## Applied Mathematics Graduate Theses & Dissertations

Summer 8-10-2018

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Applied Mathematics

Gregory Beylkin

#### Abstract

We consider a fast deterministic algorithm to identify the "best" linearly independent terms in multivariate mixtures and use them to compute an equivalent representation with fewer terms, up to user-selected accuracy. Our algorithm employs the well-known pivoted Cholesky decomposition of the Gram matrix constructed using terms of the mixture. Importantly, the multivariate mixtures do not have to be a separated representation of a function and complexity of the algorithm is independent of the number of variables (dimensions). The algorithm requires $\mathcal{O}\left(r^{2}N\right)$ operations, where $N$ is the initial number of terms in a multivariate mixture and $r$ is the number of selected terms. Due to the condition number of the Gram matrix, the resulting accuracy is limited to about $1/2$ digits of the used floating point arithmetic. We also consider two additional reduction algorithms for the same purpose. The first algorithm is based on orthogonalization of the multivariate mixture and have a similar performance as the approach based on Cholesky factorization. The second algorithm yields a better accuracy, but currently in high dimensions is only applicable to multivariate mixtures in a separated representation.

We use the reduction algorithm to develop a new adaptive numerical method for solving differential and integral equations in quantum chemistry. We demonstrate the performance of this approach by solving the Hartree-Fock equations in two cases of small molecules. We also describe a number of initial applications of the reduction algorithm to solve partial differential and integral equations and to address several problems in data sciences. For data science applications in high dimensions we consider kernel density estimation (KDE) approach for constructing a probability density function (PDF) of a cloud of points, a far-field kernel summation method and the construction of equivalent sources for non-oscillatory kernels (used in both, computational physics and data science) and, finally, show how to use the reduction algorithm to produce seeds for subdividing a cloud of points into groups.

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