In their classical 1937 paper, Kolmogorov, Petrovsky and Piskunov proved that for a particular class of reaction-diffusion equations on the real line the solution of the initial value problem with the initial data in the form of a unit step propagates at long times with constant velocity equal to that of a certain special traveling wave solution. This type of a propagation result has since been established for a number of general classes of reaction-diffusion-advection problems in cylinders. Here we show that in problems without advection or in the presence of transverse advection by a potential flow these results do not rely on the specifics of the problem. Instead, they are a consequence of the fact that the equation considered is a gradient flow in an exponentially weighted L²-space generated by a certain functional, when the dynamics is considered in the reference frame moving with constant velocity along the cylinder axis. We show that independently of the details of the problem only three propagation scenarios are possible in the above context: no propagation, a “pulled” front, or a “pushed” front. The choice of the scenario is completely characterized via a minimization problem.

In this paper, we present a class of explicit numerical methods for stiff Itô stochastic differential equations (SDEs). These methods are as simple to program and to use as the well-known Euler-Maruyama method, but much more efficient for stiff SDEs. For such problems, it is well known that standard explicit methods face step-size reduction. While semi-implicit methods can avoid these problems at the cost of solving (possibly large) nonlinear systems, we show that the step- size reduction phenomena can be reduced significantly for explicit methods by using stabilization techniques. Stabilized explicit numerical methods called S-ROCK (for stochastic orthogonal Runge- Kutta Chebyshev) have been introduced as an alternative to (semi-) implicit methods for the solution of stiff stochastic systems. In this paper we discuss a genuine Itô version of the S-ROCK methods which avoid the use of transformation formulas from Stratonovich to Itô calculus. This is important for many applications. We present two families of methods for one-dimensional and multi-dimensional Wiener processes. We show that for stiff problems, significant improvement over classical explicit methods can be obtained. Convergence and stability properties of the methods are discussed and numerical examples as well as applications to the simulation of stiff chemical Langevin equations are presented.

We say that the vanishing viscosity limit holds in the classical sense if the velocity for a solution to the Navier-Stokes equations converges in the energy norm uniformly in time to the velocity for a solution to the Euler equations. We prove, for a bounded domain in dimension 2 or higher, that the vanishing viscosity limit holds in the classical sense if and only if a vortex sheet forms on the boundary.

We present an algorithm for interpolating the visible portions of a point loud that are sampled from opaque objects in the environment. Our algorithm projects point clouds onto a sphere entered at the observing location and performs essentially non-oscillatory (ENO) interpolation of the projected data. Curvatures of the occluding objects can be approximated and used in many ways. We demonstrate how our visibility formulation can be incorporated into novel algorithms for mapping unknown environments with single or multiple observers, and target finding problems. A convergence proof is provided indicating suitability of our algorithm for some canonical types of environments. Various postprocessing optimization techniques are considered to obtain a more uniform exposure of the region along the path.

Bregman divergences are generalizations of the well known Kullback-Leibler divergence. They are based on convex functions and have recently received great attention. We present a class of “squared root metrics” based on Bregman divergences. They can be regarded as natural generalization of Euclidean distance. We provide necessary and sufficient conditions for a convex function so that the square root of its associated average Bregman divergence is a metric.

Bregman divergences have played an important role in many research areas. Divergence is a measure of dissimilarity and by itself is not a metric. If a function of the divergence is a metric, then it becomes much more powerful. In Part 1 we have given necessary and sufficient conditions on the convex function in order that the square root of the averaged associated divergence is a metric. In this paper we provide a min-max approach to getting a metric from Bregman divergence. We show that the “capacity” to the power 1/e is a metric.

We study equilibrium states of the Smoluchowski equation for rigid, rod-like polymer ensembles. We start with several cases in the three dimensional space: a) nematic polymers where the only intermolecular interaction is the excluded volume effect, modeled using the Maier-Saupe potential, b) dipolar nematic polymers where the intermolecular interaction consists of the dipole- dipole potential and the Maier-Saupe potential, c) dipolar nematic polymers in the presence of a stretching elongational flow, and d) nematic polymers in higher dimensional space. For each of the cases a), b) and c), it has been established separately with various mathematical manipulations that all stable equilibrium states have rotational symmetry. In this study, we present a unified view of the rotational symmetry of cases a), b) and c). Specifically, in cases a), b) and c), the rotational symmetry is determined by a key inequality. The inequality, once established for case a), is extended elegantly to cases b) and c). Furthermore, this inequality is used in case d) to establish the rotational symmetry of equilibrium states of nematic polymers in higher dimensional space. In three dimensional space, rotational symmetry simply means axisymmetry. In higher dimensional space, rotational symmetry is more complex in structure. For example, in four dimensional space, rotational symmetry may be around a one dimensional sub-space (i.e., axisymmetry) or it may be around a two dimensional sub-space. Nevertheless, the rotational symmetry significantly simplifies the classification of equilibrium states. We calculate and present phase diagrams of nematic polymers in higher dimensional spaces.

The existence of inertial manifolds for a Smoluchowski equation arising in the 2D Doi-Hess model for liquid crystalline polymers subjected to a shear flow is investigated. The presence of a non-variational drift term dramatically complicates the long-term dynamics from the variational gradient case, in which it is solely characterized by the steady states. Several transformations are used in order to transform the equation into a form suitable for application of the standard theory of inertial manifolds. A nonlinear and nonlocal transformation developed in Inertial manifolds for a Smoluchowski equation on a circle and Inertial manifolds for a Smoluchowski equation on the unit sphere,, to appear, is used to eliminate the first-order derivative from the micro-micro interaction term. A traveling wave transformation eliminates the first-order derivative from the non-variational term, transforming the equation into a nonautonomous one for which the theory of nonautonomous inertial manifolds applies.

The solution to the Schrödinger equation is highly oscillatory when the rescaled Planck constant $\varepsilon$ is small in the semiclassical regime. A direct numerical simulation requires the mesh size to be $\emph{O}(\varepsilon)$. The Gaussian beam method is an efficient way to solve the high frequency wave equations asymptotically, outperforming the geometric optics method in that the Gaussian beam method is accurate even at caustics.

In this paper, we solve the Schrödinger equation using both the Lagrangian and Eulerian formulations of the Gaussian beam methods. A new Eulerian Gaussian beam method is developed using the level set method based only on solving the (complex-valued) homogeneous Liouville equations. A major contribution here is that we are able to construct the Hessian matrices of the beams by using the level set function’s first derivatives. This greatly reduces the computational cost in computing the Hessian of the phase function in the Eulerian framework, yielding an Eulerian Gaussian beam method with computational complexity comparable to that of the geometric optics but with a much better accuracy around caustics.

We verify through several numerical experiments that our Gaussian beam solutions are good approximations to Schrödinger solutions even at caustics. We also numerically study the optimal relation between the number of beams and the rescaled Planck constant $\varepsilon$ in the Gaussian beam summation.

We consider the multi-objective optimal dopant profiling of semiconductor devices. The two objectives are to gain a higher on-state current while the off-state current is kept small. This design question is treated as a constrained optimization problem, where the constraints are given by the stationary drift-diffusion model for the on-state and the linearized drift-diffusion model for the off-state. Using the doping profile as a state variable and the electrostatic potential as the new design variable, we obtain a simpler optimization problem, whose Karush-Kuhn-Tucker conditions partially decouple. Based on this observation we can construct a very efficient iterative optimization algorithm, which avoids solving the fully coupled drift-diffusion system. Due to the simple structure of the adjoint equations, this algorithm can be easily included into existing semiconductor simulation tools. The efficiency and success of this multi-objective design approach is underlined by various numerical examples.

We introduce and discuss a kinetic model for wealth distribution in a simple market economy which is built of a number of countries or social groups. Our approach is based on the model with risky investments introduced by Cordier, Pareschi, and Toscani in On a kinetic model for a simple market economy, and borrows ideas from the kinetic theory of mixtures of rarefied gases. Wealth is exchanged by individuals inside these countries (domestic trade) as well as in between different countries (international trade). Under a suitable scaling we derive a system of Fokker-Planck type equations and discuss its extension to a two-dimensional model with distributed trading propensity. Theoretical and numerical results for two groups show that the wealth distribution develops a bimodal (and in general, a polymodal) shape.

We investigate the numerical approximation of (discontinuous) entropy solutions to nonlinear hyperbolic conservation laws posed on a Lorentzian manifold. Our main result establishes the convergence of monotone and first-order finite volume schemes for a large class of (space and time) triangulations. The proof relies on a discrete version of entropy inequalities and an entropy dissipation bound, which take into account the manifold geometry and were originally discovered by Cockburn, Coquel, and LeFloch in the (flat) Euclidian setting.

We prove that the force-based quasicontinuum method converges uniformly with first order accuracy.