous Hodge decomposition. Using the orthogonality, we present a novel analysis which shows that the convergence order is 1.5 in the $L^2$-norm for approximating the divergence-free vector field. Numerical results are presented to validate our analyses.

This paper studies the non-autonomous globally modified Navier-Stokes equations. The authors first prove that the associated process possesses a pullback attractor. Then they establish that there exists a unique family of Borel invariant probability measures on the pullback attractor.

A time-dependent method is coupled with the method of approximate particular solutions (MAPS) and the method of approximate fundamental solutions (MAFS) of Delta-shaped basis functions to solve a nonlinear Poisson-type boundary value problem on an irregular shaped domain. The problem is first converted into a sequence of time-dependent nonhomogeneous modified Helmholtz boundary value problems through a fictitious time integration method. Then the superposition princi- ple is applied to split the numerical solution at each time step into an approximate particular solution and a homogeneous solution. A Delta-shaped basis function is used to provide an approximation of the source function at each time step. This allows for an easy derivation of an approximate particular solution. The corresponding homogeneous boundary value problem is solved using MAFS, and also with the method of fundamental solutions (MFS) for comparison purposes. Numerical results support the accuracy and validity of this computational method.

In this paper we prove an exact controllability result for the Vlasov-Stokes system in the two-dimensional torus with small data by means of an internal control. We show that one can steer, in arbitrarily small time, any initial datum of class C^1 satisfying a smallness condition in certain weighted spaces to any final state satisfying the same conditions. The proof of the main result is achieved thanks to the return method and a Leray-Schauder fixed-point argument.

In this paper, we study the homogenization of a thermal diffusion problem in a highly heterogeneous medium formed by two constituents. The main characteristics of the medium are the discontinuity of the thermal conductivity over the domain as we go from one constituent to another and the presence of an imperfect interface between the two constituents, where both the temperature and the flux exhibit jumps. The limit problem, obtained via the periodic unfolding method, captures the influence of the jumps in the limit temperature field, in an additional source term, and in the correctors, as well.

We study Gaussian approximations to the distribution of a diffusion. The approximations are easy to compute: they are defined by two simple ordinary differential equations for the mean and the covariance. Time correlations can also be computed via solution of a linear stochastic differential equation. We show, using the Kullback-Leibler divergence, that the approximations are accurate in the small noise regime. An analogous discrete time setting is also studied. The results provide both theoretical support for the use of Gaussian processes in the approximation of diffusions, and methodological guidance in the construction of Gaussian approximations in applications.

We show that if the motion of a particle in a linear viscoelastic liquid is described by a Generalized Langevin Equation with generalized Rouse kernel, then the resulting velocity process satisfies equipartition of energy. In doing so, we present a closed formula for the improper integration along the positive line of the product of a rational polynomial function and of even powers of the sinc function. The only requirements on the rational function are sufficient decay at infinity, no purely real poles and only simple nonzero poles. In such, our results are applicable to a family of exponentially decaying kernels. The proof of the integral result follows from the residue theorem and equipartition of energy is a natural consequence thereof. Furthermore, we apply the integral result to obtain an explicit formulae for the covariance of the position process both in the general case and for the Rouse kernel. We also discuss a numerical algorithm based on residue calculus to evaluate the covariance for the Rouse kernel at arbitrary times.

This work focuses on the numerical approximation of the Shallow Water Equations (SWE) using a Lagrange-Projection type approach. We propose to extend to this context the recent implicitexplicit schemes developed in [C. Chalons, M. Girardin, and S. Kokh, SIAM J. Sci. Comput., 35(6):pp. a2874–a2902, 2013], [C. Chalons, M. Girardin, and S. Kokh, Communications in Computational Physics, 2016, to appear.] in the framework of compressible flows, with or without stiff source terms. These methods enable the use of time steps that are no longer constrained by the sound velocity thanks to an implicit treatment of the acoustic waves, and maintain accuracy in the subsonic regime thanks to an explicit treatment of the material waves. In the present setting, a particular attention will be also given to the discretization of the non-conservative terms in SWE and more specifically to the well-known well-balanced property. We prove that the proposed numerical strategy enjoys important non linear stability properties and we illustrate its behaviour past several relevant test cases.

In this paper, we establish the existence of time periodic solutions to the two and three dimensional compressible Navier-Stokes-Poisson equations with the linear damping, under some smallness and symmetry assumptions on the time periodic external force. Based on the uniform estimates and the topological degree theory, we prove the existence of a time periodic solution in a bounded domain. Finally, the existence result in the whole space is obtained by a limiting process.

We present a high-order Lagrange-projection like method for the approximation of the compressible Euler equations with a general equation of state. We extend the method introduced in Renac (Numer. Math., 2016, DOI 10.1007/s00211-016-0807-0) in the case of the isentropic gas dynamics to the compressible Euler equations and minimize the numerical dissipation by quantifying it from a parameter evaluated locally in each element of the mesh. The method is based on a decomposition between acoustic and transport operators associated to an implicit-explicit time integration, thus relaxing the constraint of acoustic waves on the time step as proposed in Coquel et. al. (Math. Comput., 79 (2010), pp. 1493-1533) in the context of a first-order finite volume method. We derive conditions on the time step and on a local numerical dissipation parameter to keep positivity of the mean value of the discrete density and internal energy in each element of the mesh and to satisfy a discrete inequality for the physical entropy at any approximation order in space. These results are then used to design limiting procedures in order to restore these properties at nodal values within elements. Moreover, the scheme is designed to avoid over-resolution in space and time in the low Mach number regime. Numerical experiments support the conclusions of the analysis and highlight stability and robustness of the present method when applied to either discontinuous flows or vacuum. Large time steps are allowed while keeping accuracy on smooth solutions even for low Mach number flows.

Highly-accurate numerical methods that can efficiently handle problems with interfaces and/or problems in domains with complex geometry are essential for the resolution of a wide range of temporal and spatial scales in many partial differential equations based models from Biology, Materials Science and Physics. In this paper we continue our work started in 1D, and we develop high-order accurate methods based on the Difference Potentials for 2D parabolic interface/composite domain problems. Extensive numerical experiments are provided to illustrate high-order accuracy and efficiency of the developed schemes.

We study a non-convex low-rank promoting penalty function, the transformed Schatten-1 (TS1), and its applications in matrix completion. The TS1 penalty, as a matrix quasi-norm defined on its singular values, interpolates the rank and the nuclear norm through a nonnegative parameter a \in (0, +\infty). We consider the unconstrained TS1 regularized low-rank matrix recovery problem and develop a fixed point representation for its global minimizer. The TS1 thresholding functions are in closed analytical form for all parameter values. The TS1 threshold values differ in subcritical (supercritical) parameter regime where the TS1 threshold functions are continuous (discontinuous). We propose TS1 iterative thresholding algorithms and compare them with some state-of-the-art algorithms on matrix completion test problems. For problems with known rank, a fully adaptive TS1 iterative thresholding algorithm consistently performs the best under different conditions, where ground truth matrices are generated by multivariate Gaussian, (0,1) uniform and Chi-square distributions. For problems with unknown rank, TS1 algorithms with an additional rank estimation procedure approach the level of IRucL-q which is an iterative reweighted algorithm, non-convex in nature and best in performance.

We study the long-time asymptotics for the so-called McKendrick-Von Foerster or renewal equation, a simple model frequently considered in structured population dynamics. In contrast to previous works, we can admit a bounded measure as initial data. To this end, we apply techniques from the calculus of variations that have not been employed previously in this context. We demonstrate how the generalized relative entropy method can be refined in the Radon measure framework.

Multi-scale modeling and numerical simulations of the isothermal chemical vapor infiltration (CVI) process for the fabrication of carbon fiber reinforced silicon carbide (C/SiC) composites was presented in [Y. Bai, X. Yue and Q. Zeng, Commun. Comput. Phys., 7:3 (2010), 597-612]. The homogenization theory, which played a fundamental role in the multi-scale algorithm, will be rigorously established in this paper. The governing system, which is a multi-scale reaction-diffusion equation, is different in the two stages of CVI process, so we will consider the homogenization for the two stages respectively. One of the main features is that the reaction only occurs on the surface of fiber, so it behaves as a singular surface source. The other feature is that in the second stage of the process when the micro pores inside the fiber bundles are all closed, the diffusion only occurs in the macro pores between fiber bundles and we face up with a problem in a locally periodic perforated domain.

We consider a one-dimensional version of a model obtained in [Engwer, C., Hunt, A., Surulescu, IMA J. Math. Med. Bio, 2015] and describing the anisotropic spread of tumor cells in a tissue network. The model consists of a reaction-diffusion-taxis equation for the density of tumor cells coupled with an ODE for the density of tissue fibers and allows for strong degeneracy both in the diffusion and the haptotaxis terms. In this setting we prove the global existence of weak solutions to an associated no-flux initial-boundary value problem. Numerical simulations are performed in order to illustrate the model behavior.

We consider the full time-harmonic Maxwell equations in the presence of a thin, high-contrast dielectric object. As an extension of previous work by two of the authors, we continue to study limits of the electric field as the thickness of the scatterer goes to zero simultaneously as the contrast goes to infinity. We present both analytical and computational results, including simulations which demonstrate that the interior transverse component of the electric field has limit zero, and a rigorous asymptotic approximation accurate outside of the scatterer. Finally, we propose an inversion method to recover the geometry of the scatterer given its two-dimensional plane and we present numerical simulations using this method.

The logarithmic KdV (log-KdV) equation admits global solutions in an energy space and exhibits Gaussian solitary waves. Orbital stability of Gaussian solitary waves is known to be an open problem. We address properties of solutions to the linearized log-KdV equation at the Gaussian solitary waves. By using the decomposition of solutions in the energy space in terms of Hermite functions, we show that the time evolution of the linearized log-KdV equation is related to a Jacobi difference operator with a limit circle at infinity. This exact reduction allows us to characterize both spectral and linear orbital stability of Gaussian solitary waves. We also introduce a convolution representation of solutions to the linearized log-KdV equation with the Gaussian weight and show that the time evolution in such a weighted space is dissipative with the exponential rate of decay.

We extend Brenier's transport collapse scheme on the initial-boundary value problem for scalar conservation laws. It is based on averaging out the solution to the corresponding kinetic equation, and it leads to a new solution concept for the problem under consideration. We also provide numerical examples.

In this paper, we obtain global well-posedness for the 2D damped Boussinesq equations. Based on the estimate of the damped Euler equations leading to the uniform corresponding bound (do not grow in time), we can achieve this goal by using a new decomposition technic. Comparing with the previous works, we do not need any small assumptions of the initial velocity. As an application of our method, we obtain a similar result for the 2D damped MHD equations.

This paper studies the stability of smooth traveling wave solutions to a nonlinear PDE problem in reducing image noise. Specifically, we prove that the solution to the Cauchy problem approaches to the traveling wave solution if the initial data is a small perturbation of the traveling wave. We use a weighted energy method to show that if the initial perturbation decays algebraically or exponentially as |x| \to \infty, then the Cauchy problem solution approaches to the traveling wave at corresponding rates as t\to \infty.