Optimal Jittered Sampling for two Points in the Unit Square

Jittered Sampling is a refinement of the classical Monte Carlo sampling method. Instead of picking $n$ points randomly from $[0,1]^2$, one partitions the unit square into $n$ regions of equal measure and then chooses a point randomly from each partition. Currently, no good rules for how to partition the space are available. In this paper, we present a solution for the special case of subdividing the unit square by a decreasing function into two regions so as to minimize the expected squared $\mathcal{L}_2-$discrepancy. The optimal partitions are given by a \textit{highly} nonlinear integral equation for which we determine an approximate solution. In particular, there is a break of symmetry and the optimal partition is not into two sets of equal measure. We hope this stimulates further interest in the construction of good partitions

A fluctuating boundary integral method for Brownian suspensions

We present a fluctuating boundary integral method (FBIM) for overdamped Brownian Dynamics (BD) of two-dimensional periodic suspensions of rigid particles of complex shape immersed in a Stokes fluid. We develop a novel approach for generating Brownian displacements that arise in response to the thermal fluctuations in the fluid. Our approach relies on a first-kind boundary integral formulation of a mobility problem in which a random surface velocity is prescribed on the particle surface, with zero mean and covariance proportional to the Green's function for Stokes flow (Stokeslet). This approach yields an algorithm that scales linearly in the number of particles for both deterministic and stochastic dynamics, handles particles of complex shape, achieves high order of accuracy, and can be generalized to three dimensions and other boundary conditions. We show that Brownian displacements generated by our method obey the discrete fluctuation-dissipation balance relation (DFDB). Based on a recently-developed Positively Split Ewald method [A. M. Fiore, F. Balboa Usabiaga, A. Donev and J. W. Swan, J. Chem. Phys., 146, 124116, 2017], near-field contributions to the Brownian displacements are efficiently approximated by iterative methods in real space, while far-field contributions are rapidly generated by fast Fourier-space methods based on fluctuating hydrodynamics. FBIM provides the key ingredient for time integration of the overdamped Langevin equations for Brownian suspensions of rigid particles. We demonstrate that FBIM obeys DFDB by performing equilibrium BD simulations of suspensions of starfish-shaped bodies using a random finite difference temporal integrator.Comment: Submitted to J. Comp. Phy

FMM-accelerated solvers for the Laplace-Beltrami problem on complex surfaces in three dimensions

The Laplace-Beltrami problem on closed surfaces embedded in three dimensions arises in many areas of physics, including molecular dynamics (surface diffusion), electromagnetics (harmonic vector fields), and fluid dynamics (vesicle deformation). Using classical potential theory,the Laplace-Beltrami operator can be pre-/post-conditioned with integral operators whose kernel is translation invariant, resulting in well-conditioned Fredholm integral equations of the second-kind. These equations have the standard Laplace kernel from potential theory, and therefore the equations can be solved rapidly and accurately using a combination of fast multipole methods (FMMs) and high-order quadrature corrections. In this work we detail such a scheme, presenting two alternative integral formulations of the Laplace-Beltrami problem, each of whose solution can be obtained via FMM acceleration. We then present several applications of the solvers, focusing on the computation of what are known as harmonic vector fields, relevant for many applications in electromagnetics. A battery of numerical results are presented for each application, detailing the performance of the solver in various geometries.Comment: 18 pages, 5 tables, 3 figure

On the robustness of inverse scattering for penetrable, homogeneous objects with complicated boundary

The acoustic inverse obstacle scattering problem consists of determining the shape of a domain from measurements of the scattered far field due to some set of incident fields (probes). For a penetrable object with known sound speed, this can be accomplished by treating the boundary alone as an unknown curve. Alternatively, one can treat the entire object as unknown and use a more general volumetric representation, without making use of the known sound speed. Both lead to strongly nonlinear and nonconvex optimization problems for which recursive linearization provides a useful framework for numerical analysis. After extending our shape optimization approach developed earlier for impenetrable bodies, we carry out a systematic study of both methods and compare their performance on a variety of examples. Our findings indicate that the volumetric approach is more robust, even though the number of degrees of freedom is significantly larger. We conclude with a discussion of this phenomenon and potential directions for further research.Comment: 24 pages, 9 figure