19,296 research outputs found
Green's functions for Sturm-Liouville problems
Green function for inhomogeneous Strum-Liouville differential equation proble
Spatiospectral concentration of vector fields on a sphere
We construct spherical vector bases that are bandlimited and spatially
concentrated, or, alternatively, spacelimited and spectrally concentrated,
suitable for the analysis and representation of real-valued vector fields on
the surface of the unit sphere, as arises in the natural and biomedical
sciences, and engineering. Building on the original approach of Slepian,
Landau, and Pollak we concentrate the energy of our function bases into
arbitrarily shaped regions of interest on the sphere, and within certain
bandlimits in the vector spherical-harmonic domain. As with the concentration
problem for scalar functions on the sphere, which has been treated in detail
elsewhere, a Slepian vector basis can be constructed by solving a
finite-dimensional algebraic eigenvalue problem. The eigenvalue problem
decouples into separate problems for the radial and tangential components. For
regions with advanced symmetry such as polar caps, the spectral concentration
kernel matrix is very easily calculated and block-diagonal, lending itself to
efficient diagonalization. The number of spatiospectrally well-concentrated
vector fields is well estimated by a Shannon number that only depends on the
area of the target region and the maximal spherical-harmonic degree or
bandwidth. The spherical Slepian vector basis is doubly orthogonal, both over
the entire sphere and over the geographic target region. Like its scalar
counterparts it should be a powerful tool in the inversion, approximation and
extension of bandlimited fields on the sphere: vector fields such as gravity
and magnetism in the earth and planetary sciences, or electromagnetic fields in
optics, antenna theory and medical imaging.Comment: Submitted to Applied and Computational Harmonic Analysi
A General Approach to Regularizing Inverse Problems with Regional Data using Slepian Wavelets
Slepian functions are orthogonal function systems that live on subdomains
(for example, geographical regions on the Earth's surface, or bandlimited
portions of the entire spectrum). They have been firmly established as a useful
tool for the synthesis and analysis of localized (concentrated or confined)
signals, and for the modeling and inversion of noise-contaminated data that are
only regionally available or only of regional interest. In this paper, we
consider a general abstract setup for inverse problems represented by a linear
and compact operator between Hilbert spaces with a known singular-value
decomposition (svd). In practice, such an svd is often only given for the case
of a global expansion of the data (e.g. on the whole sphere) but not for
regional data distributions. We show that, in either case, Slepian functions
(associated to an arbitrarily prescribed region and the given compact operator)
can be determined and applied to construct a regularization for the ill-posed
regional inverse problem. Moreover, we describe an algorithm for constructing
the Slepian basis via an algebraic eigenvalue problem. The obtained Slepian
functions can be used to derive an svd for the combination of the regionalizing
projection and the compact operator. As a result, standard regularization
techniques relying on a known svd become applicable also to those inverse
problems where the data are regionally given only. In particular, wavelet-based
multiscale techniques can be used. An example for the latter case is elaborated
theoretically and tested on two synthetic numerical examples
Internal and external potential-field estimation from regional vector data at varying satellite altitude
When modeling global satellite data to recover a planetary magnetic or
gravitational potential field and evaluate it elsewhere, the method of choice
remains their analysis in terms of spherical harmonics. When only regional data
are available, or when data quality varies strongly with geographic location,
the inversion problem becomes severely ill-posed. In those cases, adopting
explicitly local methods is to be preferred over adapting global ones (e.g., by
regularization). Here, we develop the theory behind a procedure to invert for
planetary potential fields from vector observations collected within a
spatially bounded region at varying satellite altitude. Our method relies on
the construction of spatiospectrally localized bases of functions that mitigate
the noise amplification caused by downward continuation (from the satellite
altitude to the planetary surface) while balancing the conflicting demands for
spatial concentration and spectral limitation. Solving simultaneously for
internal and external fields in the same setting of regional data availability
reduces internal-field artifacts introduced by downward-continuing unmodeled
external fields, as we show with numerical examples. The AC-GVSF are optimal
linear combinations of vector spherical harmonics. Their construction is not
altogether very computationally demanding when the concentration domains (the
regions of spatial concentration) have circular symmetry, e.g., on spherical
caps or rings - even when the spherical-harmonic bandwidth is large. Data
inversion proceeds by solving for the expansion coefficients of truncated
function sequences, by least-squares analysis in a reduced-dimensional space.
Hence, our method brings high-resolution regional potential-field modeling from
incomplete and noisy vector-valued satellite data within reach of contemporary
desktop machines.Comment: Under revision for Geophys. J. Int. Supported by NASA grant
NNX14AM29
New results on q-positivity
In this paper we discuss symmetrically self-dual spaces, which are simply
real vector spaces with a symmetric bilinear form. Certain subsets of the space
will be called q-positive, where q is the quadratic form induced by the
original bilinear form. The notion of q-positivity generalizes the classical
notion of the monotonicity of a subset of a product of a Banach space and its
dual. Maximal q-positivity then generalizes maximal monotonicity. We discuss
concepts generalizing the representations of monotone sets by convex functions,
as well as the number of maximally q-positive extensions of a q-positive set.
We also discuss symmetrically self-dual Banach spaces, in which we add a Banach
space structure, giving new characterizations of maximal q-positivity. The
paper finishes with two new examples.Comment: 18 page
- …