Computing with functions in spherical and polar geometries I. The sphere

A collection of algorithms is described for numerically computing with smooth functions defined on the unit sphere. Functions are approximated to essentially machine precision by using a structure-preserving iterative variant of Gaussian elimination together with the double Fourier sphere method. We show that this procedure allows for stable differentiation, reduces the oversampling of functions near the poles, and converges for certain analytic functions. Operations such as function evaluation, differentiation, and integration are particularly efficient and can be computed by essentially one-dimensional algorithms. A highlight is an optimal complexity direct solver for Poisson's equation on the sphere using a spectral method. Without parallelization, we solve Poisson's equation with $100$ million degrees of freedom in one minute on a standard laptop. Numerical results are presented throughout. In a companion paper (part II) we extend the ideas presented here to computing with functions on the disk.Comment: 23 page

On the Spectral Properties of Matrices Associated with Trend Filters

This paper is concerned with the spectral properties of matrices associated with linear filters for the estimation of the underlying trend of a time series. The interest lies in the fact that the eigenvectors can be interpreted as the latent components of any time series that the filter smooths through the corresponding eigenvalues. A difficulty arises because matrices associated with trend filters are finite approximations of Toeplitz operators and therefore very little is known about their eigenstructure, which also depends on the boundary conditions or, equivalently, on the filters for trend estimation at the end of the sample. Assuming reflecting boundary conditions, we derive a time series decomposition in terms of periodic latent components and corresponding smoothing eigenvalues. This decomposition depends on the local polynomial regression estimator chosen for the interior. Otherwise, the eigenvalue distribution is derived with an approximation measured by the size of the perturbation that different boundary conditions apport to the eigenvalues of matrices belonging to algebras with known spectral properties, such as the Circulant or the Cosine. The analytical form of the eigenvectors is then derived with an approximation that involves the extremes only. A further topic investigated in the paper concerns a strategy for a filter design in the time domain. Based on cut-off eigenvalues, new estimators are derived, that are less variable and almost equally biased as the original estimator, based on all the eigenvalues. Empirical examples illustrate the effectiveness of the method

Diffuse Optical Imaging Using Decomposition Methods

Diffuse optical imaging (DOI) for detecting and locating targets in a highly scattering turbid medium is treated as a blind source separation (BSS) problem. Three matrix decomposition methods, independent component analysis (ICA), principal component analysis (PCA), and nonnegative matrix factorization (NMF) were used to study the DOI problem. The efficacy of resulting approaches was evaluated and compared using simulated and experimental data. Samples used in the experiments included Intralipid-10% or Intralipid-20% suspension in water as the medium with absorptive or scattering targets embedded