3,970 research outputs found
Stochastic collocation on unstructured multivariate meshes
Collocation has become a standard tool for approximation of parameterized
systems in the uncertainty quantification (UQ) community. Techniques for
least-squares regularization, compressive sampling recovery, and interpolatory
reconstruction are becoming standard tools used in a variety of applications.
Selection of a collocation mesh is frequently a challenge, but methods that
construct geometrically "unstructured" collocation meshes have shown great
potential due to attractive theoretical properties and direct, simple
generation and implementation. We investigate properties of these meshes,
presenting stability and accuracy results that can be used as guides for
generating stochastic collocation grids in multiple dimensions.Comment: 29 pages, 6 figure
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GeD spline estimation of multivariate Archimedean copulas
A new multivariate Archimedean copula estimation method is proposed in a non-parametric setting. The method uses the so called Geometrically Designed splines (GeD splines), recently introduced by Kaishev et al. (2006 a,b) [10] and [11], to represent the cdf of a random variable Wµ, obtained through the probability integral transform of an Archimedean copula with parameter µ. Sufficient conditions for the GeD spline estimator to posses the properties of the underlying theoretical cdf, K(µ; t), of Wµ, are given. The latter conditions allow for defining a three-step estimation procedure for solving the resulting non-linear regression problem with linear inequality constraints. In the proposed procedure,finding the number and location of the knots and the coefficients of the unconstrained GeD spline estimator and solving the constraint least-squares optimisation problem, are separated. Thus, the resulting spline estimator ^K (^µ; t) is used to recover the generator and the related Archimedean copula by solving an ordinary differential equation. The proposed method is truly multivariate, it brings about numerical efficiency and as a result can be applied with large volumes of data and for dimensions d ¸ 2, as illustrated by the numerical examples presented
Structure-Aware Sampling: Flexible and Accurate Summarization
In processing large quantities of data, a fundamental problem is to obtain a
summary which supports approximate query answering. Random sampling yields
flexible summaries which naturally support subset-sum queries with unbiased
estimators and well-understood confidence bounds.
Classic sample-based summaries, however, are designed for arbitrary subset
queries and are oblivious to the structure in the set of keys. The particular
structure, such as hierarchy, order, or product space (multi-dimensional),
makes range queries much more relevant for most analysis of the data.
Dedicated summarization algorithms for range-sum queries have also been
extensively studied. They can outperform existing sampling schemes in terms of
accuracy on range queries per summary size. Their accuracy, however, rapidly
degrades when, as is often the case, the query spans multiple ranges. They are
also less flexible - being targeted for range sum queries alone - and are often
quite costly to build and use.
In this paper we propose and evaluate variance optimal sampling schemes that
are structure-aware. These summaries improve over the accuracy of existing
structure-oblivious sampling schemes on range queries while retaining the
benefits of sample-based summaries: flexible summaries, with high accuracy on
both range queries and arbitrary subset queries
Robust Adaptive Least Squares Polynomial Chaos Expansions in High-Frequency Applications
We present an algorithm for computing sparse, least squares-based polynomial
chaos expansions, incorporating both adaptive polynomial bases and sequential
experimental designs. The algorithm is employed to approximate stochastic
high-frequency electromagnetic models in a black-box way, in particular, given
only a dataset of random parameter realizations and the corresponding
observations regarding a quantity of interest, typically a scattering
parameter. The construction of the polynomial basis is based on a greedy,
adaptive, sensitivity-related method. The sequential expansion of the
experimental design employs different optimality criteria, with respect to the
algebraic form of the least squares problem. We investigate how different
conditions affect the robustness of the derived surrogate models, that is, how
much the approximation accuracy varies given different experimental designs. It
is found that relatively optimistic criteria perform on average better than
stricter ones, yielding superior approximation accuracies for equal dataset
sizes. However, the results of strict criteria are significantly more robust,
as reduced variations regarding the approximation accuracy are obtained, over a
range of experimental designs. Two criteria are proposed for a good
accuracy-robustness trade-off.Comment: 17 pages, 7 figures, 2 table
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