A stochastic mode reduction strategy is applied to multiscale models with a deterministic energy-conserving fast sub-system. Specically, we consider situations where the slow variables are driven stochastically and interact with the fast sub-system in an energy-conserving fashion. Since the stochastic terms only affect the slow variables, the fast-subsystem evolves deter- ministically on a sphere of constant energy. However, in the full model the radius of the sphere slowly changes due to the coupling between the slow and fast dynamics. Therefore, the energy of the fast sub-system becomes an additional hidden slow variable that must be accounted for in order to apply the stochastic mode reduction technique to systems of this type.

The linear Poisson-Boltzmann equation (LPBE) is one well-known implicit solvent continuum model for computing the electrostatic potential of biomolecules in ionic solvent. To overcome its singular difficulty caused by Dirac delta distributions of point charges and to further improve its solution accuracy, we developed a new scheme for solving the current LPBE model, a new LPBE model, and a new LPBE finite element program package based on our previously proposed PBE solution decomposition. Numerical tests on biomolecules and a nonlinear Born ball model with an analytical solution validate the new LPBE solution decomposition schemes, demonstrate the effectiveness and efficiency of the new program package, and confirm that the new LPBE model can signicantly improve the solution accuracy of the current LPBE model.

This paper is concerned with the viscous polytropic uids in the two-dimensional (2D) space with vacuum as far field density. By means of weighted initial density, we obtain the local existence of classical solution to the Cauchy problem, in the case that the initial data satisfy a natural compatibility condition and the heat conduction coefficient is zero. Remember the blowup result of Xin [Z. Xin, Comm Pure Appl Math 51, 229-240, 1998], one should not expect the global smooth solution because the compactly supported initial density is included in our case.

We present an algorithm which computes the value function and optimal paths for a two-player static game, where the goal of one player is to maintain visibility of an adversarial player for as long as p ossible, and that of the adversarial player is to minimize this time. In a static game both players cho ose their controls at initial time and run in open-loop for t>0 until the end-game condition is met. Closed-loop (feedback strategy) games typically require solving PDEs in high dimensions and thus pose unsurmountable computational challenges. We demonstrate that, at the expense of a simpler information pattern that is more conservative towards one player, more memory and computationally efficient static games can be solved iteratively in the state space by the proposed PDE-based technique. In addition, we describe how this algorithm can be easily generalized to games with multiple evaders. Applications to target tracking and an extension to a feedback control game are also presented.

We study the Strang splitting scheme for quasilinear Schrodinger equations. We establish the convergence of the scheme for solutions with small initial data. We analyze the linear instability of the numerical scheme, which explains the numerical blow-up of large data solutions and connects to analytical breakdown of regularity of solutions to quasilinear Schrodinger equations. Numerical tests are performed for a modified version of the superfluid thin film equation.

We show that a smooth compactly supported solution to the relativistic Vlasov-Maxwell system exists as long as the L^6 norm of the macroscopic density of particles remains bounded.

We investigate a nonlocal wave equation with damping term and singular nonlinearity, which models an electrostatic micro-electro-mechanical system (MEMS) device. In the case of the relative strength parameter \lambda being small, the existence and uniqueness of the global solution are established. Moreover,the asymptotic result that the solution exponentially converges to the steady state solution is also proved. For large \lambda, quenching results of the solution are obtained.

We present a nonlinear predator-prey system consisting of a nonlocal conservation law for predators coupled with a parabolic equation for preys. The drift term in the predators' equation is a nonlocal function of the prey density, so that the movement of predators can be directed towards region with high prey density. Moreover, Lotka-Volterra type right hand sides describe the feeding. A theorem ensuring existence, uniqueness, continuous dependence of weak solutions and various stability estimates is proved, in any space dimension. Numerical integrations show a few qualitative features of the solutions.

We prove the existence and uniqueness of global strong solutions to the one dimensional, compressible Navier-Stokes system for the viscous and heat conducting ideal polytropic gas flow, when heat conductivity depends on temperature in power law of Chapman-Enskog. The results reported in this article is valid for initial boundary value problem with non-slip and heat insulated boundary along with smooth initial data with positive temperature and density without smallness assumption.

We prove the existence of piecewise polynomials strictly convex smooth functions which converge uniformly on compact subsets to the Aleksandrov solution of the Monge-Ampe quation. We extend the Aleksandrov theory to right hand side only locally integrable and on convex bounded domains not necessarily strictly convex. The result suggests that for the numerical resolution of the equation, it is enough to assume that the solution is convex and piecewise smooth.

In [C. De Lellis and L. Szekelyhidi, Ann. of Math. 170, 1417-1436m 2009] C. De Lellis and L. Szekelyhidi Jr. constructed wild solutions of the incompressible Euler equations using a reformulation of the Euler equations as a differential inclusion together with convex integration. In this article we adapt their construction to the system consisting of adding the transport of a passive scalar to the two-dimensional incompressible Euler equations.

We consider the large time behavior of solutions to defocusing nonlinear Schrodinger equation in the presence of a time dependent external potential. The main assumption on the potential is that it grows at most quadratically in space, uniformly with respect to the time variable. We show a general exponential control of first order derivatives and momenta, which yields a double exponential bound for higher Sobolev norms and momenta. On the other hand, we show that if the potential is an isotropic harmonic potential with a time dependent frequency which decays sufficiently fast, then Sobolev norms are bounded, and momenta grow at most polynomially in time, because the potential becomes negligible for large time: there is scattering, even though the potential is unbounded in space for fixed time.

We obtain new regularity criteria and smallness condition for the global regularity of the N-dimensional supercritical porous media equation. In particular, it is shown that in order to obtain global regularity result, one only needs to bound a partial derivative in one direction or the pressure scalar eld. Our smallness condition is also in terms of one direction, dropping conditions on (N-1) other directions completely, or the pressure scalar eld. The proof relies on key observations concerning the incompressibility of the velocity vector eld and the special identity derived from Darcy's law.

The vanishing viscosity limit of the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity c(\rho)=\epsilon \rho^\alpha (\alpha >0) is considered in the present paper. It is proved that given a rarefaction wave with one-side vacuum state to the compressible Euler equations, we can construct a sequence of solutions to the compressible Navier-Stokes equations which converge to the above rarefaction wave with vacuum as the viscosity tends to zero. Moreover, the convergence rate depending on \alpha is obtained for all \alpha >0. The main difficulty in our proof lies in the degeneracies of the density and the density-dependent viscosity at the vacuum region in the vanishing viscosity limit.

We consider the problem of marketing a new product in a population modelled as a random graph, in which each individual (node) has a random number of connections to other individuals. Marketing can occur via word of mouth along edges, or via advertising. Our main result is adaptation of the Miller model, describing the spread of an infectious disease, to this setting, leading to a generalized Bass marketing model. The Miller model can be directly applied to word- of-mouth marketing. The main challenge lies in revising the Miller model to incorporate advertisement, which we solve by introducing a marketing node that is connected to every individual in the popula- tion. We tested this model for Poisson and scale free random networks, and found excellent agreement with microscopic simulations. In the homogeneous limit where the number of individuals goes to \infty and the network is completely connected our model becomes the classical Bass model. We further present the generalization of this model to two competing products. For a completely connected network this model is again consistent with the known continuum limit. Numerical simulations show excellent agreement with microscopic simulations ob- tained via an adaptation of the Gillespie algorithm. Our model shows that, if the two products have the same word-of-mouth marketing rate on the network, then the ratio of their market shares is exactly the ratio of their advertisement rates.

We address the question of how a neuron integrates excitatory (E) and inhibitory (I ) synaptic inputsolutions. Using these asymptotic solutions, in the presence of E and I inputs, we can successfully reveal the underlying mechanisms of a dendritic integration rule, which was discovered in a recent experiment. Our analysis can be extended to the multi-branch case to characterize the E-I dendritic integration on any branches. The novel characterization is confirmed by the numerical simulation of a biologically realistic neuron.

This paper is concerned with problems of scattering of time-harmonic electromagnetic and acoustic waves from an infinite penetrable medium with a finite height modeled by the Helmholtz equation. On the lower boundary of the rough layer the Neumann or generalized impedance boundary condition is imposed. The scattered field in the unbounded homogeneous medium is required to satisfy the upward angular-spectrum representation. Using the variational approach, we prove uniqueness and existence of solutions in the standard space of finite energy for inhomogeneous source terms, and in appropriate weighted Sobolev spaces for incident point source waves in R^m (m=2,3) and incident plane waves in R^2. To avoid guided waves, we assume that the penetrable medium satisfies certain non-trapping and geometric conditions.

Asymptotic behaviors of stochastic long-short equations driven by random force, which is smooth enough in space and white noise in time, are mainly considered. The existence and uniqueness of solutions for stochastic long-short equations are obtained via Galerkin approximation by the stopping time and Borel-Cantelli Lemma on the basis of a priori estimates in the sense of expectation. A global random attractor and the existence of a stationary measure are investigated by Birkhoff ergodic theorem and Chebyshev inequality.

We aim to present a relaxation model that can be used in real simulations of dilute multicomponent reacting gases. The kinetic framework is the semi-classical approach with only one variable for the internal energy modes. The relaxation times for the internal energy modes are assumed to be smaller than the chemistry characteristic times. The strategy is the same as in [S. Brull, J. Schneider, Comm. Math. Sci. to appear]. That is a sum of operators for respectively the mechanical and chemical processes. The mechanical operator(s) is the "natural" extension to polyatomic gases of the method of moment relaxations presented in [S. Brull, J. Schneider, Cont. Mech. Thermodyn. 20, 63-74, 2008] and [S. Brull, V. Pavan and J. Schnieder, Eur. J. Mech. (B-Fluids) 33, 74-86, 2012]. The derivation of the chemical model lies on the chemical processes at thermal equilibria. It is shown that this BGK approach features the same properties as the Boltzmann equation: conservations and entropy production. Moreover null entropy production states are characterized by vanishing chemical production rates. We also study the hydrodynamic limit in the slow chemistry regime. Finally we show that the whole set of parameters that are used in the derivation of the model can be calculated by softwares such as EGlib or STANJAN.

In this note we examine the dynamical role played by inertial forces on the sur face temperature (or buoyancy) variance in strongly rotating, stratified flows with uniform potential vorticity fields and fractional dissipation. In particu lar, using a dynamic, multi-scale averaging process, we identify a sufficient c ondition for the existence of a direct temperature variance cascade across an i nertial range. While the result is consistent with the physical and numerical t heories of SQG turbulence, the condition triggering the cascade is more exotic, a fact reflecting the non-locality introduced by fractional dissipation. A comment regarding the scale-locality of the temperature variance flux is also included.

We consider relaxation systems of transport equations with heterogeneous source terms and with boundary conditions, which limits are scalar conservation laws. Classical bounds fail in this context and in particular BV estimates. They are the most standard and simplest way to prove compactness and convergence. We provide a novel and simple method to obtain partial BV regularity and strong compactness in this framework. The standard notion of entropy is not convenient either and we also indicate another, but closely related, notion. We give two examples motivated by renal flows which consist of 2 by 2 and 3 by 3 relaxation systems with 2-velocities but the method is more general.

Epitaxially grown heterogeneous nanowires present dislocations at the interface between the phases if their radius is big. We consider a corresponding variational discrete model with quadratic pairwise atomic interaction energy. By employing the notion of Gamma-convergence and a geometric rigidity estimate, we perform a discrete to continuum limit and a dimension reduction to a one-dimensional system. Moreover, we compare a defect-free model and models with dislocations at the interface and show that the latter are energetically convenient if the thickness of the wire is sufficiently large.

We investigate the time evolution of spin densities in a two-dimensional electron gas subjected to Rashba spin-orbit coupling on the basis of the quantum drift-diffusive model derived in [L. barletti and F. Mehats, J. Math. Phys. 51, 053304, 2010]. This model assumes the electrons to be in a quantum equilibrium state in the form of a Maxwellian operator. The resulting quantum drift-diffusion equations for spin-up and spin-down densities are coupled in a non-local manner via two spin chemical potentials (Lagrange multipliers) and via off-diagonal elements of the equilibrium spin density and spin current matrices, respectively. We present two space-time discretizations of the model, one semi-implicit and one explicit, which comprise also the Poisson equation in order to account for electron-electron interactions. In a first step pure time discretization is applied in order to prove the well-posedness of the two schemes, both of which are based on a functional formalism to treat the non-local relations between spin densities. We then use the fully space-time discrete schemes to simulate the time evolution of a Rashba electron gas confined in a bounded domain and subjected to spin-dependent external potentials. Finite difference approximations are first order in time and second order in space. The discrete functionals introduced are minimized with the help of a conjugate gradient-based algorithm, where the Newton method is applied in order to find the respective line minima. The numerical convergence in the long-time limit of a Gaussian initial condition towards the solution of the corresponding stationary Schr\"odinger-Poisson problem is demonstrated for different values of the parameters $\eps$ (semiclassical parameter), $\alpha$ (Rashba coupling parameter), $\Delta x$ (grid spacing) and $\Delta t$ (time step). Moreover, the performances of the semi-implicit and the explicit scheme are compared.

In this paper, we address the issue of designing a theoretically well- motivated and computationally efficient method ensuring topology preservation on image-registration-related deformation elds. The model is motivated by a mathematical characterization of topology preservation for a deformation eld mapping two subsets of Z^2, namely, positivity of the four approximations to the Jacobian determinant of the deformation on a square patch. The first step of the proposed algorithm thus consists in correcting the gradient vector field of the deformation (that does not comply with the topology preservation criteria) at the discrete level in order to fulfill this positivity condition. Once this step is achieved, it thus remains to reconstruct the deformation field, given its full set of discrete gradient vectors. We propose to decompose the reconstruction problem into independent problems of smaller dimensions, yielding a natural parallelization of the computations and enabling us to reduce drastically the computational time (up to 80 in some applications). For each subdomain, a functional minimization problem under Lagrange interpolation constraints is introduced and its well-posedness is studied: existence/uniqueness of the solution, characterization of the solution, convergence of the method when the number of data increases to infinity, discretization with the Finite Element Method and discussion on the properties of the matrix involved in the linear system. Numerical simulations based on OpenMP parallelization and MKL multi-threading demonstrating the ability of the model to handle large deformations (contrary to classical methods) and the interest of having decomposed the problem into smaller ones are provided.

We study orbital stability of solitary wave of the least energy for a nonlinear 2D Benney-Luke model of higher order related with long water waves with small amplitude in the presence of strong surface tension. We follow a variational approach which includes the characterization of the ground state solution set associated with solitary waves. We use the Hamiltonian structure of this model to establish the existence of an energy functional conserved in time for the modulated equation associated with this Benney-Luke type model. For wave speed near to zero or one, and in the regime of strong surface tension, we prove the orbital stability result by following a variational approach.

We study a system of self-propelled particles which interact with their neighbors via alignment and repulsion. The particle velocities result from self-propulsion and repulsion by close neighbors. The direction of self-propulsion is continuously aligned to that of the neighbors, up to some noise. A continuum model is derived starting from a mean-field kinetic description of the particle system. It leads to a set of non conservative hydrodynamic equations. We provide a numerical validation of the continuum model by comparison with the particle model. We also provide comparisons with other self-propelled particle models with alignment and repulsion.

In this paper an optimal control problem for a large system of interacting agents is considered using a kinetic perspective. As a prototype model we analyze a microscopic model of opinion formation under constraints. For this problem a Boltzmann type equation based on a model predictive control formulation is introduced and discussed. In particular, the receding horizon strategy permits to embed the minimization of suitable cost functional into binary particle interactions. The corresponding Fokker-Planck asymptotic limit is also derived and explicit expressions of stationary solutions are given. Several numerical results showing the robustness of the present approach are finally reported.

Owing to the Rosenau argument [Phys. Rev. A 46, 12-15, 1992], originally proposed to obtain a regularized version of the Chapman-Enskog expansion of hydrodynamics, we introduce a non-local linear kinetic equation which approximates a fractional diffusion equation. We then show that the solution to this approximation, apart of a rapidly vanishing in time perturbation, approaches the fundamental solution of the fractional diffusion (a Levy stable law) at large times.

We study a non-local parabolic Lotka-Volterra type equation describing a population struc- tured by a space variable x\in R^d and a phenotypical trait \theta \in {\Cal \Theta}. Considering diffusion, mu- tations and space-local competition between the individuals, we analyze the asymptotic (long time/longreal phase WKB ansatz, we prove that the propagation of the population in space can be de- scribed by a Hamilton-Jacobi equation with obstacle which is independent of \theta. The effective Hamiltonian is derived from an eigenvalue problem. The main difficulties are the lack of regularity estimates in the space variable, and the lack of comparison principle due to the non-local term.

In this paper, we study the asymptotic behavior of a state-based multiscale heterogeneous peridynamic model. The model involves nonlocal interaction forces with highly oscillatory perturbations representing the presence of heterogeneities on a finer spatial length scale. The two-scale convergence theory is established for a steady state variational problem associated with the multiscale linear model. We also examine the regularity of the limit nonlocal equation and present the strong approximation to the solution of the peridyanmic model via a suitably scaled two-scale limit.

This paper deals with the derivation of macroscopic equations from the underlying mesoscopic description that is suitable to capture the main features of pedestrian crowd dynamics. The interactions are modeled by means of theoretical tools of game theory, while the macroscopic equations are derived from asymptotic limits.

This paper concentrates on a (1+1)-dimensional nonlinear Dirac (NLD) equation with a general self-interaction, being a linear combination of the scalar, pseudoscalar, vector and axial vector self-interactions to the power of the integer k+1. The solitary wave solutions to the NLD equation are analytically derived, and the upper bounds of the hump number in the charge, energy and momentum densities for the solitary waves are proved analytically in theory. The results show that: (1) for a given integer k, the hump number in the charge density is not bigger than 4, while that in the energy density is not bigger than 3; (2) those upper bounds can only be achieved in the situation of higher nonlinearity, namely k\in {5,6,7, ...} for the charge density and k \in {3,5,7 ...} for the energy density; (3) the momentum density has the same multi-hump structure as the energy density; (4) more than two humps (resp. one hump) in the charge (resp. energy) density can only happen under the linear combination of the pseudoscalar self-interaction and at least one of the scalar and vector (or axial vector) self-interactions. Our results on the multi-hump structure will be interesting in the interaction dynamics for the NLD solitary waves.

We consider a diffuse interface model for phase separation of an isothermal incompressible binary fluid in a Brinkman porous medium. The coupled system consists of a convective Cahn-Hilliard equation for the phase field \phi, i.e., the difference of the (relative) concentrations of the two phases, coupled with a modified Darcy equation proposed by H.C. Brinkman in 1947 for the fluid velocity u. This equation incorporates a diffuse interface surface force proportional to \phi \nabla \mu, where \mu where 5 is the so-called chemical potential. We analyze the well-posedness of the resulting Cahn-Hilliard-Brinkman (CHB) system for (\phi, u). Then we establish the existence of a global attractor and the convergence of a given (weak) solution to a single equilibrium via Lojasiewicz-Simon inequality. Furthermore, we study the behavior of the solutions as the viscosity goes to zero, that is, when the CHB system approaches the Cahn-Hilliard-Hele-Shaw (CHHS) system. We first prove the existence of a weak solution to the CHHS system as limit of CHB solutions. Then, in dimension two, we estimate the difference of the solutions to CHB and CHHS systems in terms of the viscosity constant appearing in CHB.

This paper deals with the long time behavior of solutions to a "fractional Fokker-Planck" equation of the form \partial_t f= I[f] + \div(xf) where the operator I stands for a fractional Laplacian. We prove an exponential in time convergence towards equilibrium in new spaces. Indeed, such a result was already obtained in a L^2 space with a weight prescribed by the equilibrium in [Gentil, I. and ]Imbert, C., Asymp. Anal. 59, 3-4 (2008), 125-138] . We improve this result obtaining the convergence in a L^1 space with a polynomial weight. To do that, we take advantage of the recent paper [Gualdani, M.P., Mischler, S. and Mouhot, C., http://hal.archives-ouvertes.fr/ccsd-0049578 (2010).] in which an abstract theory of enlargement of the functional space of the semigroup decay is developed.

We study pathwise entropy solutions for scalar conservation laws with inhomogeneous fluxes and quasilinear multiplicative rough path dependence. This extends the previous work of Lions, Perthame and Souganidis who considered spatially independent and inhomogeneous fluxes with multiple paths and a single driving singular path respectively. The approach is motivated by the theory of stochastic viscosity solutions which relies on special test functions constructed by inverting locally the flow of the stochastic characteristics. For conservation laws this is best implemented at the level of the kinetic formulation which we follow here.

Selective image segmentation is the task of extracting one object of interest from an image,
based on minimal user input. Recent level-set based variational models have shown to
be effective and reliable, although they can be sensitive to initialization due to the minimization problems being nonconvex. This sometimes means that successful segmentation
relies too heavily on user input or a solution found is only a local minimizer, i.e. not the
correct solution. The same principle applies to variational models that extract all objects
in an image (global segmentation); however, in recent years, some have been successfully
reformulated as convex optimization problems, allowing global minimizers to be found.

There are, however, problems associated with extending the convex formulation to the
current selective models, which provides the motivation for the proposal of
a new selective
model. In this paper we propose a new selective segmentation model combining ideas from
global segmentation that can be reformulated as convex such that a global minimizer can
be found independent of initialization. Numerical results are given that demonstrate its
reliability in terms of removing the sensitivity to initialization present in previous models,
and its robustness to user input.

The present paper is devoted to the study of the Cauchy problem for the magnetic-curvature-driven electromagnetic fluid equation with random effects in a bounded domain of R^3. We first obtain a crucial property of the solution to O.U. process, thanks to the lemma, the local well-posedness of the equation with the initial and boundary value is established by the contraction mapping argument. Finally, by virtue of a priori estimates, the existence and uniqueness of global solution to the stochastic plasma equation is proved.

This paper addresses the problem of global well-posedness of a cou- pled system of Korteweg-de Vries equations, derived by Majda and Biello in the context of nonlinear resonant interaction of Rossby waves, in a periodic setting in homogeneous Sobolev spaces H^s, for s\ge 0. Our approach is based on a sussessive time-averaging method developed by Babin, Ilyin and Titi [A.V. Babin, A.A. Ilyin and E.S. Titi, Comm. Pure Appl. Math. 64, 591-648, 2011].

We study the large time behavior of solutions near a constant equilibrium state to the compressible Euler-Maxwell system in R^3. We first refine the global existence of solutions by assuming that the initial data is small in the H^3 norm but its higher order derivatives could be large. If further the initial data belongs to H^{-s} (0\le s \le 3/2) or B^{-s}_{2, \infty} (0\le s \le 3/2), then we obtain the various time decay rates of the solution and its higher order derivatives. As an immediate byproduct the L^p-L^2 (1\le p \le 2) type of the decay rates follows without requiring the smallness for L^p norm of the initial data. So far, our decay results are most comprehensive ones for the bipolar Euler-Maxwell system in R^3.

The main objective of this article is to study the order-disorder phase transition and pattern formation for systems with long-range repulsive interactions. The main focus is on a Cahn-Hilliard model with a nonlocal term in the corresponding energy functional, representing certain long-range repulsive interaction. We show that as soon as the trivial steady state loses its linear stability, the system always undergoes a dynamic transition to one of the three types-- continuous, catastrophic, or random-- forming different patterns/structures, such as lamellae, hexagonally packed cylinders, rectangles, and spheres. The types of transitions are dictated by a non-dimensional parameter, measuring the interactions between the long-range repulsive term and the quadratic and cubic nonlinearities in the model. In particular, the hexagonal pattern is unique to this long-range interaction, and it is captured by the corresponding two-dimensional reduced equations on the center manifold, which involve (degenerate) quadratic terms and non-degenerate cubic terms. Explicit information on the metastability and basins of attraction of different ordered states, corresponding to different patterns, are derived as well.

We study a general Ericksen-Leslie system with non-constant density, which describes the flow of nematic liquid crystal. In particular the model investigated here is associated with Parodi's relation. We prove that: in two dimension, the solutions are globally regular with general data; in three dimension, the solutions are globally regular with small initial data, or for short time with large data. Moreover, a weak-strong type of uniqueness result is obtained.

In recent work, Li et al. (Comm.\ Math.\ Sci., 7:81-107, 2009) developed a diffuse-domain method (DDM) for solving partial differential equations in complex, dynamic geometries with Dirichlet, Neumann, and Robin boundary conditions. The diffuse-domain method uses an implicit representation of the geometry where the sharp boundary is replaced by a diffuse layer with thickness $\epsilon$ that is typically proportional to the minimum grid size. The original equations are reformulated on a larger regular domain and the boundary conditions are incorporated via singular source terms. The resulting equations can be solved with standard finite difference and finite element software packages. Here, we present a matched asymptotic analysis of general diffuse-domain methods for Neumann and Robin boundary conditions. Our analysis shows that for certain choices of the boundary condition approximations, the DDM is second-order accurate in $\epsilon$. However, for other choices the DDM is only first-order accurate. This helps to explain why the choice of boundary-condition approximation is important for rapid global convergence and high accuracy. Our analysis also suggests correction terms that may be added to yield more accurate diffuse-domain methods. Simple modifications of first-order boundary condition approximations are proposed to achieve asymptotically second-order accurate schemes. Our analytic results are confirmed numerically in the $L^2$ and $L^\infty$ norms for selected test problems.

Global geometrical optics method is a new semi-classical approach for the high frequency linear waves proposed by the author in [Commun. Math. Sci., 11(1), 105-140, 2013]. In this paper, we rederive it in a more concise way. It is shown that the right candidate of solution ansatz for the high frequency wave equations is the extended WKB function, other than the WKB function used in the classical geometrical optics approximation. A new and main contribution of this paper is an interface analysis for the Helmholtz equation when the incident wave is of extended WKB-type. We derive asymptotic expressions for the reflected and/or transmitted propagating waves in the general case. These expressions are valid even when the incident rays include caustic points.

MUSCL extensions (Monotone Upstream-centered Schemes for Conservation Laws) of the Godunov numerical scheme for scalar conservation laws are shown to admit a rather simple reformulation when recast in the formalism of the Harr multi-resolution analysis of L^2(R). By pursuing this wavelet reformulation, a seeimingly new MUSCI-WB scheme is derived for advection-reaction equations which is stable for a Courant number up to 1 (instead of roughly 1/2). However these high-order reconstructions aren't likely to improve the handling of delication nonlinear wave interactions in the involved case of systems of Conservation/Balance laws.

Phase retrieval aims to recover a signal x\in C^m from its amplitude measurements |(x, a_i)|^2, i=1,2,...m, where a_i's are over-complete basis vectors, with m at least 3n-2 to ensure a unique solution up to a constant phase factor. The quadratic measurement becomes linear in terms of the rank-one matrix X=xx^*. Phase retrieval is then a rank- one minimization problem subject to linear constraint for which a convex relaxation based on trace-norm minimization (PhaseLift) has been extensively studied recently. At m=O(n), PhaseLift recovers with high probability the rank-one solution. In this paper, we present a precise proxy of rank-one condition via the dierence of trace and Frobenius norms which we call PhaseLiftO. The associated least squares minimization with this penalty as regularization is equivalent to the rank-one least squares problem under a mild condition on the measurement noise. Stable recovery error estimates are valid at m=O(n) with high probability. Computation of PhaseLiftO minimization is carried out by a convergent dierence of convex functions algorithm. In our numerical example, a_i's are Gaussian distributed. Numerical results show that PhaseLiftO outperforms PhaseLift and its nonconvex variant (log-determinant regularization), and successfully recovers signals near the theoretical lower limit on the number of measurements without the noise.

We study the Godunov scheme for a nonlinear Maxwell model arising in nonlinear optics, the Kerr model. This is a hyperbolic system of conservation laws with some eigenvalues of variable multiplicity, neither genuinely nonlinear nor linearly degenerate. The solution of the Riemann problem for the full-vector 6\times 6 system is constrcuted and proved to exist for all data. This solution is comprated to the one of the reduced Transverse Magnetic model. The scheme is implemented in one and two space dimensions. The results are very close to the ones obtained with a Kerr-Debye relaxation approximation.