4,618 research outputs found
Hermite matrix in Lagrange basis for scaling static output feedback polynomial matrix inequalities
Using Hermite's formulation of polynomial stability conditions, static output
feedback (SOF) controller design can be formulated as a polynomial matrix
inequality (PMI), a (generally nonconvex) nonlinear semidefinite programming
problem that can be solved (locally) with PENNON, an implementation of a
penalty method. Typically, Hermite SOF PMI problems are badly scaled and
experiments reveal that this has a negative impact on the overall performance
of the solver. In this note we recall the algebraic interpretation of Hermite's
quadratic form as a particular Bezoutian and we use results on polynomial
interpolation to express the Hermite PMI in a Lagrange polynomial basis, as an
alternative to the conventional power basis. Numerical experiments on benchmark
problem instances show the substantial improvement brought by the approach, in
terms of problem scaling, number of iterations and convergence behavior of
PENNON
Nodal bases for the serendipity family of finite elements
Using the notion of multivariate lower set interpolation, we construct nodal
basis functions for the serendipity family of finite elements, of any order and
any dimension. For the purpose of computation, we also show how to express
these functions as linear combinations of tensor-product polynomials.Comment: Pre-print of version that will appear in Foundations of Computational
Mathematic
Ellipse-preserving Hermite interpolation and subdivision
We introduce a family of piecewise-exponential functions that have the
Hermite interpolation property. Our design is motivated by the search for an
effective scheme for the joint interpolation of points and associated tangents
on a curve with the ability to perfectly reproduce ellipses. We prove that the
proposed Hermite functions form a Riesz basis and that they reproduce
prescribed exponential polynomials. We present a method based on Green's
functions to unravel their multi-resolution and approximation-theoretic
properties. Finally, we derive the corresponding vector and scalar subdivision
schemes, which lend themselves to a fast implementation. The proposed vector
scheme is interpolatory and level-dependent, but its asymptotic behaviour is
the same as the classical cubic Hermite spline algorithm. The same convergence
properties---i.e., fourth order of approximation---are hence ensured
Sparse-grid polynomial interpolation approximation and integration for parametric and stochastic elliptic PDEs with lognormal inputs
By combining a certain approximation property in the spatial domain, and
weighted -summability of the Hermite polynomial expansion coefficients
in the parametric domain obtained in [M. Bachmayr, A. Cohen, R. DeVore and G.
Migliorati, ESAIM Math. Model. Numer. Anal. (2017), 341-363] and [M.
Bachmayr, A. Cohen, D. D\~ung and C. Schwab, SIAM J. Numer. Anal. (2017), 2151-2186], we investigate linear non-adaptive methods of fully
discrete polynomial interpolation approximation as well as fully discrete
weighted quadrature methods of integration for parametric and stochastic
elliptic PDEs with lognormal inputs. We explicitly construct such methods and
prove corresponding convergence rates in of the approximations by them,
where is a number characterizing computation complexity. The linear
non-adaptive methods of fully discrete polynomial interpolation approximation
are sparse-grid collocation methods. Moreover, they generate in a natural way
discrete weighted quadrature formulas for integration of the solution to
parametric and stochastic elliptic PDEs and its linear functionals, and the
error of the corresponding integration can be estimated via the error in the
Bochner space norm of the generating methods
where is the Gaussian probability measure on and
is the energy space. We also briefly consider similar problems for
parametric and stochastic elliptic PDEs with affine inputs, and by-product
problems of non-fully discrete polynomial interpolation approximation and
integration. In particular, the convergence rate of non-fully discrete obtained
in this paper improves the known one
Exponential Splines and Pseudo-Splines: Generation versus reproduction of exponential polynomials
Subdivision schemes are iterative methods for the design of smooth curves and
surfaces. Any linear subdivision scheme can be identified by a sequence of
Laurent polynomials, also called subdivision symbols, which describe the linear
rules determining successive refinements of coarse initial meshes. One
important property of subdivision schemes is their capability of exactly
reproducing in the limit specific types of functions from which the data is
sampled. Indeed, this property is linked to the approximation order of the
scheme and to its regularity. When the capability of reproducing polynomials is
required, it is possible to define a family of subdivision schemes that allows
to meet various demands for balancing approximation order, regularity and
support size. The members of this family are known in the literature with the
name of pseudo-splines. In case reproduction of exponential polynomials instead
of polynomials is requested, the resulting family turns out to be the
non-stationary counterpart of the one of pseudo-splines, that we here call the
family of exponential pseudo-splines. The goal of this work is to derive the
explicit expressions of the subdivision symbols of exponential pseudo-splines
and to study their symmetry properties as well as their convergence and
regularity.Comment: 25 page
Sampling and interpolation in de Branges spaces with doubling phase
The de Branges spaces of entire functions generalise the classical
Paley-Wiener space of square summable bandlimited functions. Specifically, the
square norm is computed on the real line with respect to weights given by the
values of certain entire functions. For the Paley-Wiener space, this can be
chosen to be an exponential function where the phase increases linearly. As our
main result, we establish a natural geometric characterisation, in terms of
densities, for real sampling and interpolating sequences in the case when the
derivative of the phase function merely gives a doubling measure on the real
line. Moreover, a consequence of this doubling condition, is that the spaces we
consider are one component model spaces. A novelty of our work is the
application to de Branges spaces of techniques developed by Marco, Massaneda
and Ortega-Cerd\'a for Fock spaces satisfying a doubling condition analogue to
ours.Comment: 31 pages, 1 figur
Divided Differences
Starting with a novel definition of divided differences, this essay derives
and discusses the basic properties of, and facts about, (univariate) divided
differences.Comment: 24 page
From approximating to interpolatory non-stationary subdivision schemes with the same generation properties
In this paper we describe a general, computationally feasible strategy to
deduce a family of interpolatory non-stationary subdivision schemes from a
symmetric non-stationary, non-interpolatory one satisfying quite mild
assumptions. To achieve this result we extend our previous work [C.Conti,
L.Gemignani, L.Romani, Linear Algebra Appl. 431 (2009), no. 10, 1971-1987] to
full generality by removing additional assumptions on the input symbols. For
the so obtained interpolatory schemes we prove that they are capable of
reproducing the same exponential polynomial space as the one generated by the
original approximating scheme. Moreover, we specialize the computational
methods for the case of symbols obtained by shifted non-stationary affine
combinations of exponential B-splines, that are at the basis of most
non-stationary subdivision schemes. In this case we find that the associated
family of interpolatory symbols can be determined to satisfy a suitable set of
generalized interpolating conditions at the set of the zeros (with reversed
signs) of the input symbol. Finally, we discuss some computational examples by
showing that the proposed approach can yield novel smooth non-stationary
interpolatory subdivision schemes possessing very interesting reproduction
properties
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