Nail A. Gumerov

Ross Adelman, Nail A. Gumerov, Ramani Duraiswami

The spheroidal wave functions, which are the solutions to the Helmholtz equation in spheroidal coordinates, are notoriously difficult to compute. Because of this, practically no programming language comes equipped with the means to compute them. This makes problems that require their use hard to tackle. We have developed computational software for calculating these special functions. Our software is called spheroidal and includes several novel features, such as: using arbitrary precision arithmetic; adaptively choosing the number of expansion coefficients to compute and use; and using the Wronskian to choose from several different methods for computing the spheroidal radial functions to improve their accuracy. There are two types of spheroidal wave functions: the prolate kind when prolate spheroidal coordinates are used; and the oblate kind when oblate spheroidal coordinate are used. In this paper, we describe both, methods for computing them, and our software. We have made our software freely available on our webpage.

Balaji Vasan Srinivasan, Qi Hu, Nail A. Gumerov et al.

A primary computational problem in kernel regression is solution of a dense linear system with the $N\times N$ kernel matrix. Because a direct solution has an O($N^3$) cost, iterative Krylov methods are often used with fast matrix-vector products. For poorly conditioned problems, convergence of the iteration is slow and preconditioning becomes necessary. We investigate preconditioning from the viewpoint of scalability and efficiency. The problems that conventional preconditioners face when applied to kernel methods are demonstrated. A \emph{novel flexible preconditioner }that not only improves convergence but also allows utilization of fast kernel matrix-vector products is introduced. The performance of this preconditioner is first illustrated on synthetic data, and subsequently on a suite of test problems in kernel regression and geostatistical kriging.

Nail A. Gumerov, Ramani Duraiswami

Non-uniform fast Fourier Transform (NUFFT) and inverse NUFFT (INUFFT) algorithms, based on the Fast Multipole Method (FMM) are developed and tested. Our algorithms are based on a novel factorization of the FFT kernel, and are implemented with attention to data structures and error analysis. Note: This unpublished manuscript was available on our web pages and has been referred to by others in the literature. To provide a proper archival reference we are placing it on arXiv.

Z. Zhang, K. Duraisamy, N. A. Gumerov

Standard Gaussian Process (GP) regression, a powerful machine learning tool, is computationally expensive when it is applied to large datasets, and potentially inaccurate when data points are sparsely distributed in a high-dimensional feature space. To address these challenges, a new multiscale, sparsified GP algorithm is formulated, with the goal of application to large scientific computing datasets. In this approach, the data is partitioned into clusters and the cluster centers are used to define a reduced training set, resulting in an improvement over standard GPs in terms of training and evaluation costs. Further, a hierarchical technique is used to adaptively map the local covariance representation to the underlying sparsity of the feature space, leading to improved prediction accuracy when the data distribution is highly non-uniform. A theoretical investigation of the computational complexity of the algorithm is presented. The efficacy of this method is then demonstrated on smooth and discontinuous analytical functions and on data from a direct numerical simulation of turbulent combustion.

Ross Adelman, Nail A. Gumerov, Ramani Duraiswami

Analytical solutions to acoustic scattering problems involving nonspherical shapes, such as spheroids and disks, have long been known and have many applications. However, these solutions require special functions that are not easily computable. For this reason, their asymptotic forms are typically used since they are more readily available. We explore these solutions and provide computational software for calculating their nonasymptotic forms, which are accurate over a wide range of frequencies and distances. This software, which runs in MATLAB, computes the solutions to acoustic scattering problems involving spheroids and disks by semi-analytical means, and is freely available from our webpage.