5 research outputs found
On Equivalence of Anchored and ANOVA Spaces; Lower Bounds
We provide lower bounds for the norms of embeddings between
-weighted Anchored and ANOVA spaces of -variate
functions with mixed partial derivatives of order one bounded in norm
(). In particular we show that the norms behave polynomially in
for Finite Order Weights and Finite Diameter Weights if , and increase
faster than any polynomial in for Product Order-Dependent Weights and any
On efficient weighted integration via a change of variables
In this paper, we study the approximation of -dimensional -weighted
integrals over unbounded domains or using a
special change of variables, so that quasi-Monte Carlo (QMC) or sparse grid
rules can be applied to the transformed integrands over the unit cube. We
consider a class of integrands with bounded norm of mixed partial
derivatives of first order, where
The main results give sufficient conditions on the change of variables
which guarantee that the transformed integrand belongs to the standard Sobolev
space of functions over the unit cube with mixed smoothness of order one. These
conditions depend on and .
The proposed change of variables is in general different than the standard
change based on the inverse of the cumulative distribution function. We stress
that the standard change of variables leads to integrands over a cube; however,
those integrands have singularities which make the application of QMC and
sparse grids ineffective. Our conclusions are supported by numerical
experiments
Very Low Truncation Dimension for High Dimensional Integration Under Modest Error Demand
We consider the problem of numerical integration for weighted anchored and
ANOVA Sobolev spaces of -variate functions. Here is large including
. Under the assumption of sufficiently fast decaying weights, we
prove in a constructive way that such integrals can be approximated by
quadratures for functions with only variables, where
depends solely on the error demand and is
surprisingly small when is sufficiently large relative to .
This holds, in particular, for and arbitrary since
then for all . Moreover
does not depend on the function being integrated, i.e., is the same for all
functions from the unit ball of the space
Effective dimension of some weighted pre-Sobolev spaces with dominating mixed partial derivatives
This paper considers two notions of effective dimension for quadrature in
weighted pre-Sobolev spaces with dominating mixed partial derivatives. We begin
by finding a ball in those spaces just barely large enough to contain a
function with unit variance. If no function in that ball has more than
of its variance from ANOVA components involving interactions of
order or more, then the space has effective dimension at most in the
superposition sense. A similar truncation sense notion replaces the cardinality
of the ANOVA component by the largest index it contains. Some Poincar\'e type
inequalities are used to bound variance components by multiples of these
space's squared norm and those in turn provide bounds on effective dimension.
Very low effective dimension in the superposition sense holds for some spaces
defined by product weights in which quadrature is strongly tractable. The
superposition dimension is
just like the superposition dimension used in the multidimensional
decomposition method. Surprisingly, even spaces where all subset weights are
equal, regardless of their cardinality or included indices, have low
superposition dimension in this sense. This paper does not require periodicity
of the integrands
Efficient implementations of the Multivariate Decomposition Method for approximating infinite-variate integrals
In this paper we focus on efficient implementations of the Multivariate
Decomposition Method (MDM) for approximating integrals of -variate
functions. Such -variate integrals occur for example as expectations in
uncertainty quantification. Starting with the anchored decomposition , where the sum is over all
finite subsets of and each depends only on the
variables with , our MDM algorithm approximates the
integral of by first truncating the sum to some `active set' and then
approximating the integral of the remaining functions
term-by-term using Smolyak or (randomized) quasi-Monte Carlo (QMC) quadratures.
The anchored decomposition allows us to compute explicitly by
function evaluations of . Given the specification of the active set and
theoretically derived parameters of the quadrature rules, we exploit structures
in both the formula for computing and the quadrature rules to
develop computationally efficient strategies to implement the MDM in various
scenarios. In particular, we avoid repeated function evaluations at the same
point. We provide numerical results for a test function to demonstrate the
effectiveness of the algorithm