143 research outputs found
Successive Coordinate Search and Component-by-Component Construction of Rank-1 Lattice Rules
The (fast) component-by-component (CBC) algorithm is an efficient tool for
the construction of generating vectors for quasi-Monte Carlo rank-1 lattice
rules in weighted reproducing kernel Hilbert spaces. We consider product
weights, which assigns a weight to each dimension. These weights encode the
effect a certain variable (or a group of variables by the product of the
individual weights) has. Smaller weights indicate less importance. Kuo (2003)
proved that the CBC algorithm achieves the optimal rate of convergence in the
respective function spaces, but this does not imply the algorithm will find the
generating vector with the smallest worst-case error. In fact it does not. We
investigate a generalization of the component-by-component construction that
allows for a general successive coordinate search (SCS), based on an initial
generating vector, and with the aim of getting closer to the smallest
worst-case error. The proposed method admits the same type of worst-case error
bounds as the CBC algorithm, independent of the choice of the initial vector.
Under the same summability conditions on the weights as in [Kuo,2003] the error
bound of the algorithm can be made independent of the dimension and we
achieve the same optimal order of convergence for the function spaces from
[Kuo,2003]. Moreover, a fast version of our method, based on the fast CBC
algorithm by Nuyens and Cools, is available, reducing the computational cost of
the algorithm to operations, where denotes the number
of function evaluations. Numerical experiments seeded by a Korobov-type
generating vector show that the new SCS algorithm will find better choices than
the CBC algorithm and the effect is better when the weights decay slower.Comment: 13 pages, 1 figure, MCQMC2016 conference (Stanford
Adaptive Multidimensional Integration Based on Rank-1 Lattices
Quasi-Monte Carlo methods are used for numerically integrating multivariate
functions. However, the error bounds for these methods typically rely on a
priori knowledge of some semi-norm of the integrand, not on the sampled
function values. In this article, we propose an error bound based on the
discrete Fourier coefficients of the integrand. If these Fourier coefficients
decay more quickly, the integrand has less fine scale structure, and the
accuracy is higher. We focus on rank-1 lattices because they are a commonly
used quasi-Monte Carlo design and because their algebraic structure facilitates
an error analysis based on a Fourier decomposition of the integrand. This leads
to a guaranteed adaptive cubature algorithm with computational cost ,
where is some fixed prime number and is the number of data points
Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules
We show how to obtain a fast component-by-component construction algorithm
for higher order polynomial lattice rules. Such rules are useful for
multivariate quadrature of high-dimensional smooth functions over the unit cube
as they achieve the near optimal order of convergence. The main problem
addressed in this paper is to find an efficient way of computing the worst-case
error. A general algorithm is presented and explicit expressions for base~2 are
given. To obtain an efficient component-by-component construction algorithm we
exploit the structure of the underlying cyclic group.
We compare our new higher order multivariate quadrature rules to existing
quadrature rules based on higher order digital nets by computing their
worst-case error. These numerical results show that the higher order polynomial
lattice rules improve upon the known constructions of quasi-Monte Carlo rules
based on higher order digital nets
Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains
We explore the connection between fractional order partial differential
equations in two or more spatial dimensions with boundary integral operators to
develop techniques that enable one to efficiently tackle the integral
fractional Laplacian. In particular, we develop techniques for the treatment of
the dense stiffness matrix including the computation of the entries, the
efficient assembly and storage of a sparse approximation and the efficient
solution of the resulting equations. The main idea consists of generalising
proven techniques for the treatment of boundary integral equations to general
fractional orders. Importantly, the approximation does not make any strong
assumptions on the shape of the underlying domain and does not rely on any
special structure of the matrix that could be exploited by fast transforms. We
demonstrate the flexibility and performance of this approach in a couple of
two-dimensional numerical examples
Regularized least squares approximations on the sphere using spherical designs
2011-2012 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe
Well conditioned spherical designs for integration and interpolation on the two-sphere
2010-2011 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe
Hot new directions for quasi-Monte Carlo research in step with applications
This article provides an overview of some interfaces between the theory of
quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC
theoretical settings: first order QMC methods in the unit cube and in
, and higher order QMC methods in the unit cube. One important
feature is that their error bounds can be independent of the dimension
under appropriate conditions on the function spaces. Another important feature
is that good parameters for these QMC methods can be obtained by fast efficient
algorithms even when is large. We outline three different applications and
explain how they can tap into the different QMC theory. We also discuss three
cost saving strategies that can be combined with QMC in these applications.
Many of these recent QMC theory and methods are developed not in isolation, but
in close connection with applications
Application of quasi-Monte Carlo methods to PDEs with random coefficients -- an overview and tutorial
This article provides a high-level overview of some recent works on the
application of quasi-Monte Carlo (QMC) methods to PDEs with random
coefficients. It is based on an in-depth survey of a similar title by the same
authors, with an accompanying software package which is also briefly discussed
here. Embedded in this article is a step-by-step tutorial of the required
analysis for the setting known as the uniform case with first order QMC rules.
The aim of this article is to provide an easy entry point for QMC experts
wanting to start research in this direction and for PDE analysts and
practitioners wanting to tap into contemporary QMC theory and methods.Comment: arXiv admin note: text overlap with arXiv:1606.0661
Smolyak's algorithm: A powerful black box for the acceleration of scientific computations
We provide a general discussion of Smolyak's algorithm for the acceleration
of scientific computations. The algorithm first appeared in Smolyak's work on
multidimensional integration and interpolation. Since then, it has been
generalized in multiple directions and has been associated with the keywords:
sparse grids, hyperbolic cross approximation, combination technique, and
multilevel methods. Variants of Smolyak's algorithm have been employed in the
computation of high-dimensional integrals in finance, chemistry, and physics,
in the numerical solution of partial and stochastic differential equations, and
in uncertainty quantification. Motivated by this broad and ever-increasing
range of applications, we describe a general framework that summarizes
fundamental results and assumptions in a concise application-independent
manner
Is There a Valence-Specific Pattern in Emotional Conflict in Major Depressive Disorder? An Exploratory Psychological Study
Objective: Patients with major depressive disorder (MDD) clinically exhibit a deficit in positive emotional processing and are often distracted by especially negative emotional stimuli. Such emotional-cognitive interference in turn hampers the cognitive abilities of patients in their ongoing task. While the psychological correlates of such emotional conflict have been well identified in healthy subjects, possible alterations of emotional conflict in depressed patients remain to be investigated. We conducted an exploratory psychological study to investigate emotional conflict in MDD. We also distinguished depression-related stimuli from negative stimuli in order to check whether the depression-related distractors will induce enhanced conflict in MDD. Methods: A typical word-face Stroop paradigm was adopted. In order to account for valence-specificities in MDD, we included positive and general negative as well as depression-related words in the study. Results: MDD patients demonstrated a specific pattern of emotional conflict clearly distinguishable from the healthy control group. In MDD, the positive distractor words did not significantly interrupt the processing of the negative target faces, while they did in healthy subjects. On the other hand, the depression-related distractor words induced significant emotional conflict to the positive target faces in MDD patients but not in the healthy control group. Conclusion: Our findings demonstrated for the first time an altered valence-specific pattern in emotional conflict in MD
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