A fast high-order solver for problems of scattering by heterogeneous bodies

A new high-order integral algorithm for the solution of scattering problems by heterogeneous bodies is presented. Here, a scatterer is described by a (continuously or discontinuously) varying refractive index n(x) within a two-dimensional (2D) bounded region; solutions of the associated Helmholtz equation under given incident fields are then obtained by high-order inversion of the Lippmann-Schwinger integral equation. The algorithm runs in O(Nlog(N)) operations where N is the number of discretization points. A wide variety of numerical examples provided include applications to highly singular geometries, high-contrast configurations, as well as acoustically/electrically large problems for which supercomputing resources have been used recently. Our method provides highly accurate solutions for such problems on small desktop computers in CPU times of the order of seconds

Inverse scattering problem for optical coherence tomography

We deal with the imaging problem of determining the internal structure of a body from backscattered laser light and low-coherence interferometry. Specifically, using the interference fringes that result when the backscattering of low-coherence light is made to interfere with the reference beam, we obtain maps detailing the values of the refractive index within the sample. Our approach accounts fully for the statistical nature of the coherence phenomenon; the numerical experiments that we present, which show image reconstructions of high quality, were obtained under noise floors exceeding those present for various experimental setups reported in the literature

Rise of correlations of transformation strains in random polycrystals

We investigate the statistics of the transformation strains that arise in random martensitic polycrystals as boundary conditions cause its component crystallites to undergo martensitic phase transitions. In our laminated polycrystal model the orientation of the n grains (crystallites) is given by an uncorrelated random array of the orientation angles θ_i, i = 1, . . . ,n. Under imposed boundary conditions the polycrystal grains may undergo a martensitic transformation. The associated transformation strains ε_i, i = 1, . . . ,n depend on the array of orientation angles, and they can be obtained as a solution to a nonlinear optimization problem. While the random variables θ_i, i = 1, . . . ,n are uncorrelated, the random variables ε_i, i = 1, . . . ,n may be correlated. This issue is central in our considerations. We investigate it in following three different scaling limits: (i) Infinitely long grains (laminated polycrystal of height L = ∞); (ii) Grains of finite but large height (L » 1); and (iii) Chain of short grains (L = l_0/(2n), l_0 « 1). With references to de Finetti’s theorem, Riesz’ rearrangement inequality, and near neighbor approximations, our analyses establish that under the scaling limits (i), (ii), and (iii) the arrays of transformation strains arising from given boundary conditions exhibit no correlations, long-range correlations, and exponentially decaying short-range correlations, respectivel

Asymptotic Behavior for a Coalescence Problem

Consider spherical particles of volume x having paint on a fraction y of their surface area. The particles are assumed to be homogeneously distributed at each time t, so that one can introduce the density number n (x, y, t). When collision between two particles occurs, the particles will coalesce if and only if they happen to touch each other, at impact, at points which do not belong to the painted portions of their surfaces. Introducing a dynamics for this model, we study the evolution of n (x, y, t) and, in particular, the asymptotic behavior of the mass x n (x, y, t) dx as t → ∞

On the O(1) Solution of Multiple-Scattering Problems

In this paper, we present a multiple-scattering solver for nonconvex geometries such as those obtained as the union of a finite number of convex surfaces. For a prescribed error tolerance, this algorithm exhibits a fixed computational cost for arbitrarily high frequencies. At the core of the method is an extension of the method of stationary phase, together with the use of an ansatz for the unknown density in a combined-field boundary integral formulation