14,365 research outputs found
Optimal Uniform Convergence Rates and Asymptotic Normality for Series Estimators Under Weak Dependence and Weak Conditions
We show that spline and wavelet series regression estimators for weakly
dependent regressors attain the optimal uniform (i.e. sup-norm) convergence
rate of Stone (1982), where is the number of
regressors and is the smoothness of the regression function. The optimal
rate is achieved even for heavy-tailed martingale difference errors with finite
th absolute moment for . We also establish the asymptotic
normality of t statistics for possibly nonlinear, irregular functionals of the
conditional mean function under weak conditions. The results are proved by
deriving a new exponential inequality for sums of weakly dependent random
matrices, which is of independent interest.Comment: forthcoming in Journal of Econometric
Sum-factorization techniques in Isogeometric Analysis
The fast assembling of stiffness and mass matrices is a key issue in
isogeometric analysis, particularly if the spline degree is increased. We
present two algorithms based on the idea of sum factorization, one for matrix
assembling and one for matrix-free methods, and study the behavior of their
computational complexity in terms of the spline order . Opposed to the
standard approach, these algorithms do not apply the idea element-wise, but
globally or on macro-elements. If this approach is applied to Gauss quadrature,
the computational complexity grows as instead of as
previously achieved.Comment: 34 pages, 8 figure
A continuous analogue of the tensor-train decomposition
We develop new approximation algorithms and data structures for representing
and computing with multivariate functions using the functional tensor-train
(FT), a continuous extension of the tensor-train (TT) decomposition. The FT
represents functions using a tensor-train ansatz by replacing the
three-dimensional TT cores with univariate matrix-valued functions. The main
contribution of this paper is a framework to compute the FT that employs
adaptive approximations of univariate fibers, and that is not tied to any
tensorized discretization. The algorithm can be coupled with any univariate
linear or nonlinear approximation procedure. We demonstrate that this approach
can generate multivariate function approximations that are several orders of
magnitude more accurate, for the same cost, than those based on the
conventional approach of compressing the coefficient tensor of a tensor-product
basis. Our approach is in the spirit of other continuous computation packages
such as Chebfun, and yields an algorithm which requires the computation of
"continuous" matrix factorizations such as the LU and QR decompositions of
vector-valued functions. To support these developments, we describe continuous
versions of an approximate maximum-volume cross approximation algorithm and of
a rounding algorithm that re-approximates an FT by one of lower ranks. We
demonstrate that our technique improves accuracy and robustness, compared to TT
and quantics-TT approaches with fixed parameterizations, of high-dimensional
integration, differentiation, and approximation of functions with local
features such as discontinuities and other nonlinearities
- …