19,434 research outputs found
C-1-Cubic Quasi-Interpolation Splines over a CT Refinement of a Type-1 Triangulation
C1 continuous quasi-interpolating splines are constructed over CloughâTocher refinement
of a type-1 triangulation. Their BernsteinâBĂ©zier coefficients are directly defined from the known
values of the function to be approximated, so that a set of appropriate basis functions is not required.
The resulting quasi-interpolation operators reproduce cubic polynomials. Some numerical tests are
given in order to show the performance of the approximation scheme
From discretization to regularization of composite discontinuous functions
Discontinuities between distinct regions, described by different equation sets, cause difficulties for PDE/ODE solvers. We present a new algorithm that eliminates integrator discontinuities through regularizing discontinuities. First, the algorithm determines the optimum switch point between two functions spanning adjacent or overlapping domains. The optimum switch point is determined by searching for a âjump pointâ that minimizes a discontinuity between adjacent/overlapping functions. Then, discontinuity is resolved using an interpolating polynomial that joins the two discontinuous functions.
This approach eliminates the need for conventional integrators to either discretize and then link discontinuities through generating interpolating polynomials based on state variables or to reinitialize state variables when discontinuities are detected in an ODE/DAE system. In contrast to conventional approaches that handle discontinuities at the state variable level only, the new approach tackles discontinuity at both state variable and the constitutive equations level. Thus, this approach eliminates errors associated with interpolating polynomials generated at a state variable level for discontinuities occurring in the constitutive equations.
Computer memory space requirements for this approach exponentially increase with the dimension of the discontinuous function hence there will be limitations for functions with relatively high dimensions. Memory availability continues to increase with price decreasing so this is not expected to be a major limitation
Discovery of statistical equivalence classes using computer algebra
Discrete statistical models supported on labelled event trees can be
specified using so-called interpolating polynomials which are generalizations
of generating functions. These admit a nested representation. A new algorithm
exploits the primary decomposition of monomial ideals associated with an
interpolating polynomial to quickly compute all nested representations of that
polynomial. It hereby determines an important subclass of all trees
representing the same statistical model. To illustrate this method we analyze
the full polynomial equivalence class of a staged tree representing the best
fitting model inferred from a real-world dataset.Comment: 26 pages, 9 figure
Equidistribution of the Fekete points on the sphere
The Fekete points are the points that maximize a Vandermonde-type determinant
that appears in the polynomial Lagrange interpolation formula. They are well
suited points for interpolation formulas and numerical integration. We prove
the asymptotic equidistribution of the Fekete points in the sphere. The way we
proceed is by showing their connection with other array of points, the
Marcinkiewicz-Zygmund arrays and the interpolating arrays, that have been
studied recently
Polynomial-based non-uniform interpolatory subdivision with features control
Starting from a well-known construction of polynomial-based interpolatory 4-point schemes, in this paper we present
an original affine combination of quadratic polynomial samples that leads to a non-uniform 4-point scheme with edge
parameters. This blending-type formulation is then further generalized to provide a powerful subdivision algorithm
that combines the fairing curve of a non-uniform refinement with the advantages of a shape-controlled interpolation
method and an arbitrary point insertion rule. The result is a non-uniform interpolatory 4-point scheme that is unique
in combining a number of distinctive properties. In fact it generates visually-pleasing limit curves where special
features ranging from cusps and flat edges to point/edge tension effects may be included without creating undesired
undulations. Moreover such a scheme is capable of inserting new points at any positions of existing intervals, so that
the most convenient parameter values may be chosen as well as the intervals for insertion.
Such a fully flexible curve scheme is a fundamental step towards the construction of high-quality interpolatory subdivision surfaces with features control
- âŠ