PROBLEMAS DE EVOLUCION: MODELOS, APLICACIONES Y NUEVAS TECNICAS ASINTOTICAS Y NUMERICAS DE RESOLUCION

MTM2014-52859-P

Nombre agencia financiadora Ministerio de Economía y Competitividad
Acrónimo agencia financiadora MINECO
Programa Programa Estatal de Fomento de la Investigación Científica y Técnica de Excelencia
Subprograma Subprograma Estatal de Generación del Conocimiento
Convocatoria Proyectos de I+D dentro del Subprograma Estatal de Generación del Conocimiento (2014)
Año convocatoria 2014
Unidad de gestión Dirección General de Investigación Científica y Técnica
Centro beneficiario UNIVERSIDAD PÚBLICA DE NAVARRA (UPNA)
Centro realización UNIVERSIDAD PUBLICA DE NAVARRA
Identificador persistente http://dx.doi.org/10.13039/501100003329

Publicaciones

Found(s) 6 result(s)
Found(s) 1 page(s)

Orthogonal basis for the optical transfer function

Zaguán. Repositorio Digital de la Universidad de Zaragoza
  • Ferreira, C.
  • López, J.L.
  • Navarro, R.
  • Sinusa, E.P.
We propose systems of orthogonal functions qn to represent optical transfer functions (OTF) characterized by including the diffraction-limited OTF as the first basis function q0 = OTFperfect. To this end, we apply a powerful and rigorous theoretical framework based on applying the appropriate change of variables to well-known orthogonal systems. Here we depart from Legendre polynomials for the particular case of rotationally symmetric OTF and from spherical harmonics for the general case. Numerical experiments with different examples show that the number of terms necessary to obtain an accurate linear expansion of the OTF mainly depends on the image quality. In the rotationally symmetric case we obtained a reasonable accuracy with approximately 10 basis functions, but in general, for cases of poor image quality, the number of basis functions may increase and hence affect the efficiency of the method. Other potential applications, such as new image quality metrics are also discussed.




The use of two-point Taylor expansions in singular one-dimensional boundary value problems I

Zaguán. Repositorio Digital de la Universidad de Zaragoza
  • Ferreira, C.
  • López, J.L.
  • Pérez Sinusía, E.
We consider the second-order linear differential equation (x+1)y¿+f(x)y'+g(x)y=h(x) in the interval (-1, 1) with initial conditions or boundary conditions (Dirichlet, Neumann or mixed Dirichlet–Neumann). The functions f(x), g(x) and h(x) are analytic in a Cassini disk Dr with foci at x=±1 containing the interval [-1, 1]. Then, the end point of the interval x=-1 may be a regular singular point of the differential equation. The two-point Taylor expansion of the solution y(x) at the end points ±1 is used to study the space of analytic solutions in Dr of the differential equation, and to give a criterion for the existence and uniqueness of analytic solutions of the boundary value problem. This method is constructive and provides the two-point Taylor approximation of the analytic solutions when they exist.




An efficient numerical method for singularly perturbed time dependent parabolic 2D convection–diffusion systems

Zaguán. Repositorio Digital de la Universidad de Zaragoza
  • Clavero, C.
  • Jorge, J. C.
In this paper we deal with solving efficiently 2D linear parabolic singularly perturbed systems of convection–diffusion type. We analyze only the case of a system of two equations where both of them feature the same diffusion parameter. Nevertheless, the method is easily extended to systems with an arbitrary number of equations which have the same diffusion coefficient. The fully discrete numerical method combines the upwind finite difference scheme, to discretize in space, and the fractional implicit Euler method, together with a splitting by directions and components of the reaction–convection–diffusion operator, to discretize in time. Then, if the spatial discretization is defined on an appropriate piecewise uniform Shishkin type mesh, the method is uniformly convergent and it is first order in time and almost first order in space. The use of a fractional step method in combination with the splitting technique to discretize in time, means that only tridiagonal linear systems must be solved at each time level of the discretization. Moreover, we study the order reduction phenomenon associated with the time dependent boundary conditions and we provide a simple way of avoiding it. Some numerical results, which corroborate the theoretical established properties of the method, are shown.




Zernike-like systems in polygons and polygonal facets

Digital.CSIC. Repositorio Institucional del CSIC
  • Ferreira, Chelo
  • López, José L.
  • Navarro, Rafael
  • Pérez Sinusía, Ester
arXiv:1506.07396v1, Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit disk. In [Opt. Lett. 32, 74 (2007)] we introduced a new Zernike basis for elliptic and annular optical apertures based on an appropriate diffeomorphism between the unit disk and the ellipse and the annulus. Here, we present a generalization of this Zernike basis for a variety of important optical apertures, paying special attention to polygons and the polygonal facets present in segmented mirror telescopes. On the contrary to ad hoc solutions, most of them based on the Gram-Smith orthonormalization method, here we consider a piecewise diffeomorphism that transforms the unit disk into the polygon under consideration. We use this mapping to define a Zernike-like orthonormal system over the polygon. We also consider ensembles of polygonal facets that are essential in the design of segmented mirror telescopes. This generalization, based on in-plane warping of the basis functions, provides a unique solution, and what is more important, it guarantees a reasonable level of invariance of the mathematical properties and the physical meaning of the initial basis functions. Both the general form and the explicit expressions for a typical example of telescope optical aperture are provided., European Union (FIS2011-22496); Government of Aragón (E99); Spanish Ministry of Economía y Competitividad (FIS2014-58303), (MTM2014-52859); State University of Navarra., Peer Reviewed




Orthogonal basis for the optical transfer function

Digital.CSIC. Repositorio Institucional del CSIC
  • Ferreira, Chelo
  • López, José L.
  • Navarro, Rafael
  • Pérez Sinusía, Ester
We propose systems of orthogonal functions qn to represent optical transfer functions (OTF) characterized by including the diffraction-limited OTF as the first basis function q0 = OTFperfect. To this end, we apply a powerful and rigorous theoretical framework based on applying the appropriate change of variables to well-known orthogonal systems. Here we depart from Legendre polynomials for the particular case of rotationally symmetric OTF and from spherical harmonics for the general case. Numerical experiments with different examples show that the number of terms necessary to obtain an accurate linear expansion of the OTF mainly depends on the image quality. In the rotationally symmetric case we obtained a reasonable accuracy with approximately 10 basis functions, but in general, for cases of poor image quality, the number of basis functions may increase and hence affect the efficiency of the method. Other potential applications, such as new image quality metrics are also discussed., Ministerio de Economía y Competitividad (MINECO) (MTM2014-52859-P, FIS2014-58303-P)., Peer Reviewed




Orthogonal basis with a conicoid first mode for shape specification of optical surfaces

Digital.CSIC. Repositorio Institucional del CSIC
  • Ferreira, Chelo
  • López, José L.
  • Navarro, Rafael
  • Pérez Sinusía, Ester
Open Access., A rigorous and powerful theoretical framework is proposed to obtain systems of orthogonal functions (or shape modes) to represent optical surfaces. The method is general so it can be applied to different initial shapes and different polynomials. Here we present results for surfaces with circular apertures when the first basis function (mode) is a conicoid. The system for aspheres with rotational symmetry is obtained applying an appropriate change of variables to Legendre polynomials, whereas the system for general freeform case is obtained applying a similar procedure to spherical harmonics. Numerical comparisons with standard systems, such as Forbes and Zernike polynomials, are performed and discussed., This research was supported by the Spanish Ministry of Economía y Competitividad and the European Union MTM2014-52859 and FIS2014-58303., Peer Reviewed