New nonlinear evolution equations are derived that generalize the system by Matsuno [16] and a terrain-following Boussinesq system by Nachbin [23]. The regime considers finite-amplitude surface gravity waves on a two-dimensional incompressible and inviscid fluid of, highly variable, finite depth. The asymptotic simplification of the nonlinear potential theory equations is performed through a perturbation anaylsis of the Dirichlet-to-Neumann operator on a highly corrugated strip. This is achieved through the use of a curvilinear coordinate system. Rather than doing a long wave expansion for the velocity potential, a Fourier-type operator is expanded in a wave steepness parameter. The novelty is that the topography can vary on a broad range of scales. It can also have a complex profile including that of a multiply-valued function. The resulting evolution equations are variable coefficient Boussinesq-type equations. These equations represent a fully dispersive system in the sense that the original (hyperbolic tangent) dispersion relation is not truncated. The formulation is done over a periodically extended domain so that, as an application, it produces efficient Fourier (FFT) solvers. A preliminary communication of this work has been published in the Physical Review Letters [1].

Magnetotellurics (MT) is a method of determining the electrical resistivity of the earth’s subsurface as a function of position by analyzing the electromagnetic (EM) field on the earth’s surface. It is a passive method, in that ambient EM radiation is used as a source. In this paper we consider model subsurfaces for MT that contain small scale random stratification; that is, we introduce random microlayers and allow the earth’s electrical properties to vary rapidly and randomly in space as the layer boundaries are crossed. The layers are not assumed to be plane, but are allowed to vary laterally in space in a direction that changes smoothly on the scale of an EM wavelength. By asymptotic analysis of the resulting stochastic differential equations with a small parameter we generalize previous results of White, Kohler and Srnka for plane layered media; we show that the resulting EM field may be approximated using a non-random effective medium theory, but with random corrections. These corrections are a gaussian random process which represents multiple scattering from the random microlayers. We show how the effective medium theory differs from the plane layered case, and derive a spatially varying correction for the EM field on the surface of the earth, which accounts for stratifications that are not planar.

The motion of a collection of vertical strings subject to horizontal linear vibrations in the plane can be described by a system of first order nonlinear conservations laws. This system -that we call the Chaplygin-Born-Infeld (CBI) system- is related to Magnetohydrodynamics and more specifically to its shallow water version. Then, each vibrating string can be interpreted as a magnetic line. The CBI system is also related to the Born-Infeld theory for the electromagnetic field, a nonlinear correction to the classical Maxwell’s equations.

Due to the linearity of vibrations, there is a priori no mechanism to prevent the strings to cross each other, at least for sufficiently large initial impulse. These crossings generate concentration sin- gularities in the CBI system. A numerical scheme is introduced to maintain order preserving strings beyond singularities. This order preserving scheme is shown to be convergent to a distinguished limit, which can be interpreted, through maximal monotone operator theory, as a vanishing viscosity limit of the CBI system. Finally, models of pressureless gas with sticky particles are revisited and a new formulation is provided.

In this paper, we present two methods in order to calibrate the local volatility with American put options. Both calibration methods use a least-square formulation and a descent algorithm. Pricing is done by solving parabolic variational inequalities, for which solution procedures by active set methods are discussed.

The first strategy consists in computing the optimality conditions and the descent direction needed by the optimization loop. This approach has been implemented both at the continuous and discrete levels. It requires a careful analysis of the underlying variational inequalities and of their discrete counterparts. In the numerical example presented here (American options on the FTSE index), the squared volatility is parameterized by a bicubic spline.

In the second approach, which works in low dimension, the descent directions are computed with Automatic Differentiation of computer programs implemented in C++.

We carry out an error analysis for the heterogeneous multi-scale method for the case when the macroscale process is that of gas dynamics or more generally nonlinear conservation laws and the microscale process is an atomistic model such as kinetic Monte Carlo methods or molecular dynamics (MD). We will consider problems of type B as defined in [4], i.e. the macroscale constitutive relations are unknown and are extracted from the microscopic model. In addition to the standard error in the macroscale solver, a new error term occurs in estimating the data, here the fluxes. This new error term consists of three parts: the relaxation error, the sampling error and the error due to the finite size of the atomistic simulation. Our results serve as guidelines for designing multiscale methods, as was done in [13, 16].

We introduce a modified particle method for semi-linear hyperbolic systems with highly oscillatory solutions. The main feature of this modified particle method is that we do not require different families of characteristics to meet at one point. In the modified particle method, we update the ith component of the solution along its own characteristics, and interpolate the other components of the solution from their own characteristic points to the ith characteristic point. We prove the convergence of the modified particle method essentially independent of the small scale for the variable coefficient Carleman model. The same result also applies to the non-resonant Broadwell model. Numerical evidence suggests that the modified particle method also converges essentially independent of the small scale for the original Broadwell model if a cubic spline interpolation is used.

Non blow-up of the 3D incompressible Euler Equations is proven for a class of three- dimensional initial data characterized by uniformly large vorticity in bounded cylindrical domains. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach of proving regularity is based on investigation of fast singular oscillating limits and nonlinear averaging methods in the context of almost periodic functions. We establish the global regularity of the 3D limit resonant Euler equations without any restriction on the size of 3D initial data. After establishing strong convergence to the limit resonant equations, we bootstrap this into the regularity on arbitrary large time intervals of the solutions of 3D Euler Equations with weakly aligned uniformly large vorticity at $t = 0$.

Speed ensemble of bistable (combustion) fronts in mean zero stationary Gaussian shear flows inside two and three dimensional channels is studied with a min-max variational principle. In the small root mean square regime of shear flows, a new class of multi-scale test functions are found to yield speed asymptotics. The quadratic speed enhancement law holds with probability arbitrarily close to one under the almost sure continuity (dimension two) and mean square Hölder regularity (dimension three) of the shear flows. Remarks are made on the quadratic and linear laws of front speed expectation in the small and large root mean square regimes.