arXiv subjects

Gregory Beylkin

Publications and source records attributed to Gregory Beylkin.

At least 19 recordsLinked to original sources

arXiv2025

Fast convolution algorithm for state space models

We present an unconditionally stable algorithm for applying matrix transfer function of a linear time invariant system (LTI) in time domain. The state matrix of an LTI system used for modeling long range dependencies in state space models (SSMs) has eigenvalues close to $1$. The standard recursion defining LTI system becomes unstable if the $m\times m$ state matrix has just one eigenvalue with absolute value even slightly greater than 1. This may occur when approximating a state matrix by a structured matrix to reduce the cost of matrix-vector multiplication from $\mathcal{O}\left(m^{2}\right)$ to $\mathcal{O}\left(m\right)$ or $\mathcal{O}\left(m\log m\right).$ We introduce an unconditionally stable algorithm that uses an approximation of the rational transfer function in the z-domain by a matrix polynomial of degree $2^{N+1}-1$, where $N$ is chosen to achieve any user-selected accuracy. Using a cascade implementation in time domain, applying such transfer function to compute $L$ states requires no more than $2L$ matrix-vector multiplications (whereas the standard recursion requires $L$ matrix-vector multiplications). However, using unconditionally stable algorithm, it is not necessary to assure that an approximate state matrix has all eigenvalues with absolute values strictly less than 1 i.e., within the desired accuracy, the absolute value of some eigenvalues may possibly exceed $1$. Consequently, this algorithm allows one to use a wider variety of structured approximations to reduce the cost of matrix-vector multiplication and we briefly describe several of them to be used for this purpose.

math.NA↗

arXiv2023

On representations of the Helmholtz Green's function

We consider the free space Helmholtz Green's function and split it into the sum of oscillatory and non-oscillatory (singular) components. The goal is to separate the impact of the singularity of the real part at the origin from the oscillatory behavior controlled by the wave number k. The oscillatory component can be chosen to have any finite number of continuous derivatives at the origin and can be applied to a function in the Fourier space in $\mathcal{O}\left(k^{d}\log k\right)$ operations. The non-oscillatory component has a multiresolution representation via a linear combination of Gaussians and is applied efficiently in space.

math-ph↗

arXiv2022

Fast exchange with Gaussian basis set using robust pseudospectral method

In this article we present an algorithm to efficiently evaluate the exchange matrix in periodic systems when Gaussian basis set with pseudopotentials are used. The usual algorithm for evaluating exchange matrix scales cubically with the system size because one has to perform O(N2) fast Fourier transforms (FFT). Here we introduce an algorithm that retains the cubic scaling but reduces the prefactor significantly by eliminating the need to do FFTs during each exchange build. This is accomplished by representing the products of Gaussian basis function using a linear combination of an auxiliary basis the number of which scales linearly with the size of the system. We store the potential due to these auxiliary functions in memory which allows us to obtain the exchange matrix without the need to do FFT, albeit at the cost of additional memory requirement. Although the basic idea of using auxiliary functions is not new, our algorithm is cheaper due to a combination of three ingredients: (a) we use robust Pseudospectral method that allows us to use a relatively small number of auxiliary basis to obtain high accuracy (b) we use occ-RI exchange which eliminates the need to construct the full exchange matrix and (c) we use the (interpolative separable density fitting) ISDF algorithm to construct these auxiliary basis that are used in the robust pseudospectral method. The resulting algorithm is accurate and we note that the error in the final energy decreases exponentially rapidly with the number of auxiliary functions.

cond-mat.str-el↗

arXiv2021

A fast algorithm for computing the Boys function

We present a new fast algorithm for computing the Boys function using a nonlinear approximation of the integrand via exponentials. The resulting algorithms evaluate the Boys function with real and complex valued arguments and are competitive with previously developed algorithms for the same purpose.

math.NA↗

arXiv2021

On computing bound states of the Dirac and Schrödinger Equations

We cast the quantum chemistry problem of computing bound states as that of solving a set of auxiliary eigenvalue problems for a family of parameterized compact integral operators. The compactness of operators assures that their spectrum is discrete and bounded with the only possible accumulation point at zero. We show that, by changing the parameter, we can always find the bound states, i.e., the eigenfunctions that satisfy the original equations and are normalizable. While for the non-relativistic equations these properties may not be surprising, it is remarkable that the same holds for the relativistic equations where the spectrum of the original relativistic operators does not have a lower bound. We demonstrate that starting from an arbitrary initialization of the iteration leads to the solution, as dictated by the properties of compact operators.

quant-ph↗

arXiv2021

Efficient evaluation of two-center Gaussian integrals in periodic systems

By using Poisson's summation formula, we calculate periodic integrals over Gaussian basis functions by partitioning the lattice summations between the real and reciprocal space, where both sums converge exponentially fast with a large exponent. We demonstrate that the summation can be performed efficiently to calculate 2-center Gaussian integrals over various kernels including overlap, kinetic, and Coulomb. The summation in real space is performed using an efficient flavor of the McMurchie-Davidson Recurrence Relation (MDRR). The expressions for performing summation in the reciprocal space are also derived and implemented. The algorithm for reciprocal space summation allows us to reuse several terms and leads to significant improvement in efficiency when highly contracted basis functions with large exponents are used. We find that the resulting algorithm is only between a factor of 5 to 15 slower than that for molecular integrals, indicating the very small number of terms needed in both the real and reciprocal space summations. An outline of the algorithm for calculating 3-center Coulomb integrals is also provided.

cond-mat.str-el↗

arXiv2019

Efficient Fourier Basis Particle Simulation

The standard particle-in-cell algorithm suffers from grid heating. There exists a gridless alternative which bypasses the deposition step and calculates each Fourier mode of the charge density directly from the particle positions. We show that a gridless method can be computed efficiently through the use of an Unequally Spaced Fast Fourier Transform (USFFT) algorithm. After a spectral field solve, the forces on the particles are calculated via the inverse USFFT (a rapid solution of an approximate linear system). We provide one and two dimensional implementations of this algorithm with an asymptotic runtime of $O(N_p + N_m^D \log N_m^D)$ for each iteration, identical to the standard PIC algorithm (where $N_p$ is the number of particles and $N_m$ is the number of Fourier modes, and $D$ is the spatial dimensionality of the problem) We demonstrate superior energy conservation and reduced noise, as well as convergence of the energy conservation at small time steps.

physics.plasm-ph↗

arXiv2018

The periodic table of the elements with $4n^{2}$ $n=2,3,\dots$ periods

A modification of the standard periodic table of the elements reveals $4n^{2}$ periods, where $n=2,3,\dots$. The new arrangement places hydrogen with halogens and keeps the rare-earth elements in the table proper (without separating them as they are in the standard table). Effectively, periods in the modified table are defined by the halogens rather than by the noble gases. The graph of ionization energy of the elements is presented for comparison of periods in the standard and the modified tables.

physics.gen-ph↗

arXiv2018

Adaptive algorithm for electronic structure calculations using reduction of Gaussian mixtures

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.

math.NA↗

arXiv2018

Reduction of multivariate mixtures and its applications

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.

math.NA↗

arXiv2018

On computing distributions of products of non-negative independent random variables

We introduce a new functional representation of probability density functions (PDFs) of non-negative random variables via a product of a monomial factor and linear combinations of decaying exponentials with complex exponents. This approximate representation of PDFs is obtained for any finite, user-selected accuracy. Using a fast algorithm involving Hankel matrices, we develop a general numerical method for computing the PDF of the sums, products, or quotients of any number of non-negative random variables yielding the result in the same type of functional representation. We present several examples to demonstrate the accuracy of the approach.

math.PR↗

arXiv2016

On computing distributions of products of random variables via Gaussian multiresolution analysis

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.

math.NA↗

arXiv2016

Optimization via Separated Representations and the Canonical Tensor Decomposition

We introduce a new, quadratically convergent algorithm for finding maximum absolute value entries of tensors represented in the canonical format. The computational complexity of the algorithm is linear in the dimension of the tensor. We show how to use this algorithm to find global maxima of non-convex multivariate functions in separated form. We demonstrate the performance of the new algorithms on several examples.

math.NA↗

arXiv2015

Randomized Alternating Least Squares for Canonical Tensor Decompositions: Application to a PDE with Random Data

This paper introduces a randomized variation of the alternating least squares (ALS) algorithm for rank reduction of canonical tensor formats. The aim is to address the potential numerical ill-conditioning of least squares matrices at each ALS iteration. The proposed algorithm, dubbed randomized ALS, mitigates large condition numbers via projections onto random tensors, a technique inspired by well-established randomized projection methods for solving overdetermined least squares problems in a matrix setting. A probabilistic bound on the condition numbers of the randomized ALS matrices is provided, demonstrating reductions relative to their standard counterparts. Additionally, results are provided that guarantee comparable accuracy of the randomized ALS solution at each iteration. The performance of the randomized algorithm is studied with three examples, including manufactured tensors and an elliptic PDE with random inputs. In particular, for the latter, tests illustrate not only improvements in condition numbers, but also improved accuracy of the iterative solver for the PDE solution represented in a canonical tensor format.

math.NA↗

arXiv2015

MADNESS: A Multiresolution, Adaptive Numerical Environment for Scientific Simulation

MADNESS (multiresolution adaptive numerical environment for scientific simulation) is a high-level software environment for solving integral and differential equations in many dimensions that uses adaptive and fast harmonic analysis methods with guaranteed precision based on multiresolution analysis and separated representations. Underpinning the numerical capabilities is a powerful petascale parallel programming environment that aims to increase both programmer productivity and code scalability. This paper describes the features and capabilities of MADNESS and briefly discusses some current applications in chemistry and several areas of physics.

cs.MS↗

arXiv2013

Randomized Interpolative Decomposition of Separated Representations

We introduce tensor Interpolative Decomposition (tensor ID) for the reduction of the separation rank of Canonical Tensor Decompositions (CTDs). Tensor ID selects, for a user-defined accuracy ε, a near optimal subset of terms of a CTD to represent the remaining terms via a linear combination of the selected terms. Tensor ID can be used as an alternative to or a step of the Alternating Least Squares (ALS) algorithm. In addition, we briefly discuss Q-factorization to reduce the size of components within an ALS iteration. Combined, tensor ID and Q-factorization lead to a new paradigm for the reduction of the separation rank of CTDs. In this context, we also discuss the spectral norm as a computational alternative to the Frobenius norm. We reduce the problem of finding tensor IDs to that of constructing Interpolative Decompositions of certain matrices. These matrices are generated via either randomized projection or randomized sampling of the given tensor. We provide cost estimates and several examples of the new approach to the reduction of separation rank.

math.NA↗

arXiv2012

ODE Solvers Using Bandlimited Approximations

We use generalized Gaussian quadratures for exponentials to develop a new ODE solver. Nodes and weights of these quadratures are computed for a given bandlimit $c$ and user selected accuracy $\epsilon$, so that they integrate functions $e^{ibx}$, for all $|b|\le c$, with accuracy $\epsilon$. Nodes of these quadratures do not concentrate excessively near the end points of an interval as those of the standard, polynomial-based Gaussian quadratures. Due to this property, the usual implicit Runge Kutta (IRK) collocation method may be used with a large number of nodes, as long as the method chosen for solving the nonlinear system of equations converges. We show that the resulting ODE solver is symplectic and demonstrate (numerically) that it is A-stable. We use this solver, dubbed Band-limited Collocation (BLC-IRK), in the problem of orbit determination. Since BLC-IRK minimizes the number of nodes needed to obtain the solution, in this problem we achieve speed close to that of explicit multistep methods.

math.NA↗

arXiv2007

Approximating a Wavefunction as an Unconstrained Sum of Slater Determinants

The wavefunction for the multiparticle Schr\"odinger equation is a function of many variables and satisfies an antisymmetry condition, so it is natural to approximate it as a sum of Slater determinants. Many current methods do so, but they impose additional structural constraints on the determinants, such as orthogonality between orbitals or an excitation pattern. We present a method without any such constraints, by which we hope to obtain much more efficient expansions, and insight into the inherent structure of the wavefunction. We use an integral formulation of the problem, a Green's function iteration, and a fitting procedure based on the computational paradigm of separated representations. The core procedure is the construction and solution of a matrix-integral system derived from antisymmetric inner products involving the potential operators. We show how to construct and solve this system with computational complexity competitive with current methods.

math-ph↗