808 research outputs found
On ANOVA expansions and strategies for choosing the anchor point
The classic Lebesgue ANOVA expansion offers an elegant way to represent functions that depend on a high-dimensional set of parameters and it often enables a substantial reduction in the evaluation cost of such functions once the ANOVA representation is constructed. Unfortunately, the construction of the expansion itself is expensive due to the need to evaluate high-dimensional integrals. A way around this is to consider an alternative formulation, known as the anchored ANOVA expansion. This formulation requires no integrals but has an accuracy that depends sensitively on the choice of a special parameter, known as the anchor point. We present a comparative study of several strategies for the choice of this anchor point and argue that the optimal choice of this anchor point is the center point of a sparse grid quadrature. This choice induces no additional cost and, as we shall show, results in a natural truncation of the ANOVA expansion. The efficiency and accuracy is illustrated through several standard benchmarks and this choice is shown to outperform the alternatives over a range of applications. (C) 2010 Elsevier Inc. All rights reserved
Retail Concentration and Shopping Center Rents - A Comparison of Two Cities
This study aims primarily at testing whether, and to what extent, retail concentration within regional and super-regional shopping centers affects rent levels as well as the differential impact it may exert for various goods categories and sub-categories and in different urban contexts. In this paper, 1,499 leases distributed among eleven regional and super-regional shopping centers in Montreal and Quebec City, Canada, and negotiated over the 2000-2003 period are being considered. Unit base rents (base rent per sq. ft.) are regressed on a series of descriptors that include percentage rent rate, retail unit size (GLA), lease duration, shopping center age as well as 31 retail categories while the Herfindahl index is used as a measure of intra-category retail concentration. Findings suggest that while, overall, intra-category retail concentration affects base rent negatively, the magnitude and, eventually, direction of the impact varies depending on the nature of the activity and the market dynamics that prevail for the category considered.
Clustering based Multiple Anchors High-Dimensional Model Representation
In this work, a cut high-dimensional model representation (cut-HDMR)
expansion based on multiple anchors is constructed via the clustering method.
Specifically, a set of random input realizations is drawn from the parameter
space and grouped by the centroidal Voronoi tessellation (CVT) method. Then for
each cluster, the centroid is set as the reference, thereby the corresponding
zeroth-order term can be determined directly. While for non-zero order terms of
each cut-HDMR, a set of discrete points is selected for each input component,
and the Lagrange interpolation method is applied. For a new input, the cut-HDMR
corresponding to the nearest centroid is used to compute its response.
Numerical experiments with high-dimensional integral and elliptic stochastic
partial differential equation as backgrounds show that the CVT based multiple
anchors cut-HDMR can alleviate the negative impact of a single inappropriate
anchor point, and has higher accuracy than the average of several expansions
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Numerical methods for high-dimensional probability density function equations
In this paper we address the problem of computing the numerical solution to kinetic partial differential equations involving many phase variables. These types of equations arise naturally in many different areas of mathematical physics, e.g., in particle systems (Liouville and Boltzmann equations), stochastic dynamical systems (Fokker-Planck and Dostupov-Pugachev equations), random wave theory (Malakhov-Saichev equations) and coarse-grained stochastic systems (Mori-Zwanzig equations). We propose three different classes of new algorithms addressing high-dimensionality: The first one is based on separated series expansions resulting in a sequence of low-dimensional problems that can be solved recursively and in parallel by using alternating direction methods. The second class of algorithms relies on truncation of interaction in low-orders that resembles the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) framework of kinetic gas theory and it yields a hierarchy of coupled probability density function equations. The third class of algorithms is based on high-dimensional model representations, e.g., the ANOVA method and probabilistic collocation methods. A common feature of all these approaches is that they are reducible to the problem of computing the solution to high-dimensional equations via a sequence of low-dimensional problems. The effectiveness of the new algorithms is demonstrated in numerical examples involving nonlinear stochastic dynamical systems and partial differential equations, with up to 120 variables
Nonlinear approximation in bounded orthonormal product bases
We present a dimension-incremental algorithm for the nonlinear approximation
of high-dimensional functions in an arbitrary bounded orthonormal product
basis. Our goal is to detect a suitable truncation of the basis expansion of
the function, where the corresponding basis support is assumed to be unknown.
Our method is based on point evaluations of the considered function and
adaptively builds an index set of a suitable basis support such that the
approximately largest basis coefficients are still included. For this purpose,
the algorithm only needs a suitable search space that contains the desired
index set. Throughout the work, there are various minor modifications of the
algorithm discussed as well, which may yield additional benefits in several
situations. For the first time, we provide a proof of a detection guarantee for
such an index set in the function approximation case under certain assumptions
on the sub-methods used within our algorithm, which can be used as a foundation
for similar statements in various other situations as well. Some numerical
examples in different settings underline the effectiveness and accuracy of our
method
An adaptive ANOVA stochastic Galerkin method for partial differential equations with random inputs
It is known that standard stochastic Galerkin methods encounter challenges
when solving partial differential equations with high dimensional random
inputs, which are typically caused by the large number of stochastic basis
functions required. It becomes crucial to properly choose effective basis
functions, such that the dimension of the stochastic approximation space can be
reduced. In this work, we focus on the stochastic Galerkin approximation
associated with generalized polynomial chaos (gPC), and explore the gPC
expansion based on the analysis of variance (ANOVA) decomposition. A concise
form of the gPC expansion is presented for each component function of the ANOVA
expansion, and an adaptive ANOVA procedure is proposed to construct the overall
stochastic Galerkin system. Numerical results demonstrate the efficiency of our
proposed adaptive ANOVA stochastic Galerkin method
An adaptive reduced basis ANOVA method forhigh-dimensional Bayesian inverse problems
In Bayesian inverse problems sampling the posterior distribution is often a
challenging task when the underlying models are computationally intensive. To
this end, surrogates or reduced models are often used to accelerate the
computation. However, in many practical problems, the parameter of interest can
be of high dimensionality, which renders standard model reduction techniques
infeasible. In this paper, we present an approach that employs the ANOVA
decomposition method to reduce the model with respect to the unknown
parameters, and the reduced basis method to reduce the model with respect to
the physical parameters. Moreover, we provide an adaptive scheme within the
MCMC iterations, to perform the ANOVA decomposition with respect to the
posterior distribution. With numerical examples, we demonstrate that the
proposed model reduction method can significantly reduce the computational cost
of Bayesian inverse problems, without sacrificing much accuracy
Analysis of Reactor Simulations Using Surrogate Models.
The relatively recent abundance of computing resources has driven computational scientists to build more complex and approximation-free computer models of physical phenomenon. Often times, multiple high fidelity computer codes are coupled together in hope of improving the predictive powers of simulations with respect to experimental data. To improve the
predictive capacity of computer codes experimental data should be folded back into the parameters processed by the codes through optimization and calibration algorithms. However, application of such algorithms may be prohibitive since they generally require thousands of evaluations of computationally expensive, coupled, multiphysics codes. Surrogates models
for expensive computer codes have shown promise towards making optimization and calibration feasible.
In this thesis, non-intrusive surrogate building techniques are investigated for their applicability in nuclear engineering applications. Specifically, Kriging and the coupling of the anchored-ANOVA decomposition with collocation are utilized as surrogate building approaches. Initially, these approaches are applied and naively tested on simple reactor applications
with analytic solutions. Ultimately, Kriging is applied to construct a surrogate to analyze fission gas release during the Risø AN3 power ramp experiment using the fuel performance modeling code Bison. To this end, Kriging is extended from building surrogates for scalar quantities to entire time series using principal component analysis. A surrogate model is built for fission gas kinetics time series and the true values of relevant parameters are inferred by folding experimental data with the surrogate. Sensitivity analysis is also performed on the fission gas release parameters to gain insight into the underlying physics.PhDNuclear Engineering and Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/111485/1/yankovai_1.pd
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