1,326 research outputs found
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
Manifold interpolation and model reduction
One approach to parametric and adaptive model reduction is via the
interpolation of orthogonal bases, subspaces or positive definite system
matrices. In all these cases, the sampled inputs stem from matrix sets that
feature a geometric structure and thus form so-called matrix manifolds. This
work will be featured as a chapter in the upcoming Handbook on Model Order
Reduction (P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A.
Schilders, L.M. Silveira, eds, to appear on DE GRUYTER) and reviews the
numerical treatment of the most important matrix manifolds that arise in the
context of model reduction. Moreover, the principal approaches to data
interpolation and Taylor-like extrapolation on matrix manifolds are outlined
and complemented by algorithms in pseudo-code.Comment: 37 pages, 4 figures, featured chapter of upcoming "Handbook on Model
Order Reduction
LMI Representations of Convex Semialgebraic Sets and Determinantal Representations of Algebraic Hypersurfaces: Past, Present, and Future
10 years ago or so Bill Helton introduced me to some mathematical problems
arising from semidefinite programming. This paper is a partial account of what
was and what is happening with one of these problems, including many open
questions and some new results
A framework for structured linearizations of matrix polynomials in various bases
We present a framework for the construction of linearizations for scalar and
matrix polynomials based on dual bases which, in the case of orthogonal
polynomials, can be described by the associated recurrence relations. The
framework provides an extension of the classical linearization theory for
polynomials expressed in non-monomial bases and allows to represent polynomials
expressed in product families, that is as a linear combination of elements of
the form , where and
can either be polynomial bases or polynomial families
which satisfy some mild assumptions. We show that this general construction can
be used for many different purposes. Among them, we show how to linearize sums
of polynomials and rational functions expressed in different bases. As an
example, this allows to look for intersections of functions interpolated on
different nodes without converting them to the same basis. We then provide some
constructions for structured linearizations for -even and
-palindromic matrix polynomials. The extensions of these constructions
to -odd and -antipalindromic of odd degree is discussed and
follows immediately from the previous results
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