1. Near Optimal Reconstruction of Spherical Harmonic Expansions(arXiv)
Abstract : We propose an algorithm for robust recovery of the spherical harmonic expansion of functions defined on the d-dimensional unit sphere Sd−1 using a near-optimal number of function evaluations. We show that for any f∈L2(Sd−1), the number of evaluations of f needed to recover its degree-q spherical harmonic expansion equals the dimension of the space of spherical harmonics of degree at most q up to a logarithmic factor. Moreover, we develop a simple yet efficient algorithm to recover degree-q expansion of f by only evaluating the function on uniformly sampled points on Sd−1. Our algorithm is based on the connections between spherical harmonics and Gegenbauer polynomials and leverage score sampling methods. Unlike the prior results on fast spherical harmonic transform, our proposed algorithm works efficiently using a nearly optimal number of samples in any dimension d. We further illustrate the empirical performance of our algorithm on numerical examples.
2. Spherical harmonic shape descriptors of nodal force demands for quantifying spatial truss connection complexity(arXiv)
Abstract : The connections of a spatial truss structure play a critical role in the safe and efficient transfer of axial forces between members. For discrete connections, they can also improve construction efficiency by acting as registration devices that lock members in precise orientations. As more geometrically complex spatial trusses are enabled by computational workflows and the demand for material-efficient spanning systems, there is a need to understand the effects of global form on the demands at the connections. For large-scale structures with irregular geometry, customizing individual nodes to meet exact member orientations and force demands may be infeasible; conversely, standardizing all connections results in oversized nodes and a compromise in registration potential. We propose a method for quantifying the complexity of spatial truss designs by the variation in nodal force demands. By representing nodal forces as a geometric object, we leverage the spherical harmonic shape descriptor, developed for applications in computational geometry, to characterize each node by a rotation and translation-invariant fixed-length vector. We define a complexity score for spatial truss design by the variance in the positions of the feature vectors in higher-dimensional space, providing an additional performance metric during early stage design exploration. We then develop a pathway towards reducing complexity by clustering nodes with respect to their feature vectors to reduce the number of unique connectors for design while minimizing the effects of mass standardization
3.On the correlation between critical points and critical values for random spherical harmonics (arXiv)
Abstract : We study the correlation between the total number of critical points of random spherical harmonics and the number of critical points with value in any interval I⊂R. We show that the correlation is asymptotically zero, while the partial correlation, after controlling the random L2-norm on the sphere of the eigenfunctions, is asymptotically one. Our findings complement the results obtained by Wigman (2012) and Marinucci and Rossi (2021) on the correlation between nodal and boundary length of random spherical harmonics.