7,241 research outputs found
High order operator splitting methods based on an integral deferred correction framework
Integral deferred correction (IDC) methods have been shown to be an efficient
way to achieve arbitrary high order accuracy and possess good stability
properties. In this paper, we construct high order operator splitting schemes
using the IDC procedure to solve initial value problems (IVPs). We present
analysis to show that the IDC methods can correct for both the splitting and
numerical errors, lifting the order of accuracy by with each correction,
where is the order of accuracy of the method used to solve the correction
equation. We further apply this framework to solve partial differential
equations (PDEs). Numerical examples in two dimensions of linear and nonlinear
initial-boundary value problems are presented to demonstrate the performance of
the proposed IDC approach.Comment: 33 pages, 22 figure
Bayesian Posterior Contraction Rates for Linear Severely Ill-posed Inverse Problems
We consider a class of linear ill-posed inverse problems arising from
inversion of a compact operator with singular values which decay exponentially
to zero. We adopt a Bayesian approach, assuming a Gaussian prior on the unknown
function. If the observational noise is assumed to be Gaussian then this prior
is conjugate to the likelihood so that the posterior distribution is also
Gaussian. We study Bayesian posterior consistency in the small observational
noise limit. We assume that the forward operator and the prior and noise
covariance operators commute with one another. We show how, for given
smoothness assumptions on the truth, the scale parameter of the prior can be
adjusted to optimize the rate of posterior contraction to the truth, and we
explicitly compute the logarithmic rate.Comment: 25 pages, 2 figure
Embeddings and immersions of tropical curves
We construct immersions of trivalent abstract tropical curves in the
Euclidean plane and embeddings of all abstract tropical curves in higher
dimensional Euclidean space. Since not all curves have an embedding in the
plane, we define the tropical crossing number of an abstract tropical curve to
be the minimum number of self-intersections, counted with multiplicity, over
all its immersions in the plane. We show that the tropical crossing number is
at most quadratic in the number of edges and this bound is sharp. For curves of
genus up to two, we systematically compute the crossing number. Finally, we use
our immersed tropical curves to construct totally faithful nodal algebraic
curves via lifting results of Mikhalkin and Shustin.Comment: 23 pages, 14 figures, final submitted versio
Positivity-Preserving Finite Difference WENO Schemes with Constrained Transport for Ideal Magnetohydrodynamic Equations
In this paper, we utilize the maximum-principle-preserving flux limiting
technique, originally designed for high order weighted essentially
non-oscillatory (WENO) methods for scalar hyperbolic conservation laws, to
develop a class of high order positivity-preserving finite difference WENO
methods for the ideal magnetohydrodynamic (MHD) equations. Our schemes, under
the constrained transport (CT) framework, can achieve high order accuracy, a
discrete divergence-free condition and positivity of the numerical solution
simultaneously. Numerical examples in 1D, 2D and 3D are provided to demonstrate
the performance of the proposed method.Comment: 21 pages, 28 figure
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