1,269 research outputs found
Efficient spatial modelling using the SPDE approach with bivariate splines
Gaussian fields (GFs) are frequently used in spatial statistics for their
versatility. The associated computational cost can be a bottleneck, especially
in realistic applications. It has been shown that computational efficiency can
be gained by doing the computations using Gaussian Markov random fields (GMRFs)
as the GFs can be seen as weak solutions to corresponding stochastic partial
differential equations (SPDEs) using piecewise linear finite elements. We
introduce a new class of representations of GFs with bivariate splines instead
of finite elements. This allows an easier implementation of piecewise
polynomial representations of various degrees. It leads to GMRFs that can be
inferred efficiently and can be easily extended to non-stationary fields. The
solutions approximated with higher order bivariate splines converge faster,
hence the computational cost can be alleviated. Numerical simulations using
both real and simulated data also demonstrate that our framework increases the
flexibility and efficiency.Comment: 26 pages, 7 figures and 3 table
Multilevel refinable triangular PSP-splines (Tri-PSPS)
A multi-level spline technique known as partial shape preserving splines (PSPS) (Li and Tian, 2011) has recently been developed for the design of piecewise polynomial freeform geometric surfaces, where the basis functions of the PSPS can be directly built from an arbitrary set of polygons that partitions a giving parametric domain. This paper addresses a special type of PSPS, the triangular PSPS (Tri-PSPS), where all spline basis functions are constructed from a set of triangles. Compared with other triangular spline techniques, Tri-PSPS have several distinctive features. Firstly, for each given triangle, the corresponding spline basis function for any required degree of smoothness can be expressed in closed-form and directly written out in full explicitly as piecewise bivariate polynomials. Secondly, Tri-PSPS are an additive triangular spline technique, where the spline function built from a given triangle can be replaced with a set of refined spline functions built on a set of smaller triangles that partition the initial given triangle. In addition, Tri-PSPS are a multilevel spline technique, Tri-PSPS surfaces can be designed to have a continuously varying levels of detail, achieved simply by specifying a proper value for the smoothing parameter introduced in the spline functions. In terms of practical implementation, Tri-PSPS are a parallel computing friendly spline scheme, which can be easily implemented on modern programmable GPUs or on high performance computer clusters, since each of the basis functions of Tri-PSPS can be directly computed independent of each other in parallel
Recent Results on Near-Best Spline Quasi-Interpolants
Roughly speaking, a near-best (abbr. NB) quasi-interpolant (abbr. QI) is an
approximation operator of the form where the 's are B-splines and the 's
are linear discrete or integral forms acting on the given function . These
forms depend on a finite number of coefficients which are the components of
vectors for . The index refers to this sequence of
vectors. In order that for all polynomials belonging to some
subspace included in the space of splines generated by the 's, each
vector must lie in an affine subspace , i.e. satisfy some
linear constraints. However there remain some degrees of freedom which are used
to minimize for each . It is easy to
prove that is an upper bound of
: thus, instead of minimizing the infinite norm of
, which is a difficult problem, we minimize an upper bound of this norm,
which is much easier to do. Moreover, the latter problem has always at least
one solution, which is associated with a NB QI. In the first part of the paper,
we give a survey on NB univariate or bivariate spline QIs defined on uniform or
non-uniform partitions and already studied by the author and coworkers. In the
second part, we give some new results, mainly on univariate and bivariate
integral QIs on {\sl non-uniform} partitions: in that case, NB QIs are more
difficult to characterize and the optimal properties strongly depend on the
geometry of the partition. Therefore we have restricted our study to QIs having
interesting shape properties and/or infinite norms uniformly bounded
independently of the partition
Recent progress on univariate and multivariate polynomial and spline quasi-interpolants
Polynomial and spline quasi-interpolants (QIs) are practical and effective approximation operators. Among their remarkable properties, let us cite for example: good shape properties, easy computation and evaluation (no linear system to solve), uniform boundedness independently of the degree (polynomials) or of the partition (splines), good approximation order. We shall emphasize new results on various types of univariate and multivariate polynomial or spline QIs, depending on the nature of coefficient functionals, which can be differential, discrete or integral. We shall also present some applications of QIs to numerical methods
Representation and application of spline-based finite elements
Isogeometric analysis, as a generalization of the finite element method, employs spline methods to achieve the same representation for both geometric modeling and analysis purpose. Being one of possible tool in application to the isogeometric analysis, blending techniques provide strict locality and smoothness between elements. Motivated by these features, this thesis is devoted to the design and implementation of this alternative type of finite elements.
This thesis combines topics in geometry, computer science and engineering. The research is mainly focused on the algorithmic aspects of the usage of the spline-based finite elements in the context of developing generalized methods for solving different model problems.
The ability for conversion between different representations is significant for the modeling purpose. Methods for conversion between local and global representations are presented
Adaptive meshless centres and RBF stencils for Poisson equation
We consider adaptive meshless discretisation of the Dirichlet problem for Poisson equation based on numerical differentiation stencils obtained with the help of radial basis functions. New meshless stencil selection and adaptive refinement algorithms are proposed in 2D. Numerical experiments show that the accuracy of the solution is comparable with, and often better than that achieved by the mesh-based adaptive finite element method
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