Image reconstruction from scattered Radon data by weighted positive definite kernel functions

We propose a novel kernel-based method for image reconstruction from scattered Radon data. To this end, we employ generalized Hermite–Birkhoff interpolation by positive definite kernel functions. For radial kernels, however, a straightforward application of the generalized Hermite–Birkhoff interpolation method fails to work, as we prove in this paper. To obtain a well-posed reconstruction scheme for scattered Radon data, we introduce a new class of weighted positive definite kernels, which are symmetric but not radially symmetric. By our construction, the resulting weighted kernels are combinations of radial positive definite kernels and positive weight functions. This yields very flexible image reconstruction methods, which work for arbitrary distributions of Radon lines. We develop suitable representations for the weighted basis functions and the symmetric positive definite kernel matrices that are resulting from the proposed reconstruction scheme. For the relevant special case, where Gaussian radial kernels are combined with Gaussian weights, explicit formulae for the weighted Gaussian basis functions and the kernel matrices are given. Supporting numerical examples are finally presented

The global mass function of M15

Data obtained with the NICMOS instrument on board the Hubble Space Telescope (HST) have been used to determine the H-band luminosity function (LF) and mass function (MF) of three stellar fields in the globular cluster M15, located ~7' from the cluster centre. The data confirm that the cluster MF has a characteristic mass of ~0.3 Msolar, as obtained by Paresce & De Marchi (2000) for a stellar field at 4.6' from the centre. By combining the present data with those published by other authors for various radial distances (near the centre, at 20" and at 4.6'), we have studied the radial variation of the LF due to the effects of mass segregation and derived the global mass function (GMF) using the Michie-King approach. The model that simultaneously best fits the LF at various locations, the surface brightness profile and the velocity dispersion profile suggests that the GMF should resemble a segmented power-law with the following indices: x ~ 0.8 for stars more massive than 0.8 Msolar, x ~ 0.9 for 0.3 - 0.8 Msolar and x ~ -2.2 at smaller masses (Salpeter's IMF would have x=1.35). The best fitting model also suggests that the cluster mass is ~5.4 10^5 Msolar and that the mass-to-light ratio is on average M/L_V ~ 2.1, with M/L_V ~ 3.7 in the core. A large amount of mass (~ 44 %) is found in the cluster core in the form of stellar heavy remnants, which may be sufficient to explain the mass segregation in M15 without invoking the presence of an intermediate-mass black hole.Comment: 12 pages, 10 figures, accepted for publication in A&

Partition of unity interpolation using stable kernel-based techniques

In this paper we propose a new stable and accurate approximation technique which is extremely effective for interpolating large scattered data sets. The Partition of Unity (PU) method is performed considering Radial Basis Functions (RBFs) as local approximants and using locally supported weights. In particular, the approach consists in computing, for each PU subdomain, a stable basis. Such technique, taking advantage of the local scheme, leads to a significant benefit in terms of stability, especially for flat kernels. Furthermore, an optimized searching procedure is applied to build the local stable bases, thus rendering the method more efficient

On the Globular Cluster IMF below 1 Solar Mass

(Abridged) Accurate luminosity functions (LF) for a dozen globular clusters have now been measured at or just beyond their half-light radius using HST. They span almost the entire cluster main sequence below ~ 0.75 Msolar. All these clusters exhibit LF that rise continuously from an absolute I magnitude M_I ~ 6 to a peak at M_I ~ 8.5-9 and then drop with increasing M_I. Transformation of the LF into mass functions (MF) by means of the most recent mass luminosity relations that are consistent with all presently available data on the physical properties of low mass, low metallicity stars shows that all the LF observed so far can be obtained from MF having the shape of a log-normal distribution with characteristic mass m_c=0.33 +/- 0.03 Msolar and standard deviation sigma = 1.81 +/- 0.19. After correction for the effects of mass segregation, the variation of the ratio of the number of higher to lower mass stars with cluster mass or any simple orbital parameter or the expected time to disruption recently computed for these clusters shows no statistically significant trend over a range of this last parameter of more than a factor of 100. We conclude that the global MF of these clusters have not been measurably modified by evaporation and tidal interactions with the Galaxy and, thus, should reflect the initial distribution of stellar masses. Since the log-normal function that we find is also very similar to the one obtained independently for much younger clusters and to the form expected theoretically, the implication seems to be unavoidable that it represents the true stellar IMF for this type of stars in this mass range.Comment: Accepted for publication in The Astrophysical Journal. Contains 28 pages with 6 figure

On the convergence of the rescaled localized radial basis function method

The rescaled localized RBF method was introduced in Deparis, Forti, and Quarteroni (2014) for scattered data interpolation. It is a rational approximation method based on interpolation with compactly supported radial basis functions. It requires the solution of two linear systems with the same sparse matrix, which has a small condition number, due to the scaling of the basis function. Hence, it can be computed using an unpreconditioned conjugate gradient method in linear time. Numerical evidence provided in Deparis, Forti, and Quarteroni (2014) shows that the method produces good approximations for many examples but no theoretical results were provided. In this paper, we discuss the convergence of the rescaled localized RBF method in the case of quasi-uniform data and stationary scaling. As the method is not only interpolatory but also reproduces constants exactly, linear convergence is expected. We can show this linear convergence up to a certain conjecture

Greedy kernel methods for accelerating implicit integrators for parametric ODEs

We present a novel acceleration method for the solution of parametric ODEs by single-step implicit solvers by means of greedy kernel-based surrogate models. In an offline phase, a set of trajectories is precomputed with a high-accuracy ODE solver for a selected set of parameter samples, and used to train a kernel model which predicts the next point in the trajectory as a function of the last one. This model is cheap to evaluate, and it is used in an online phase for new parameter samples to provide a good initialization point for the nonlinear solver of the implicit integrator. The accuracy of the surrogate reflects into a reduction of the number of iterations until convergence of the solver, thus providing an overall speedup of the full simulation. Interestingly, in addition to providing an acceleration, the accuracy of the solution is maintained, since the ODE solver is still used to guarantee the required precision. Although the method can be applied to a large variety of solvers and different ODEs, we will present in details its use with the Implicit Euler method for the solution of the Burgers equation, which results to be a meaningful test case to demonstrate the method's features

Engineering model transformations with transML

The final publication is available at Springer via http://dx.doi.org/10.1007%2Fs10270-011-0211-2Model transformation is one of the pillars of model-driven engineering (MDE). The increasing complexity of systems and modelling languages has dramatically raised the complexity and size of model transformations as well. Even though many transformation languages and tools have been proposed in the last few years, most of them are directed to the implementation phase of transformation development. In this way, even though transformations should be built using sound engineering principles—just like any other kind of software—there is currently a lack of cohesive support for the other phases of the transformation development, like requirements, analysis, design and testing. In this paper, we propose a unified family of languages to cover the life cycle of transformation development enabling the engineering of transformations. Moreover, following an MDE approach, we provide tools to partially automate the progressive refinement of models between the different phases and the generation of code for several transformation implementation languages.This work has been sponsored by the Spanish Ministry of Science and Innovation with project METEORIC (TIN2008-02081), and by the R&D program of the Community of Madrid with projects “e-Madrid" (S2009/TIC-1650). Parts of this work were done during the research stays of Esther and Juan at the University of York, with financial support from the Spanish Ministry of Science and Innovation (grant refs. JC2009-00015, PR2009-0019 and PR2008-0185)

Jumping with variably scaled discontinuous kernels (VSDKs)

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Correction to: Jumping with variably scaled discontinuous kernels (VSDKs)

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