100,409 research outputs found
On Point Spread Function modelling: towards optimal interpolation
Point Spread Function (PSF) modeling is a central part of any astronomy data
analysis relying on measuring the shapes of objects. It is especially crucial
for weak gravitational lensing, in order to beat down systematics and allow one
to reach the full potential of weak lensing in measuring dark energy. A PSF
modeling pipeline is made of two main steps: the first one is to assess its
shape on stars, and the second is to interpolate it at any desired position
(usually galaxies). We focus on the second part, and compare different
interpolation schemes, including polynomial interpolation, radial basis
functions, Delaunay triangulation and Kriging. For that purpose, we develop
simulations of PSF fields, in which stars are built from a set of basis
functions defined from a Principal Components Analysis of a real ground-based
image. We find that Kriging gives the most reliable interpolation,
significantly better than the traditionally used polynomial interpolation. We
also note that although a Kriging interpolation on individual images is enough
to control systematics at the level necessary for current weak lensing surveys,
more elaborate techniques will have to be developed to reach future ambitious
surveys' requirements.Comment: Accepted for publication in MNRA
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
On point spread function modelling: towards optimal interpolation
Point spread function (PSF) modelling is a central part of any astronomy data analysis relying on measuring the shapes of objects. It is especially crucial for weak gravitational lensing, in order to beat down systematics and allow one to reach the full potential of weak lensing in measuring dark energy. A PSF modelling pipeline is made of two main steps: the first one is to assess its shape on stars, and the second is to interpolate it at any desired position (usually galaxies). We focus on the second part, and compare different interpolation schemes, including polynomial interpolation, radial basis functions, Delaunay triangulation and Kriging. For that purpose, we develop simulations of PSF fields, in which stars are built from a set of basis functions defined from a principal components analysis of a real ground-based image. We find that Kriging gives the most reliable interpolation, significantly better than the traditionally used polynomial interpolation. We also note that although a Kriging interpolation on individual images is enough to control systematics at the level necessary for current weak lensing surveys, more elaborate techniques will have to be developed to reach future ambitious surveys' requirement
A nonsymmetric version of Okounkov's BC-type interpolation Macdonald polynomials
Symmetric and nonsymmetric interpolation Laurent polynomials are introduced
with the interpolation points depending on and a -tuple of parameters
. For the principal specialization
the symmetric interpolation Laurent polynomials reduce to
Okounkov's -type interpolation Macdonald polynomials and the nonsymmetric
interpolation Laurent polynomials become their nonsymmetric variants. We expand
the symmetric interpolation Laurent polynomials in the nonsymmetric ones. We
show that Okounkov's -type interpolation Macdonald polynomials can also be
obtained from their nonsymmetric versions using a one-parameter family of
actions of the finite Hecke algebra of type in terms of Demazure-Lusztig
operators. In the Appendix we give some experimental results and conjectures
about extra vanishing.Comment: 30 pages, 9 figures; v4: experimental results and conjectures added
about extra vanishin
Regular polynomial interpolation and approximation of global solutions of linear partial differential equations
We consider regular polynomial interpolation algorithms on recursively
defined sets of interpolation points which approximate global solutions of
arbitrary well-posed systems of linear partial differential equations.
Convergence of the 'limit' of the recursively constructed family of
polynomials to the solution and error estimates are obtained from a priori
estimates for some standard classes of linear partial differential equations,
i.e. elliptic and hyperbolic equations. Another variation of the algorithm
allows to construct polynomial interpolations which preserve systems of linear
partial differential equations at the interpolation points. We show how this
can be applied in order to compute higher order terms of WKB-approximations of
fundamental solutions of a large class of linear parabolic equations. The error
estimates are sensitive to the regularity of the solution. Our method is
compatible with recent developments for solution of higher dimensional partial
differential equations, i.e. (adaptive) sparse grids, and weighted Monte-Carlo,
and has obvious applications to mathematical finance and physics.Comment: 28 page
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