6 research outputs found
Fast and spectrally accurate summation of 2-periodic Stokes potentials
We derive a Ewald decomposition for the Stokeslet in planar periodicity and a
novel PME-type O(N log N) method for the fast evaluation of the resulting sums.
The decomposition is the natural 2P counterpart to the classical 3P
decomposition by Hasimoto, and is given in an explicit form not found in the
literature. Truncation error estimates are provided to aid in selecting
parameters. The fast, PME-type, method appears to be the first fast method for
computing Stokeslet Ewald sums in planar periodicity, and has three attractive
properties: it is spectrally accurate; it uses the minimal amount of memory
that a gridded Ewald method can use; and provides clarity regarding numerical
errors and how to choose parameters. Analytical and numerical results are give
to support this. We explore the practicalities of the proposed method, and
survey the computational issues involved in applying it to 2-periodic boundary
integral Stokes problems
The exponentially convergent trapezoidal rule
It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators
FAST AUTOMATIC BAYESIAN CUBATURE USING MATCHING KERNELS AND DESIGNS
Automatic cubatures approximate multidimensional integrals to user-specified
error tolerances. In many real-world integration problems, the analytical solution is
either unavailable or difficult to compute. To overcome this, one can use numerical
algorithms that approximately estimate the value of the integral.
For high dimensional integrals, quasi-Monte Carlo (QMC) methods are very
popular. QMC methods are equal-weight quadrature rules where the quadrature
points are chosen deterministically, unlike Monte Carlo (MC) methods where the
points are chosen randomly. The families of integration lattice nodes and digital nets
are the most popular quadrature points used. These methods consider the integrand
to be a deterministic function. An alternate approach, called Bayesian cubature,
postulates the integrand to be an instance of a Gaussian stochastic process