Lucas Monzon

Gregory Beylkin, Lucas Monzon, Xinshuo Yang

We present a new adaptive method for electronic structure calculations based on novel fast algorithms for reduction of multivariate mixtures. In our calculations, spatial orbitals are maintained as Gaussian mixtures whose terms are selected in the process of solving equations. Using a fixed basis leads to the so-called "basis error" since orbitals may not lie entirely within the linear span of the basis. To avoid such an error, multiresolution bases are used in adaptive algorithms so that basis functions are selected from a fixed collection of functions, large enough as to approximate solutions within any user-selected accuracy. Our new method achieves adaptivity without using a multiresolution basis. Instead, as a part of an iteration to solve nonlinear equations, our algorithm selects the "best" subset of linearly independent terms of a Gaussian mixture from a collection that is much larger than any possible basis since the locations and shapes of the Gaussian terms are not fixed in advance. Approximating an orbital within a given accuracy, our algorithm yields significantly fewer terms than methods using multiresolution bases. We demonstrate our approach by solving the Hartree-Fock equations for two diatomic molecules, HeH+ and LiH, matching the accuracy previously obtained using multiwavelet bases.

Gregory Beylkin, Lucas Monzon, Xinshuo Yang

We consider fast deterministic algorithms to identify the "best" linearly independent terms in multivariate mixtures and use them to compute, up to a user-selected accuracy, an equivalent representation with fewer terms. One algorithm employs a pivoted Cholesky decomposition of the Gram matrix constructed from the terms of the mixture to select what we call skeleton terms and the other uses orthogonalization for the same purpose. Importantly, the multivariate mixtures do not have to be a separated representation of a function. Both algorithms require $O(r^2 N + p(d) r N) $ operations, where $N$ is the initial number of terms in the multivariate mixture, $r$ is the number of selected linearly independent terms, and $p(d)$ is the cost of computing the inner product between two terms of a mixture in $d$ variables. For general Gaussian mixtures $p(d) \sim d^3$ since we need to diagonalize a $d\times d$ matrix, whereas for separated representations $p(d) \sim d$. Due to conditioning issues, the resulting accuracy is limited to about one half of the available significant digits for both algorithms. We also describe an alternative algorithm that is capable of achieving higher accuracy but is only applicable in low dimensions or to multivariate mixtures in separated form. We describe a number of initial applications of these algorithms to solve partial differential and integral equations and to address several problems in data science. For data science applications in high dimensions,we consider the 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 new algorithm to produce seeds for subdividing a cloud of points into groups.

Lucas Monzon

Perfect reconstruction filter banks can be used to generate a variety of wavelet bases. Using IIR linear phase filters one can obtain symmetry properties for the wavelet and scaling functions. In this paper we describe all possible IIR linear phase filters generating symmetric wavelets with any prescribed number of vanishing moments. In analogy with the well known FIR case, we construct and study a new family of wavelets obtained by considering maximal number of vanishing moments for each fixed order of the IIR filter. Explicit expressions for the coefficients of numerator, denominator, zeroes, and poles are presented. This new parameterization allows one to design linear phase quadrature mirror filters with many other properties of interest such as filters that have any preassigned set of zeroes in the stopband or that satisfy an almost interpolating property. Using Beylkin's approach, it is indicated how to implement these IIR filters not as recursive filters but as FIR filters.

Gregory Beylkin, Lucas Monzon, Ignas Satkauskas

We introduce a new approximate multiresolution analysis (MRA) using a single Gaussian as the scaling function, which we call Gaussian MRA (GMRA). As an initial application, we employ this new tool to accurately and efficiently compute the probability density function (PDF) of the product of independent random variables. In contrast with Monte-Carlo (MC) type methods (the only other universal approach known to address this problem), our method not only achieves accuracies beyond the reach of MC but also produces a PDF expressed as a Gaussian mixture, thus allowing for further efficient computations. We also show that an exact MRA corresponding to our GMRA can be constructed for a matching user-selected accuracy.