611 research outputs found
Local RBF approximation for scattered data fitting with bivariate splines
In this paper we continue our earlier research [4] aimed at developing effcient methods of local approximation suitable for the first stage of a spline based two-stage scattered data fitting algorithm. As an improvement to the pure polynomial local approximation method used in [5], a hybrid polynomial/radial basis scheme was considered in [4], where the local knot locations for the RBF terms were selected using a greedy knot insertion algorithm. In this paper standard radial local approximations based on interpolation or least squares are considered and a faster procedure is used for knot selection, signicantly reducing the computational cost of the method. Error analysis of the method and numerical results illustrating its performance are given
Curve network interpolation by quadratic B-spline surfaces
In this paper we investigate the problem of interpolating a B-spline curve
network, in order to create a surface satisfying such a constraint and defined
by blending functions spanning the space of bivariate quadratic splines
on criss-cross triangulations. We prove the existence and uniqueness of the
surface, providing a constructive algorithm for its generation. We also present
numerical and graphical results and comparisons with other methods.Comment: With respect to the previous version, this version of the paper is
improved. The results have been reorganized and it is more general since it
deals with non uniform knot partitions. Accepted for publication in Computer
Aided Geometric Design, October 201
Condition number analysis and preconditioning of the finite cell method
The (Isogeometric) Finite Cell Method - in which a domain is immersed in a
structured background mesh - suffers from conditioning problems when cells with
small volume fractions occur. In this contribution, we establish a rigorous
scaling relation between the condition number of (I)FCM system matrices and the
smallest cell volume fraction. Ill-conditioning stems either from basis
functions being small on cells with small volume fractions, or from basis
functions being nearly linearly dependent on such cells. Based on these two
sources of ill-conditioning, an algebraic preconditioning technique is
developed, which is referred to as Symmetric Incomplete Permuted Inverse
Cholesky (SIPIC). A detailed numerical investigation of the effectivity of the
SIPIC preconditioner in improving (I)FCM condition numbers and in improving the
convergence speed and accuracy of iterative solvers is presented for the
Poisson problem and for two- and three-dimensional problems in linear
elasticity, in which Nitche's method is applied in either the normal or
tangential direction. The accuracy of the preconditioned iterative solver
enables mesh convergence studies of the finite cell method
Data-driven quasi-interpolant spline surfaces for point cloud approximation
In this paper we investigate a local surface approximation, the Weighted
Quasi Interpolant Spline Approximation (wQISA), specifically designed for large
and noisy point clouds. We briefly describe the properties of the wQISA
representation and introduce a novel data-driven implementation, which combines
prediction capability and complexity efficiency. We provide an extended
comparative analysis with other continuous approximations on real data,
including different types of surfaces and levels of noise, such as 3D models,
terrain data and digital environmental data
Short and long-term wind turbine power output prediction
In the wind energy industry, it is of great importance to develop models that
accurately forecast the power output of a wind turbine, as such predictions are
used for wind farm location assessment or power pricing and bidding,
monitoring, and preventive maintenance. As a first step, and following the
guidelines of the existing literature, we use the supervisory control and data
acquisition (SCADA) data to model the wind turbine power curve (WTPC). We
explore various parametric and non-parametric approaches for the modeling of
the WTPC, such as parametric logistic functions, and non-parametric piecewise
linear, polynomial, or cubic spline interpolation functions. We demonstrate
that all aforementioned classes of models are rich enough (with respect to
their relative complexity) to accurately model the WTPC, as their mean squared
error (MSE) is close to the MSE lower bound calculated from the historical
data. We further enhance the accuracy of our proposed model, by incorporating
additional environmental factors that affect the power output, such as the
ambient temperature, and the wind direction. However, all aforementioned
models, when it comes to forecasting, seem to have an intrinsic limitation, due
to their inability to capture the inherent auto-correlation of the data. To
avoid this conundrum, we show that adding a properly scaled ARMA modeling layer
increases short-term prediction performance, while keeping the long-term
prediction capability of the model
Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms
We propose a novel method for constructing Hilbert transform (HT) pairs of
wavelet bases based on a fundamental approximation-theoretic characterization
of scaling functions--the B-spline factorization theorem. In particular,
starting from well-localized scaling functions, we construct HT pairs of
biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet
filters via a discrete form of the continuous HT filter. As a concrete
application of this methodology, we identify HT pairs of spline wavelets of a
specific flavor, which are then combined to realize a family of complex
wavelets that resemble the optimally-localized Gabor function for sufficiently
large orders.
Analytic wavelets, derived from the complexification of HT wavelet pairs,
exhibit a one-sided spectrum. Based on the tensor-product of such analytic
wavelets, and, in effect, by appropriately combining four separable
biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for
constructing 2D directional-selective complex wavelets. In particular,
analogous to the HT correspondence between the components of the 1D
counterpart, we relate the real and imaginary components of these complex
wavelets using a multi-dimensional extension of the HT--the directional HT.
Next, we construct a family of complex spline wavelets that resemble the
directional Gabor functions proposed by Daugman. Finally, we present an
efficient FFT-based filterbank algorithm for implementing the associated
complex wavelet transform.Comment: 36 pages, 8 figure
Filling holes under non-linear constraints
In this paper we handle the problem of filling the hole in the graphic of a surface by means of a patch that joins the original surface with C1-smoothness and fulfills an additional non-linear geometrical constraint regarding its area or its mean curvature at some points. Furthermore, we develop a technique to estimate the optimum area that the filling patch is expected to have that will allow us to determine optimum filling patches by means of a system of linear and quadratic equations. We present several numerical and graphical examples showing the effectiveness of the proposed method.Funding for open access publishing: Universidad de Granada/CBUANational funds through the FCT - Fundação
para a Ciência e a TecnologiaProjects UIDB/00297/2020 and UIDP/00297/2020
(Center for Mathematics and Applications
Filling holes under non-linear constraints
Publisher Copyright:
© 2023, The Author(s).In this paper we handle the problem of filling the hole in the graphic of a surface by means of a patch that joins the original surface with C1-smoothness and fulfills an additional non-linear geometrical constraint regarding its area or its mean curvature at some points. Furthermore, we develop a technique to estimate the optimum area that the filling patch is expected to have that will allow us to determine optimum filling patches by means of a system of linear and quadratic equations. We present several numerical and graphical examples showing the effectiveness of the proposed method.publishersversionpublishe
Recent advances in directional statistics
Mainstream statistical methodology is generally applicable to data observed
in Euclidean space. There are, however, numerous contexts of considerable
scientific interest in which the natural supports for the data under
consideration are Riemannian manifolds like the unit circle, torus, sphere and
their extensions. Typically, such data can be represented using one or more
directions, and directional statistics is the branch of statistics that deals
with their analysis. In this paper we provide a review of the many recent
developments in the field since the publication of Mardia and Jupp (1999),
still the most comprehensive text on directional statistics. Many of those
developments have been stimulated by interesting applications in fields as
diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics,
image analysis, text mining, environmetrics, and machine learning. We begin by
considering developments for the exploratory analysis of directional data
before progressing to distributional models, general approaches to inference,
hypothesis testing, regression, nonparametric curve estimation, methods for
dimension reduction, classification and clustering, and the modelling of time
series, spatial and spatio-temporal data. An overview of currently available
software for analysing directional data is also provided, and potential future
developments discussed.Comment: 61 page
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