Space Of Spherical Harmonics Research Articles

An efficient algorithm for time-domain acoustic scattering in three dimensions by layer potentials

In this paper, we develop a simple and fast algorithm for the time-domain acoustic scattering by a sound-soft or an impedance obstacle in three dimensions. We express the solution to the scattering problem by layer potentials, and then a time-domain boundary integral equation is derived. To numerically solve the resulting boundary integral equation, we propose a full discretization scheme by combining the convolution splines with a Galerkin method. In time, we approximate the density in a backward manner in terms of the convolution splines. In space, we project the density at each time onto the space of spherical harmonics, and then use the spatial discretization of a Nyström type on the surface of an obstacle which is homeomorphic to a sphere. A gallery of numerical examples are presented to show the efficiency of our algorithm. The stability, convergence and accuracy of the algorithm are discussed.

Using spherical harmonics to solve the Boltzmann equation: an operator-based approach

ABSTRACT The transport of charged particles or photons in a scattering medium can be modelled with a Boltzmann equation. The mathematical treatment for scattering in such scenarios is often simplified if evaluated in a frame where the scattering centres are, on average, at rest. It is common therefore to use a mixed coordinate system, wherein space and time are measured in a fixed inertial frame, while momenta are measured in a ‘co-moving’ frame. To facilitate analytical and numerical solutions, the momentum dependence of the phase-space density may be expanded as a series of spherical harmonics, typically truncated at low order. A method for deriving the system of equations for the expansion coefficients of the spherical harmonics to arbitrary order is presented in the limit of isotropic, small-angle scattering. The method of derivation takes advantage of operators acting on the space of spherical harmonics. The matrix representations of these operators are employed to compute the system of equations. The computation of matrix representations is detailed and subsequently simplified with the aid of rotations of the coordinate system. The eigenvalues and eigenvectors of the matrix representations are investigated to prepare the application of standard numerical techniques, e.g. the finite volume method or the discontinuous Galerkin method, to solve the system.

A systematic study of a droplet breakup process in decaying homogeneous isotropic turbulence using a mesoscopic simulation approach

ABSTRACT The breakup of a spherical droplet in a decaying homogeneous isotropic turbulence is studied by solving the Cahn–Hilliard–Navier–Stokes equations. This flow provides a great opportunity to study the interactions of turbulent kinetic energy and interfacial free energy and their effects on the breakup dynamics. Three distinct stages of droplet evolution, namely, the deformation stage, the breakup stage, and the restoration stage, are identified and then analysed systematically from several perspectives: a geometric perspective, a dynamic perspective, a global energetic perspective, and a multiscale energy transfer perspective. It is found that the ending time of the breakup stage can be estimated by the Hinze criterion. The kinetic energy of the two-phase flow during the breakup stage is found to have a power-law decay with an exponent , compared to for the single-phase flow, mainly due to the enhanced viscous dissipation generated by the daughter droplets. Energy spectra of the two-phase flow show power-law decay, with a slope between and , at high wave numbers, both in the Fourier spectral space and in the spherical harmonics space.

Timing-residual power spectrum of a polarized stochastic gravitational-wave background in pulsar-timing-array observation

We study the observation of a stochastic gravitational-wave background (SGWB) made by a pulsar-timing array in the spherical harmonic space of the observable. The observable is a timing residual which is the time-averaged redshift fluctuation of a pulsar over the duration of observation. Using the Sachs-Wolfe line-of-sight integral for the redshift fluctuation, we derive the power spectrum of the timing residual, from which we develop a fast algorithm to compute the overlap reduction functions for the SGWB intensity and polarization anisotropies. We find that the algorithm is less complicated and more efficient than our previous work which is based on the expansion of the polarization basis tensors in terms of spherical harmonics. Also, we use the power spectrum to construct the bipolar spherical harmonic coefficients that characterize the statistical isotropy of the SGWB. In particular, the coefficients for the linear-polarization anisotropy are worked out for the first time. Our harmonic-space method is useful for the data analysis in future pulsar-timing-array observation on a large number of pulsars as well as for the measurement of the statistical isotropy of the SGWB.

VRfRNet: Volumetric ROI fODF reconstruction network for estimation of multi-tissue constrained spherical deconvolution with only single shell dMRI

Diffusion MRI (dMRI) is one of the most popular techniques for studying the brain structure, mainly the white matter region. Among several sampling methods in dMRI, the high angular resolution diffusion imaging (HARDI) technique has attracted researchers due to its more accurate fiber orientation estimation. However, the current single-shell HARDI makes the intravoxel structure challenging to estimate accurately. While multi-shell acquisition can address this problem, it takes a longer scanning time, restricting its use in clinical applications. In addition, most existing dMRI scanners with low gradient-strengths often acquire single-shell up to b=1000s/mm2 because of signal-to-noise ratio issues and severe image artefacts. Hence, we propose a novel generative adversarial network, VRfRNet, for the reconstruction of multi-shell multi-tissue fiber orientation distribution function from single-shell HARDI volumes. Such a transformation learning is performed in the spherical harmonics (SH) space, as raw input HARDI volume is transformed to SH coefficients to soften gradient directions. The proposed VRfRNet consists of several modules, such as multi-context feature enrichment module, feature level attention, and softmax level attention. In addition, three loss functions have been used to optimize network learning, including L1, adversarial, and total variation. The network is trained and tested using standard qualitative and quantitative performance metrics on the publicly available HCP data-set.

Single-shell to multi-shell dMRI transformation using spatial and volumetric multilevel hierarchical reconstruction framework

Single or Multi-shell high angular resolution diffusion imaging (HARDI) has become an important dMRI acquisition technique for studying brain white matter fibers. Existing single-shell HARDI makes it challenging to estimate the intravoxel structure up to the desired resolution. However, multi-shell acquisition (with multiple b-values) can provide higher resolution for the intravoxel structure, which further helps in getting accurate fiber tracts; But, this comes at the cost of larger acquisition time and larger setup. Hence, we propose a novel deep learning architecture for the reconstruction of diffusion MRI volumes for different b-values (degree of diffusion weighting) using acquisitions at a fixed b-value (termed as single-shell) acquisition. This reconstruction has been performed in the spherical harmonics space to better manage varying gradient directions. In this work, we have demonstrated such a reconstruction for b = 3000 s/mm2 and b = 2000 s/mm2 from b = 1000 s/mm2. The proposed Multilevel Hierarchical Spherical Harmonics Coefficients Reconstruction (MHSH) framework takes advantage of contextual information within each slice as well as across the slices by involving Slice Level ReconNet (SLRNet) network and a Volumetric ROI Level ReconNet (VPLRNet) network, respectively. Three-loss functions have been used to optimize network learning, i.e., L1, Adversarial, and Total Variation Loss. Finally, the network is trained and validated on the publicly available HCP data-set with standard qualitative and quantitative performance measures and achieves promising results.

Two-dimensional turbulence on a sphere

Following Fjørtoft (Tellus, vol. 5, 1953, pp. 225–230) we undertake a spectral analysis of a non-divergent flow on a sphere. It is shown that the spherical harmonic energy spectrum is invariant under rotations of the polar axis of the spherical harmonic system and argued that a constraint of isotropy would not simplify the analysis but only exclude low-order modes. The spectral energy equation is derived and it is shown that the viscous term has a slightly different form than given in previous studies. The relations involving energy transfer within a triad of modes, which Fjørtoft (Tellus, vol. 5, 1953, pp. 225–230) derived under the condition that energy transfer is restricted to three modes, are derived under general conditions. These relations show that there are two types of interaction within a triad. The first type is where the middle mode acts as a source for the two other modes and the second type is where it acts as a sink. The inequality indicating cascade directions which was derived by Gkioulekas & Tung (J. Fluid Mech., vol. 576, 2007, pp. 173–189) in Fourier space under the assumptions of narrow band forcing and stationarity is derived in spherical harmonic space under the assumption of dominance of first type interactions. The double cascade theory of Kraichnan (Phys. Fluids, vol. 10, 1967, pp. 1417–1423) is discussed in the light of the derived equations and it is hypothesised that in flows with limited scale separation the two cascades may, to a large extent, be produced by the same triad interactions. Finally, we conclude that the spherical geometry is the optimal test ground for exploration of two-dimensional turbulence by means of simulations.

Projection-based classification of surfaces for 3D human mesh sequence retrieval

We analyze human poses and motion by introducing three sequences of easily calculated surface descriptors that are invariant under reparametrizations and Euclidean transformations. These descriptors are obtained by associating to each finitely-triangulated surface two functions on the unit sphere: for each unit vector u we compute the weighted area of the projection of the surface onto the plane orthogonal to u and the length of its projection onto the line spanned by u. The L2 norms and inner products of the projections of these functions onto the space of spherical harmonics of order k provide us with three sequences of Euclidean and reparametrization invariants of the surface. The use of these invariants reduces the comparison of 3D+time surface representations to the comparison of polygonal curves in Rn. The experimental results on the FAUST and CVSSP3D artificial datasets are promising. Moreover, a slight modification of our method yields good results on the noisy CVSSP3D real dataset.

Precise and accurate cosmology with CMB×LSS power spectra and bispectra

With the advent of a new generation of cosmological experiments that will provide high-precision measurements of the cosmic microwave background (CMB) and galaxies in the large-scale structure, it is pertinent to examine the potential of performing a joint analysis of multiple cosmological probes. In this paper, we study the cosmological information content contained in the one-loop power spectra and tree bispectra of galaxies cross-correlated with CMB lensing. We use the FFTLog method to compute angular correlations in spherical harmonic space, applicable for wide angles that can be accessed by forthcoming galaxy surveys. We find that adding the bispectra and cross-correlations with CMB lensing offers a significant improvement in parameter constraints, including those on the total neutrino mass, Mν, and local non-Gaussianity amplitude, . In particular, our results suggest that the combination of the Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST) and CMB-S4 will be able to achieve σ(Mν)=42 meV from galaxy and CMB lensing correlations, and σ(Mν)=12 meV when further combined with the CMB temperature and polarization data, without any prior on the optical depth.

Nonlinear redshift-space distortions in the harmonic-space galaxy power spectrum

Future high spectroscopic resolution galaxy surveys will observe galaxies with nearly full-sky footprints. Modeling the galaxy clustering for these surveys, therefore, must include the wide-angle effect with narrow redshift binning. In particular, when the redshift-bin size is comparable to the typical peculiar velocity field, the nonlinear redshift-space distortion (RSD) effect becomes important. A naive projection of the Fourier-space RSD model to spherical harmonic space leads to diverging expressions. In this paper we present a general formalism of projecting the higher-order RSD terms into spherical harmonic space. We show that the nonlinear RSD effect, including the fingers-of-God (FoG), can be entirely attributed to a modification of the radial window function. We find that while linear RSD enhances the harmonic-space power spectrum, unlike the three-dimensional case, the enhancement decreases on small angular-scales. The fingers-of-God suppress the angular power spectrum on all transverse scales if the bin size is smaller than $\Delta r \lesssim \pi \sigma_u$; for example, the radial bin sizes corresponding to a spectral resolution $R=\lambda/\Delta \lambda$ of a few hundred satisfy the condition. We also provide the flat-sky approximation which reproduces the full calculation to sub-percent accuracy.

Theory of deep convolutional neural networks II: Spherical analysis

Deep learning based on deep neural networks of various structures and architectures has been powerful in many practical applications, but it lacks enough theoretical verifications. In this paper, we consider a family of deep convolutional neural networks applied to approximate functions on the unit sphere Sd−1 of Rd. Our analysis presents rates of uniform approximation when the approximated function lies in the Sobolev space W∞r(Sd−1) with r>0 or takes an additive ridge form. Our work verifies theoretically the modelling and approximation ability of deep convolutional neural networks followed by downsampling and one fully connected layer or two. The key idea of our spherical analysis is to use the inner product form of the reproducing kernels of the spaces of spherical harmonics and then to apply convolutional factorizations of filters to realize the generated linear features.

Linearizations of the Spherical Harmonic Discrete Ordinate Method (SHDOM)

Linearizations of the spherical harmonic discrete ordinate method (SHDOM) by means of a forward and a forward-adjoint approach are presented. Essentially, SHDOM is specialized for derivative calculations and radiative transfer problems involving the delta-M approximation, the TMS correction, and the adaptive grid splitting, while practical formulas for computing the derivatives in the spherical harmonics space are derived. The accuracies and efficiencies of the proposed methods are analyzed for several test problems.

(Lr,Ls) Resolvent estimate for the sphere off the line 1 r −1 s = 2 n

We extend the resolvent estimate on the sphere to exponents off the line [Formula: see text]. Since the condition [Formula: see text] on the exponents is necessary for a uniform bound, one cannot expect estimates off this line to be uniform still. The essential ingredient in our proof is an [Formula: see text] norm estimate on the operator [Formula: see text] that projects onto the space of spherical harmonics of degree [Formula: see text]. In showing this estimate, we apply an interpolation technique first introduced by Bourgain [J. Bourgain, Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris Sér. I Math. 301(10) (1985) 499–502.]. The rest of our proof parallels that in Huang–Sogge [S. Huang and C. D. Sogge, Concerning [Formula: see text] resolvent estimates for simply connected manifolds of constant curvature, J. Funct. Anal. 267(12) (2014) 4635–4666].

The reproducing kernel of $\mathcal H^2$ and radial eigenfunctions of the hyperbolic Laplacian

In the paper we characterize the reproducing kernel $\mathcal {K}_{n,h}$ for the Hardy space $\mathcal {H}^2$ of hyperbolic harmonic functions on the unit ball $\mathbb {B}$ in $\mathbb {R}^n$. Specifically we prove that \[ \mathcal {K}_{n,h}(x,y) = \sum _{\alpha =0}^\infty S_{n,\alpha }(\lvert x\rvert )S_{n,\alpha }(\lvert y\rvert ) Z_\alpha (x,y), \] where the series converges absolutely and uniformly on $K\times \mathbb {B}$ for every compact subset $K$ of $\mathbb {B}$. In the above, $S_{n,\alpha }$ is a hypergeometric function and $Z_\alpha $ is the reproducing kernel of the space of spherical harmonics of degree α. In the paper we prove that \[ 0\le \mathcal K_{n,h}(x,y) \le \frac {C_n}{(1-2\langle x,y\rangle + \lvert x \rvert^2 \lvert y \rvert^2)^{n-1}}, \] where $C_n$ is a constant depending only on $n$. It is known that the diagonal function $\mathcal K_{n,h}(x,x)$ is a radial eigenfunction of the hyperbolic Laplacian $\varDelta_h $ on $\mathbb{B} $ with eigenvalue $\lambda _2 = 8(n-1)^2$. The result for $n=4$ provides motivation that leads to an explicit characterization of all radial eigenfunctions of $\varDelta_h $ on $\mathbb{B} $. Specifically, if $g$ is a radial eigenfunction of $\varDelta_h $ with eigenvalue $\lambda _\alpha = 4(n-1)^2\alpha (\alpha -1)$, then \[ g(r) = g(0) \frac {p_{n,\alpha }(r^2)}{(1-r^2)^{(\alpha -1)(n-1)}}, \] where $p_{n,\alpha }$ is again a hypergeometric function. If α is an integer, then $p_{n,\alpha }(r^2)$ is a polynomial of degree $2(\alpha -1)(n-1)$.

Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry

We consider bifurcation problems in the presence of \begin{document}$ O(3) $\end{document} symmetry. Well known group-theoretic techniques enable the classification of all maximal isotropy subgroups of \begin{document}$ O(3) $\end{document} , with associated mode numbers \begin{document}$\ell∈\mathbb{N} $\end{document} , leading to 1-dimensional fixed-point subspaces of the \begin{document}$ (2\ell+1) $\end{document} -dimensional space of spherical harmonics. In each case the so-called equivariant branching lemma can then be used to establish the existence of a local branch of bifurcating solutions having the symmetry of the respective subgroup. To first-order, such a branch is a precise linear combination of the \begin{document}$ 2\ell+1 $\end{document} spherical harmonics, which we call the bifurcation direction. Our work here is focused on the direct construction of these bifurcation directions, complementing the above-mentioned classification. The approach is an application of a general method for constructing families of symmetric spherical harmonics, based on differentiating the fundamental solution of Laplace's equation in \begin{document}$ \mathbb{R}^3 $\end{document} .

Maps of the Southern Millimeter-wave Sky from Combined 2500 deg2 SPT-SZ and Planck Temperature Data

Abstract We present three maps of the millimeter-wave sky created by combining data from the South Pole Telescope (SPT) and the Planck satellite. We use data from the SPT-SZ survey, a survey of 2540 deg2 of the the sky with arcminute resolution in three bands centered at 95, 150, and 220 GHz, and the full-mission Planck temperature data in the 100, 143, and 217 GHz bands. A linear combination of the SPT-SZ and Planck data is computed in spherical harmonic space, with weights derived from the noise of both instruments. This weighting scheme results in Planck data providing most of the large-angular-scale information in the combined maps, with the smaller-scale information coming from SPT-SZ data. A number of tests have been done on the maps. We find their angular power spectra to agree very well with theoretically predicted spectra and previously published results.

$$(L^{r}, L^{s})$$(Lr,Ls) resolvent estimate for the sphere off the line $$\frac{1}{r}-\frac{1}{s}=\frac{2}{n}$$1r-1s=2n

We extend the resolvent estimate on the sphere to exponents off the line $$\frac{1}{r}-\frac{1}{s}=\frac{2}{n}$$ . Since the condition $$\frac{1}{r}-\frac{1}{s}=\frac{2}{n}$$ on the exponents is necessary for a uniform bound, one cannot expect estimates off this line to be uniform still. The essential ingredient in our proof is an $$(L^{r}, L^{s})$$ norm estimate on the operator $$H_{k}$$ that projects onto the space of spherical harmonics of degree k. In showing this estimate, we apply an interpolation technique first introduced by Bourgain (C R Acad Sci Paris Ser I Math 301(10):499–502, 1985). The rest of our proof parallels that in Huang–Sogge (J Funct Anal 267(12):4635–4666, 2014).

Fast and accurate computation of projected two-point functions

We present the 2-point function from Fast and Accurate Spherical Bessel Transformation (2-FAST) algorithm for a fast and accurate computation of integrals involving one or two spherical Bessel functions. These types of integrals occur when projecting the galaxy power spectrum $P(k)$ onto the configuration space, $\xi_\ell^\nu(r)$, or spherical harmonic space, $C_\ell(\chi,\chi')$. First, we employ the FFTlog transformation of the power spectrum to divide the calculation into $P(k)$-dependent coefficients and $P(k)$-independent integrations of basis functions multiplied by spherical Bessel functions. We find analytical expressions for the latter integrals in terms of special functions, for which recursion provides a fast and accurate evaluation. The algorithm, therefore, circumvents direct integration of highly oscillating spherical Bessel functions.

Energy and discrepancy of rotationally invariant determinantal point processes in high dimensional spheres

We study expected Riesz s-energies and linear statistics of some determinantal processes on the sphere Sd. In particular, we compute the expected Riesz and logarithmic energies of the determinantal processes given by the reproducing kernel of the space of spherical harmonics. This kernel defines the so called harmonic ensemble on Sd. With these computations we improve previous estimates for the discrete minimal energy of configurations of points in the sphere. We prove a comparison result for Riesz 2-energies of points defined through determinantal point processes associated with isotropic kernels. As a corollary we get that the Riesz 2-energy of the harmonic ensemble is optimal among ensembles defined by isotropic kernels with the same trace. Finally, we study the variance of smooth and rough linear statistics for the harmonic ensemble and compare the results with the variance for the spherical ensemble (in S2).

Reproducing Kernels for Polynomial Null-Solutions of Dirac Operators

It is well known that the reproducing kernel of the space of spherical harmonics of fixed homogeneity is given by a Gegenbauer polynomial. By going over to complex variables and restricting to suitable bihomogeneous subspaces, one obtains a reproducing kernel expressed as a Jacobi polynomial, which leads to Koornwinder’s celebrated result on the addition formula. In the present paper, the space of Hermitian monogenics, which is the space of polynomial bihomogeneous null-solutions of a set of two complex conjugated Dirac operators, is considered. The reproducing kernel for this space is obtained and expressed in terms of sums of Jacobi polynomials. This is achieved through use of the underlying Lie superalgebra \(\mathfrak {sl}(1|2)\), combined with the equivalence between the \(L^2\) inner product on the unit sphere and the Fischer inner product. The latter also leads to a new proof in the standard Dirac case related to the Lie superalgebra \(\mathfrak {osp}(1|2)\).