22,964 research outputs found
Multilevel Monte Carlo for Random Degenerate Scalar Convection Diffusion Equation
We consider the numerical solution of scalar, nonlinear degenerate
convection-diffusion problems with random diffusion coefficient and with random
flux functions. Building on recent results on the existence, uniqueness and
continuous dependence of weak solutions on data in the deterministic case, we
develop a definition of random entropy solution. We establish existence,
uniqueness, measurability and integrability results for these random entropy
solutions, generalizing \cite{Mishr478,MishSch10a} to possibly degenerate
hyperbolic-parabolic problems with random data. We next address the numerical
approximation of random entropy solutions, specifically the approximation of
the deterministic first and second order statistics. To this end, we consider
explicit and implicit time discretization and Finite Difference methods in
space, and single as well as Multi-Level Monte-Carlo methods to sample the
statistics. We establish convergence rate estimates with respect to the
discretization parameters, as well as with respect to the overall work,
indicating substantial gains in efficiency are afforded under realistic
regularity assumptions by the use of the Multi-Level Monte-Carlo method.
Numerical experiments are presented which confirm the theoretical convergence
estimates.Comment: 24 Page
Higher Order Estimating Equations for High-dimensional Models
We introduce a new method of estimation of parameters in semiparametric and
nonparametric models. The method is based on estimating equations that are
-statistics in the observations. The -statistics are based on higher
order influence functions that extend ordinary linear influence functions of
the parameter of interest, and represent higher derivatives of this parameter.
For parameters for which the representation cannot be perfect the method leads
to a bias-variance trade-off, and results in estimators that converge at a
slower than -rate. In a number of examples the resulting rate can be
shown to be optimal. We are particularly interested in estimating parameters in
models with a nuisance parameter of high dimension or low regularity, where the
parameter of interest cannot be estimated at -rate, but we also
consider efficient -estimation using novel nonlinear estimators. The
general approach is applied in detail to the example of estimating a mean
response when the response is not always observed
Powerful nonparametric checks for quantile regression
We address the issue of lack-of-fit testing for a parametric quantile
regression. We propose a simple test that involves one-dimensional kernel
smoothing, so that the rate at which it detects local alternatives is
independent of the number of covariates. The test has asymptotically gaussian
critical values, and wild bootstrap can be applied to obtain more accurate ones
in small samples. Our procedure appears to be competitive with existing ones in
simulations. We illustrate the usefulness of our test on birthweight data.Comment: 32 pages, 2 figure
Spin glass reflection of the decoding transition for quantum error correcting codes
We study the decoding transition for quantum error correcting codes with the
help of a mapping to random-bond Wegner spin models.
Families of quantum low density parity-check (LDPC) codes with a finite
decoding threshold lead to both known models (e.g., random bond Ising and
random plaquette gauge models) as well as unexplored earlier generally
non-local disordered spin models with non-trivial phase diagrams. The decoding
transition corresponds to a transition from the ordered phase by proliferation
of extended defects which generalize the notion of domain walls to non-local
spin models. In recently discovered quantum LDPC code families with finite
rates the number of distinct classes of such extended defects is exponentially
large, corresponding to extensive ground state entropy of these codes.
Here, the transition can be driven by the entropy of the extended defects, a
mechanism distinct from that in the local spin models where the number of
defect types (domain walls) is always finite.Comment: 15 pages, 2 figure
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