21 research outputs found
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 points randomly from , one partitions
the unit square into 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 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