8,233 research outputs found
Uniform shift estimates for transmission problems and optimal rates of convergence for the parametric Finite Element Method
Let \Omega \subset \RR^d, , be a bounded domain with
piecewise smooth boundary and let be an open subset of a
Banach space . Motivated by questions in "Uncertainty Quantification," we
consider a parametric family of uniformly strongly
elliptic, second order partial differential operators on . We
allow jump discontinuities in the coefficients. We establish a regularity
result for the solution u: \Omega \times U \to \RR of the parametric,
elliptic boundary value/transmission problem , , with
mixed Dirichlet-Neumann boundary conditions in the case when the boundary and
the interface are smooth and in the general case for . Our regularity and
well-posedness results are formulated in a scale of broken weighted Sobolev
spaces \hat\maK^{m+1}_{a+1}(\Omega) of Babu\v{s}ka-Kondrat'ev type in
, possibly augmented by some locally constant functions. This implies
that the parametric, elliptic PDEs admit a shift theorem that
is uniform in the parameter . In turn, this then leads to
-quasi-optimal rates of convergence (i.e. algebraic orders of convergence)
for the Galerkin approximations of the solution , where the approximation
spaces are defined using the "polynomial chaos expansion" of with respect
to a suitable family of tensorized Lagrange polynomials, following the method
developed by Cohen, Devore, and Schwab (2010)
Robust Distributed Parameter Estimation in Wireless Sensor Networks
abstract: Fully distributed wireless sensor networks (WSNs) without fusion center have advantages such as scalability in network size and energy efficiency in communications. Each sensor shares its data only with neighbors and then achieves global consensus quantities by in-network processing. This dissertation considers robust distributed parameter estimation methods, seeking global consensus on parameters of adaptive learning algorithms and statistical quantities.
Diffusion adaptation strategy with nonlinear transmission is proposed. The nonlinearity was motivated by the necessity for bounded transmit power, as sensors need to iteratively communicate each other energy-efficiently. Despite the nonlinearity, it is shown that the algorithm performs close to the linear case with the added advantage of power savings. This dissertation also discusses convergence properties of the algorithm in the mean and the mean-square sense.
Often, average is used to measure central tendency of sensed data over a network. When there are outliers in the data, however, average can be highly biased. Alternative choices of robust metrics against outliers are median, mode, and trimmed mean. Quantiles generalize the median, and they also can be used for trimmed mean. Consensus-based distributed quantile estimation algorithm is proposed and applied for finding trimmed-mean, median, maximum or minimum values, and identification of outliers through simulation. It is shown that the estimated quantities are asymptotically unbiased and converges toward the sample quantile in the mean-square sense. Step-size sequences with proper decay rates are also discussed for convergence analysis.
Another measure of central tendency is a mode which represents the most probable value and also be robust to outliers and other contaminations in data. The proposed distributed mode estimation algorithm achieves a global mode by recursively shifting conditional mean of the measurement data until it converges to stationary points of estimated density function. It is also possible to estimate the mode by utilizing grid vector as well as kernel density estimator. The densities are estimated at each grid point, while the points are updated until they converge to a global mode.Dissertation/ThesisDoctoral Dissertation Electrical Engineering 201
Polynomial bounds for the solutions of parametric transmission problems on smooth, bounded domains
We consider a \emph{family} of elliptic
second order differential operators on a domain
whose coefficients depend on the space variable and on a probability space. We allow the coefficients of
to have jumps over a fixed interface (independent of
). We obtain polynomial in the norms of the coefficients
estimates on the norm of the solution to the equation with transmission and mixed boundary conditions (we consider
``sign-changing'' problems as well). In particular, we show that, if and
the coefficients are smooth enough and follow a log-normal-type
distribution, then the map
is in , for all . The same is true for the norms
of the inverses of the resulting operators. We expect our estimates to be
useful in Uncertainty Quantification.Comment: We fixed a small .tex problem in the abstract on the site (the
manuscript has not changed
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