24,717 research outputs found
Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations
The Navier--Stokes equations are commonly used to model and to simulate flow
phenomena. We introduce the basic equations and discuss the standard methods
for the spatial and temporal discretization. We analyse the semi-discrete
equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index
and quantify the numerical difficulties in the fully discrete schemes, that are
induced by the strangeness of the system. By analyzing the Kronecker index of
the difference-algebraic equations, that represent commonly and successfully
used time stepping schemes for the Navier--Stokes equations, we show that those
time-integration schemes factually remove the strangeness. The theoretical
considerations are backed and illustrated by numerical examples.Comment: 28 pages, 2 figure, code available under DOI: 10.5281/zenodo.998909,
https://doi.org/10.5281/zenodo.99890
Gradient-free Hamiltonian Monte Carlo with Efficient Kernel Exponential Families
We propose Kernel Hamiltonian Monte Carlo (KMC), a gradient-free adaptive
MCMC algorithm based on Hamiltonian Monte Carlo (HMC). On target densities
where classical HMC is not an option due to intractable gradients, KMC
adaptively learns the target's gradient structure by fitting an exponential
family model in a Reproducing Kernel Hilbert Space. Computational costs are
reduced by two novel efficient approximations to this gradient. While being
asymptotically exact, KMC mimics HMC in terms of sampling efficiency, and
offers substantial mixing improvements over state-of-the-art gradient free
samplers. We support our claims with experimental studies on both toy and
real-world applications, including Approximate Bayesian Computation and
exact-approximate MCMC.Comment: 20 pages, 7 figure
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