Approximation/interpolation from spaces of positive definite or conditionally positive definite kernels is an increasingly popular tool for the analysis and synthesis of scattered data and is central ...to many meshless methods. For a set of N scattered sites, the standard basis for such a space utilizes N globally supported kernels; computing with it is prohibitively expensive for large N. Easily computable, well-localized bases with "small-footprint" basis elements—i.e., elements using only a small number of kernels—have been unavailable. Working on 𝕊2, with focus on the restricted surface spline kernels (e.g., the thin-plate splines restricted to the sphere), we construct easily computable, spatially well-localized, small-footprint, robust bases for the associated kernel spaces. Our theory predicts that each element of the local basis is constructed by using a combination of only 𝓞((log N)2) kernels, which makes the construction computationally cheap. We prove that the new basis is Lp stable and satisfies polynomial decay estimates that are stationary with respect to the density of the data sites, and we present a quasi-interpolation scheme that provides optimal Lp approximation orders. Although our focus is on 𝕊2, much of the theory applies to other manifolds—𝕊d, the rotation group, and so on. Finally, we construct algorithms to implement these schemes and use them to conduct numerical experiments, which validate our theory for interpolation problems on 𝕊2 involving over 150,000 data sites.
We present a new approach to gravitational lens mass map reconstruction. Our mass map solutions perfectly reproduce the positions, fluxes, and shears of all multiple images, and each mass map ...accurately recovers the underlying mass distribution to a resolution limited by the number of multiple images detected. We demonstrate our technique given a mock galaxy cluster similar to Abell 1689, which gravitationally lenses 19 mock background galaxies to produce 93 multiple images. We also explore cases in which as few as four multiple images are observed. Mass map solutions are never unique, and our method makes it possible to explore an extremely flexible range of physical (and unphysical) solutions, all of which perfectly reproduce the data given. Each reconfiguration of the source galaxies produces a new mass map solution. An optimization routine is provided to find those source positions (and redshifts, within uncertainties) that produce the 'most physical' mass map solution, according to a new figure of merit developed here. Our method imposes no assumptions about the slope of the radial profile or mass following light. However, unlike 'nonparametric' grid-based methods, the number of free parameters that we solve for is only as many as the number of observable constraints (or slightly greater if fluxes are constrained). For each set of source positions and redshifts, mass map solutions are obtained 'instantly' via direct matrix inversion by smoothly interpolating the deflection field using a recently developed mathematical technique. Our LensPerfect software is straightforward and easy to use, and is publicly available on our Web site.
Quadrature formulas for spheres, the rotation group, and other compact, homogeneous manifolds are important in a number of applications and have been the subject of recent research. The main purpose ...of this paper is to study coordinate independent quadrature (or cubature) formulas associated with certain classes of positive definite and conditionally positive definite kernels that are invariant under the group action of the homogeneous manifold. In particular, we show that these formulas are accurate—optimally so in many cases—and stable under an increasing number of nodes and in the presence of noise, provided the set
X
of quadrature nodes is quasi-uniform. The stability results are new in all cases. In addition, we may use these quadrature formulas to obtain similar formulas for manifolds diffeomorphic to
S
n
, oblate spheroids for instance. The weights are obtained by solving a single linear system. For
S
2
, and the restricted thin plate spline kernel
r
2
log
r
, these weights can be computed for two-thirds of a million nodes, using a preconditioned iterative technique introduced by us.
Better bases for kernel spaces Fuselier, E J; Hangelbroek, T C; Narcowich, F J ...
arXiv.org,
11/2011
Paper, Journal Article
Open access
In this article we investigate the feasibility of constructing stable, local bases for computing with kernels. In particular, we are interested in constructing families \((b_{\xi})_{\xi\in\Xi}\) that ...function as bases for kernel spaces \(S(k,\Xi)\) so that each basis function is constructed using very few kernels. In other words, each function \(b_{\zeta}(x) = \sum_{\xi\in\Xi} A_{\zeta,\xi} k(x,\xi)\) is a linear combination of samples of the kernel with few nonzero coefficients \(A_{\zeta,\xi}\). This is reminiscent of the construction of the B-spline basis from the family of truncated power functions. We demonstrate that for a large class of kernels (the Sobolev kernels as well as many kernels of polyharmonic and related type) such bases exist. In fact, the basis elements can be constructed using a combination of roughly \(O(\log N)^d\) kernels, where \(d\) is the local dimension of the manifold and \(N\) is the dimension of the kernel space (i.e. \(N=#\Xi\)). Viewing this as a preprocessing step -- the construction of the basis has computational cost \(O(N(\log N)^d)\). Furthermore, we prove that the new basis is \(L_p\) stable and satisfies polynomial decay estimates that are stationary with respect to the density of \(\Xi\).
Quadrature formulas for spheres, the rotation group, and other compact, homogeneous manifolds are important in a number of applications and have been the subject of recent research. The main purpose ...of this paper is to study coordinate independent quadrature (or cubature) formulas associated with certain classes of positive definite and conditionally positive definite kernels that are invariant under the group action of the homogeneous manifold. In particular, we show that these formulas are accurate -- optimally so in many cases -- and stable under an increasing number of nodes and in the presence of noise, provided the set X of quadrature nodes is quasi-uniform. The stability results are new in all cases. In addition, we may use these quadrature formulas to obtain similar formulas for manifolds diffeomorphic to \(\mathbb{S}^n\), oblate spheroids for instance. The weights are obtained by solving a single linear system. For \(\mathbb{S}^2\), and the restricted thin plate spline kernel \(r^2 \log r\), these weights can be computed for two-thirds of a million nodes, using a preconditioned iterative technique introduced by us.
Approximation/interpolation from spaces of positive definite or conditionally positive definite kernels is an increasingly popular tool for the analysis and synthesis of scattered data, and is ...central to many meshless methods. For a set of \(N\) scattered sites, the standard basis for such a space utilizes \(N\) \emph{globally} supported kernels; computing with it is prohibitively expensive for large \(N\). Easily computable, well-localized bases, with "small-footprint" basis elements - i.e., elements using only a small number of kernels -- have been unavailable. Working on \(\sphere\), with focus on the restricted surface spline kernels (e.g. the thin-plate splines restricted to the sphere), we construct easily computable, spatially well-localized, small-footprint, robust bases for the associated kernel spaces. Our theory predicts that each element of the local basis is constructed by using a combination of only \(\mathcal{O}((\log N)^2)\) kernels, which makes the construction computationally cheap. We prove that the new basis is \(L_p\) stable and satisfies polynomial decay estimates that are stationary with respect to the density of the data sites, and we present a quasi-interpolation scheme that provides optimal \(L_p\) approximation orders. Although our focus is on \(\mathbb{S}^2\), much of the theory applies to other manifolds - \(\mathbb{S}^d\), the rotation group, and so on. Finally, we construct algorithms to implement these schemes and use them to conduct numerical experiments, which validate our theory for interpolation problems on \(\mathbb{S}^2\) involving over one hundred fifty thousand data sites.
We present a new approach to gravitational lens massmap reconstruction. Our massmap solutions perfectly reproduce the positions, fluxes, and shears of all multiple images. And each massmap accurately ...recovers the underlying mass distribution to a resolution limited by the number of multiple images detected. We demonstrate our technique given a mock galaxy cluster similar to Abell 1689 which gravitationally lenses 19 mock background galaxies to produce 93 multiple images. We also explore cases in which far fewer multiple images are observed, such as four multiple images of a single galaxy. Massmap solutions are never unique, and our method makes it possible to explore an extremely flexible range of physical (and unphysical) solutions, all of which perfectly reproduce the data given. Each reconfiguration of the source galaxies produces a new massmap solution. An optimization routine is provided to find those source positions (and redshifts, within uncertainties) which produce the "most physical" massmap solution, according to a new figure of merit developed here. Our method imposes no assumptions about the slope of the radial profile nor mass following light. But unlike "non-parametric" grid-based methods, the number of free parameters we solve for is only as many as the number of observable constraints (or slightly greater if fluxes are constrained). For each set of source positions and redshifts, massmap solutions are obtained "instantly" via direct matrix inversion by smoothly interpolating the deflection field using a recently developed mathematical technique. Our LensPerfect software is straightforward and easy to use and is made publicly available via our website.
The Interstellar Boundary Explorer (IBEX) has obtained all-sky images of energetic neutral atoms emitted from the heliosheath, located between the solar wind termination shock and the local ...interstellar medium (LISM). These flux maps reveal distinct nonthermal (0.2 to 6 kilo-electron volts) heliosheath proton populations with spectral signatures ordered predominantly by ecliptic latitude. The maps show a globally distributed population of termination-shock-heated protons and a superimposed ribbonlike feature that forms a circular arc in the sky centered on ecliptic coordinate (longitude λ, latitude β) = (221°, 39°), probably near the direction of the LISM magnetic field. Over the IBEX energy range, the ribbon's nonthermal ion pressure multiplied by its radial thickness is in the range of 70 to 100 picodynes per square centimeter AU (AU, astronomical unit), which is significantly larger than the 30 to 60 picodynes per square centimeter AU of the globally distributed population.
The dominant feature in Interstellar Boundary Explorer (IBEX) sky maps of heliospheric energetic neutral atom (ENA) flux is a ribbon of enhanced flux that extends over a broad range of ecliptic ...latitudes and longitudes. It is narrow (approximately 20° average width) but long (extending over 300° in the sky) and is observed at energies from 0.2 to 6 kilo-electron volts. We demonstrate that the flux in the ribbon is a factor of 2 to 3 times higher than that of the more diffuse, globally distributed heliospheric ENA flux. The ribbon is most pronounced at approximately 1 kilo-electron volt. The average width of the ribbon is nearly constant, independent of energy. The ribbon is likely the result of an enhancement in the combined solar wind and pickup ion populations in the heliosheath.
Neutral gas of the local interstellar medium flows through the inner solar system while being deflected by solar gravity and depleted by ionization. The dominating feature in the energetic neutral ...atom Interstellar Boundary Explorer (IBEX) all-sky maps at low energies is the hydrogen, helium, and oxygen interstellar gas flow. The He and O flow peaked around 8 February 2009 in accordance with gravitational deflection, whereas H dominated after 26 March 2009, consistent with approximate balance of gravitational attraction by solar radiation pressure. The flow distributions arrive from a few degrees above the ecliptic plane and show the same temperature for He and O. An asymmetric O distribution in ecliptic latitude points to a secondary component from the outer heliosheath.
PDF
