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Christian Gerhards

Publications and source records attributed to Christian Gerhards.

9 recordsLinked to original sources
arXiv2023

Spatiospectral localization within the ball -- studies on the influence of the spectral shape

We investigate the Slepian spatiospectral localization problem within subdomains of the $d$-dimensional ball. Opposed to the more classical setups of the Euclidean space or the sphere, the ball lacks a standard or universally accepted definition of bandwidth. Here, we consider a Fourier-Jacobi function system, decoupling the spherical and radial contributions via spherical harmonics and Jacobi polynomials. Special cases of this setup are of interest for various inverse problems in geophysics and medical imaging, since they relate to the underlying non-uniqueness, as well as in optics, where they represent the widely used Zernike polynomials. Bandwidth can be prescribed separately for the spherical and the radial contributions, where the particular choice of coupling between the two contributions determines the spectral shape, i.e., the overall notion of bandlimit. Understanding the effects of the spectral shape on the eigenvalue distribution of the Slepian spatiospectral localization problem can provide hints on particularly suitable notions of bandwidth for different applications. We provide rigorous asymptotic results for the spectral shape being defined via the overall polynomial degree as well as for being defined via sequential limits for the spherical and radial contributions. For various other spectral shapes, we provide numerical illustrations of the asymptotic eigenvalue distribution. Furthermore, we demonstrate a direct connection of the spectral shape to common indexing schemes for Zernike polynomials.

math.FA↗
arXiv2023

Spherical Basis Functions in Hardy Spaces with Localization Constraints

Subspaces obtained by the orthogonal projection of locally supported square-integrable vector fields onto the Hardy spaces $H_+(\mathbb{S})$ and $H_-(\mathbb{S})$, respectively, play a role in various inverse potential field problems since they characterize the uniquely recoverable components of the underlying sources. Here, we consider approximation in these subspaces by a particular set of spherical basis functions. Error bounds are provided along with further considerations on norm-minimizing vector fields that satisfy the underlying localization constraint. The new aspect here is that the used spherical basis functions are themselves members of the subspaces under consideration.

math.NA↗
arXiv2022

Relation between Hardy components for locally supported vector fields on the sphere

Given a function in the Hardy space of inner harmonic gradients on the sphere, H+(S), we consider the problem of finding a corresponding function in the Hardy space of outer harmonic gradients on the sphere, H-(S), such that the sum of both functions differs from a locally supported vector field only by a tangential divergence-free contribution. We characterize the subspace of H+(S) that allows such a continuation and show that it is dense but not closed within H+(S). Furthermore, we derive the linear mapping that maps a vector field from this subspace of H+(S) to the corresponding unique vector field in H-(S). The explicit construction uses layer potentials but involves unbounded operators. We indicate some bounded extremal problems supporting a possible numerical evaluation of this mapping between the Hardy components. The original motivation to study this problem comes from an inverse magnetization problem with localization constraints.

math.FA↗
arXiv2017

On the Recovery of Core and Crustal Components of Geomagnetic Potential Fields

In Geomagnetism it is of interest to separate the Earth's core magnetic field from the crustal magnetic field. However, measurements by satellites can only sense the sum of the two contributions. In practice, the measured magnetic field is expanded in spherical harmonics and separation into crust and core contribution is achieved empirically, by a sharp cutoff in the spectral domain. In this paper, we derive a mathematical setup in which the two contributions are modeled by harmonic potentials $\Phi_0$ and $\Phi_1$ generated on two different spheres $\mathbb{S}_{R_0}$ (crust) and $\mathbb{S}_{R_1}$ (core) with radii $R_1 R_0$, we show that it becomes possible if the magnetization $\mathbf{m}$ generating $\Phi_0$ is localized in a strict subregion of $\mathbb{S}_{R_0}$. Beyond unique recoverability, we show in this case how to numerically reconstruct characteristic features of $\Phi_0$ (e.g., spherical harmonic Fourier coefficients). An alternative way of phrasing the results is that knowledge of $\mathbf{m}$ on a nonempty open subset of $\mathbb{S}_{R_0}$ allows one to perform separation.

math.AP↗
arXiv2016

On the Reconstruction of Dipole Directions from Spherical Magnetic Field Measurements

Reconstructing magnetizations from measurements of the generated magnetic potential is generally non-unique. The non-uniqueness still remains if one restricts the magnetization to those induced by an ambient magnetic dipole field (i.e., the magnetization is described by a scalar susceptibility and the dipole direction). Here, we investigate the situation under the additional constraint that the susceptibility is either spatially localized in a subregion of the sphere or that it is band-limited. If the dipole direction is known, then the susceptibility is uniquely determined under the spatial localization constraint while it is only determined up to a constant under the the assumption of band-limitedness. If the dipole direction is not known, uniqueness is lost again. However, we show that all dipole directions that could possibly generate the measured magnetic potential need to be zeros of a certain polynomial which can be computed from the given potential. We provide examples of non-uniqueness of the dipole direction and examples on how to find admissible candidates for the dipole direction under the spatial localization constraint.

physics.geo-ph↗
arXiv2016

On the Unique Reconstruction of Induced Spherical Magnetizations

Recovering spherical magnetizations $m$ from magnetic field data in the exterior is a highly non-unique problem. A spherical Hardy-Hodge decomposition supplies information on what contributions of the magnetization $m$ are recoverable but it does not supply geophysically suitable constraints on $m$ that would guarantee uniqueness for the entire magnetization. In this paper, we focus on the case of induced spherical magnetizations and show that uniqueness is guaranteed if one assumes that the magnetization is compactly supported on the sphere. The results are based on ideas presented in Baratchart et al. (2013) for the planar setting.

math.AP↗
arXiv2015

Locally Supported Wavelets for the Separation of Spherical Vector Fields with Respect to their Sources

We provide a space domain oriented separation of magnetic fields into parts generated by sources in the exterior and sources in the interior of a given sphere. The separation itself is well-known in geomagnetic modeling, usually in terms of a spherical harmonic analysis or a wavelet analysis that is spherical harmonic based. In contrast to these frequency oriented methods, we use a more spatially oriented approach in this paper. We derive integral representations with explicitly known convolution kernels. Regularizing these singular kernels allows a multiscale representation of the internal and external contributions to the magnetic field with locally supported wavelets. This representation is applied to a set of CHAMP data for crustal field modeling.

math.NA↗
arXiv2013

A Combination of Downward Continuation and Local Approximation for Harmonic Potentials

This paper presents a method for the approximation of harmonic potentials that combines downward continuation of globally available data on a sphere $\Omega_R$ of radius $R$ (e.g., a satellite's orbit) with locally available data on a sphere $\Omega_r$ of radius $r<R$ (e.g., the spherical Earth's surface). The approximation is based on a two-step algorithm motivated by spherical multiscale expansions: First, a convolution with a scaling kernel $\Phi_N$ deals with the downward continuation from $\Omega_R$ to $\Omega_r$, while in a second step, the result is locally refined by a convolution on $\Omega_r$ with a wavelet kernel $\tilde{\Psi}_N$. Different from earlier multiscale approaches, it is not the primary goal to obtain an adaptive spatial localization but to simultaneously optimize the related kernels $\Phi_N$, $\tilde{\Psi}_N$ in such a way that the former behaves well for the downward continuation while the latter shows a good localization on $\Omega_r$ in the region where data is available. The concept is indicated for scalar as well as vector potentials.

math.NA↗