The aim of this paper is to study continuum models for surface diffusion taking into account free adatoms on the surface, which is of particular importance in the (self-assembled) growth of nanostructures. The extended model yields a coupled system of parabolic differential equations for the surface morphology and the adatom density, involving a cross-diffusion structure.

We investigate two different situations, namely the growth of a film on a substrate and the growth of a crystal-like structure (a closed curve or surface). An investigation of the equilibrium situation, which can be phrased as an energy minimization problem subject to a mass constraint, shows a different behaviour in both situations: for the film the equilibrium is attained when all atoms are attached to the surface, while for a crystal the adatom density does not vanish on the surface. The latter is also a deviation from the usual equilibrium theory, since the equilibrium shape will be strictly included in the Wulff shape. Moreover, it turns out that the total energy is not lower semicontinuous and non-convex for large adatom densities and rough surfaces.

The dynamics of the adatom surface diffusion model is investigated in detail for situations close to a flat surface in the film case and the situation close to a radially symmetric curve, both with an almost spatially homogeneous adatom density, where the cross-diffusion structure of the model and the decay to equilibrium can be studied in detail.

Finally, we discuss the numerical solution of the adatom surface diffusion model in the film case and provide various simulation results.

This paper provides a numerical analysis of a procedure for determining the effects of ionization on the nonlinear susceptibility coefficients of the hydrogen atom. To solve the relevant system of Schrödinger-type equations, we have developed a multidomain pseudospectral code with high accuracy symmetric finite differences to update cell boundary points. Using a conservative time stepping, one is then able to resolve the oscillatory solutions to the underlying equations, compute the ionization probability, and accurately determine the polarization. To gain insight into the physical mechanisms involved, we have calculated the full susceptibility and one which depends solely on the electronic bound-bound transitions. Our analysis reveals that saturation of the susceptibility occurs without including bound-continuum transitions. We have also found that a linear extrapolation of the total susceptibility versus the ionization probability to obtain the instantaneous susceptibility is quantitatively unreliable.

In this paper we deal with the numerical approximation of integro-differential equations arising in financial applications in which jump processes act as the underlying stochastic processes. Our aim is to find finite differences schemes which are high-order accurate for large time regimes.Therefore, we study the asymptotic time behavior of such equations and we define as asymptotic high-order schemes those schemes that are consistent with this behavior. Numerical tests are presented to investigate the efficiency and the accuracy of such approximations.

We present a detailed theoretical framework for statistical descriptions of neuronal networks and derive $(1+1)$-dimensional kinetic equations, without introducing any new parameters, directly from conductance-based integrate-and-fire neuronal networks. We describe the details of derivation of our kinetic equation, proceeding from the simplest case of one excitatory neuron, to coupled networks of purely excitatory neurons, to coupled networks consisting of both excitatory and inhibitory neurons. The dimension reduction in our theory is achieved via novel moment closures. We also describe the limiting forms of our kinetic theory in various limits, such as the limit of mean-driven dynamics and the limit of infinitely fast conductances. We establish accuracy of our kinetic theory by comparing its prediction with the full simulations of the original point-neuron networks. We emphasize that our kinetic theory is dynamically accurate, i.e., it captures very well the instantaneous statistical properties of neuronal networks under time-inhomogeneous inputs.

We give a method to represent strong solutions of stochastic differential equations driven by Lévy processes, explicitly. Furthermore we employ these explicit representations to study strong solutions of a certain class of SDE's, whose coefficients are not necessarily Lipschitz continuous.

In this paper we consider the development of hybrid numerical methods for the solution of hyperbolic relaxation problems with multiple scales. The main ingredients in the schemes are a suitable merging of probabilistic Monte Carlo methods in non-stiff regimes with high resolution shock capturing techniques in stiff ones. The key aspect in the development of the algorithms is the choice of a suitable hybrid representation of the solution. After the introduction of the different schemes the performance of the new methods is tested in the case of the Jin-Xin relaxation system and the Broadwell model.

In this paper we generalize the iterated refinement method, introduced by the authors in a recent work, to a time-continuous inverse scale-space formulation. The iterated refinement procedure yields a sequence of convex variational problems, evolving toward the noisy image.

The inverse scale space method arises as a limit for a penalization parameter tending to zero, while the number of iteration steps tends to infinity. For the limiting flow, similar properties as for the iterated refinement procedure hold. Specifically, when a discrepancy principle is used as the stopping criterion, the error between the reconstruction and the noise-free image decreases until termination, even if only the noisy image is available and a bound on the variance of the noise is known.

The inverse flow is computed directly for one-dimensional signals, yielding high quality restorations. In higher spatial dimensions, we introduce a relaxation technique using two evolution equations. These equations allow fast, accurate, efficient and straightforward implementation. We investigate the properties of these new types of flows and show their excellent denoising capabilities, wherein noise can be well removed with minimal loss of contrast of larger objects.

We study analytically and numerically a model describing front propagation of a KPP reaction in a fluid flow. The model consists of coupled one-dimensional reaction-diffusion equations with different drift coefficients. The main rigorous results give lower bounds for the speed of propagation that are linear in the drift coefficient, which agrees very well with the numerical observations. In addition, we find the optimal constant in a functional inequality of independent interest used in the proof.

We consider in this paper a simple model of liquid-vapor phase transition. The PDE system corresponds to the isothermal $p$-system and the pressure law is defined for all positive density and involves a constant zone. Such a pressure law can occur for instance when considering the Van der Waals equation of state complemented by the Maxwell's law. The Riemann problem is globally solved for all initial states, as well in the pure phases as in the mixture zone.

Traditional transmission travel-time tomography hinges on ray tracing techniques. We propose a PDE-based Eulerian approach to travel-time tomography so that we can avoid using the cumbersome ray-tracing technique. We start from the eikonal equation, define a mismatching functional and derive the gradient of the nonlinear functional by an adjoint state method. The resulting forward and adjoint problems can be efficiently solved by using the fast sweeping method; a limited memory BFGS method is used to drive the mismatching functional to zero with quadratic convergence. 2-D and 3-D numerical results as well as Marmousi synthetic velocity model demonstrate the robustness and the accuracy of the method.

We prove a refined upper bound on the size of the mixing layer in a simplified model of gravity driven miscible fingering that quantifies diffusive slowdown. Mathematically, the system we study is a multi-dimensional system of conservation laws that admits an exact one-dimensional closure for which the Lax entropy condition is not physically appropriate.