arXiv subjects

Nail A. Gumerov

Publications and source records attributed to Nail A. Gumerov.

17 recordsLinked to original sources

arXiv2023

Efficient Fast Multipole Accelerated Boundary Elements via Recursive Computation of Multipole Expansions of Integrals

In boundary element methods (BEM) in $\mathbb{R}^3$, matrix elements and right hand sides are typically computed via analytical or numerical quadrature of the layer potential multiplied by some function over line, triangle and tetrahedral volume elements. When the problem size gets large, the resulting linear systems are often solved iteratively via Krylov subspace methods, with fast multipole methods (FMM) used to accelerate the matrix vector products needed. When FMM acceleration is used, most entries of the matrix never need be computed explicitly - {\em they are only needed in terms of their contribution to the multipole expansion coefficients.} We propose a new fast method - \emph{Quadrature to Expansion (Q2X)} - for the analytical generation of the multipole expansion coefficients produced by the integral expressions for single and double layers on surface triangles; charge distributions over line segments and over tetrahedra in the volume; so that the overall method is well integrated into the FMM, with controlled error. The method is based on the $O(1)$ per moment cost recursive computation of the moments. The method is developed for boundary element methods involving the Laplace Green's function in ${\mathbb R}^3$. The derived recursions are first compared against classical quadrature algorithms, and then integrated into FMM accelerated boundary element and vortex element methods. Numerical tests are presented and discussed.

math.NA↗

arXiv2023

Analytical Galerkin boundary integrals of Laplace kernel layer potentials in $\mathbb{R}^3$

A method for analytical computation of the double surface integrals for all layer potential kernels associated with the Laplace Green's function, in the Galerkin boundary element method (BEM) in $\mathbb{R}^3$ using piecewise constant flat elements is presented. The method uses recursive dimensionality reduction from 4D ($\mathbb{R}^2\times\mathbb{R}^2$) based on Gauss' divergence theorem. Computable analytical expressions for all cases of relative location of the source and receiver triangles are covered for the single and double layer potentials and their gradients with analytical treatment of the singular cases are presented. A trick that enables reduction of the case of gradient of the single layer to the same integrals as for the single layer is introduced using symmetry properties. The method was confirmed using analytical benchmark cases, comparisons with error-controlled computations of regular multidimensional integrals, and a convergence study for singular cases.

math.NA↗

arXiv2023

Recursive Analytical Quadrature of Laplace and Helmholtz Layer Potentials in $\mathbb{R}^3$

A method for the analytical evaluation of layer potentials arising in the collocation boundary element method for the Laplace and Helmholtz equation is developed for piecewise flat boundary elements with polynomial shape functions. The method is based on dimension-reduction via the divergence theorem and a Recursive scheme for evaluating the resulting line Integrals for Polynomial Elements (RIPE). It is used to evaluate single layer, double layer, adjoint double layer, and hypersingular potentials, for both the Laplace and the Helmholtz kernels. It naturally supports nearly singular, singular, and hypersingular integrals under a single framework without separate modifications. The developed framework exhibits accuracy and efficiency.

math.NA↗

arXiv2021

Analytical computation of boundary integrals for the Helmholtz equation in three dimensions

A key issue in the solution of partial differential equations via integral equation methods is the evaluation of possibly singular integrals involving the Green's function and its derivatives multiplied by simple functions over discretized representations of the boundary. For the Helmholtz equation, while many authors use numerical quadrature to evaluate these boundary integrals, we present analytical expressions for such integrals over flat polygons in the form of infinite series. These can be efficiently truncated based on the accurate error bounds, which is key to their integration in methods such as the Fast Multipole Method.

math.NA↗

arXiv2021

Fast multipole accelerated boundary element methods for room acoustics

The direct and indirect boundary element methods, accelerated via the fast multipole method, are applied to numerical simulation of room acoustics for large rooms of volume $\sim 150$ $m^{3}$ and frequencies up to 5 kHz on a workstation. As the parameter $kD$ (wavenumber times room diameter) is large, stabilization of the previously developed FMM algorithms is required for accuracy. A stabilization scheme is one of the key contribution of this paper. The computations are validated using well-known image source solutions for shoebox shaped rooms. Computations for L-shaped rooms are performed to illustrate the ability to capture diffractions. The ability to model in-room baffles, and boundary openings (doors/windows) is also demonstrated. The largest case has $kD>1100$ with a discretization of size 6 million elements. The performance of different boundary integral formulations was compared, and their rates of convergence using a preconditioned flexible GMRES were found to be substantially different. These promising results suggest a path to efficient simulations of room acoustics via high performance boundary element methods.

math.NA↗

arXiv2020

Boundary Element Solution of Electromagnetic Fields for Non-Perfect Conductors at Low Frequencies and Thin Skin Depths

A novel boundary element formulation for solving problems involving eddy currents in the thin skin depth approximation is developed. It is assumed that the time-harmonic magnetic field outside the scatterers can be described using the quasistatic approximation. A two-term asymptotic expansion with respect to a small parameter characterizing the skin depth is derived for the magnetic and electric fields outside and inside the scatterer, which can be extended to higher order terms if needed. The introduction of a special surface operator (the inverse surface gradient) allows the reduction of the problem complexity. A method to compute this operator is developed. The obtained formulation operates only with scalar quantities and requires computation of surface operators that are usual for boundary element (method of moments) solutions to the Laplace equation. The formulation can be accelerated using the fast multipole method. The method is much faster than solving the vector Maxwell equations. The obtained solutions are compared with the Mie solution for scattering from a sphere and the error of the solution is studied. Computations for much more complex shapes of different topologies, including for magnetic and electric field cages used in testing are also performed and discussed.

physics.comp-ph↗

arXiv2019

GPU accelerated fast multipole boundary element method for simulation of 3D bubble dynamics in potential flow

A numerical method for simulation of bubble dynamics in three-dimensional potential flows is presented. The approach is based on the boundary element method for the Laplace equation accelerated via the fast multipole method implemented on a heterogeneous CPU/GPU architecture. For mesh stabilization, a new smoothing technique using a surface filter is presented. This technique relies on spherical harmonics expansion of surface functions for bubbles topologically equivalent to a sphere (or Fourier series for toroidal bubbles). The method is validated by comparisons with solutions available in the literature and convergence studies for bubbles in acoustic fields. The accuracy and performance of the algorithm are discussed. It is demonstrated that the approach enables simulation of dynamics of bubble clusters with thousands of bubbles and millions of boundary elements on contemporary personal workstations. The algorithm is scalable and can be extended to larger systems.

physics.comp-ph↗

arXiv2016

Fast Multipole Method based filtering of non-uniformly sampled data

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.

math.NA↗

arXiv2015

Accurate computation of Galerkin double surface integrals in the 3-D boundary element method

Many boundary element integral equation kernels are based on the Green's functions of the Laplace and Helmholtz equations in three dimensions. These include, for example, the Laplace, Helmholtz, elasticity, Stokes, and Maxwell's equations. Integral equation formulations lead to more compact, but dense linear systems. These dense systems are often solved iteratively via Krylov subspace methods, which may be accelerated via the fast multipole method. There are advantages to Galerkin formulations for such integral equations, as they treat problems associated with kernel singularity, and lead to symmetric and better conditioned matrices. However, the Galerkin method requires each entry in the system matrix to be created via the computation of a double surface integral over one or more pairs of triangles. There are a number of semi-analytical methods to treat these integrals, which all have some issues, and are discussed in this paper. We present novel methods to compute all the integrals that arise in Galerkin formulations involving kernels based on the Laplace and Helmholtz Green's functions to any specified accuracy. Integrals involving completely geometrically separated triangles are non-singular and are computed using a technique based on spherical harmonics and multipole expansions and translations, which results in the integration of polynomial functions over the triangles. Integrals involving cases where the triangles have common vertices, edges, or are coincident are treated via scaling and symmetry arguments, combined with automatic recursive geometric decomposition of the integrals. Example results are presented, and the developed software is available as open source.

physics.comp-ph↗

arXiv2014

Preconditioned Krylov solvers for kernel regression

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.

math.NA↗

arXiv2014

Semi-Analytical Computation of Acoustic Scattering by Spheroids and Disks

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.

cs.MS↗

arXiv2014

Software for Computing the Spheroidal Wave Functions Using Arbitrary Precision Arithmetic

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.

cs.MS↗

arXiv2014

A method to compute periodic sums

In a number of problems in computational physics, a finite sum of kernel functions centered at $N$ particle locations located in a box in three dimensions must be extended by imposing periodic boundary conditions on box boundaries. Even though the finite sum can be efficiently computed via fast summation algorithms, such as the fast multipole method (FMM), the periodized extension is usually treated via a different algorithm, Ewald summation, accelerated via the fast Fourier transform (FFT). A different approach to compute this periodized sum just using a blackbox finite fast summation algorithm is presented in this paper. The method splits the periodized sum in to two parts. The first, comprising the contribution of all points outside a large sphere enclosing the box, and some of its neighbors, is approximated inside the box by a collection of kernel functions ("sources") placed on the surface of the sphere or using an expansion in terms of spectrally convergent local basis functions. The second part, comprising the part inside the sphere, and including the box and its immediate neighborhood, is treated via available summation algorithms. The coefficients of the sources are determined by least squares collocation of the periodicity condition of the total potential, imposed on a circumspherical surface for the box. While the method is presented in general, details are worked out for the case of evaluating electrostatic potentials and forces. Results show that when used with the FMM, the periodized sum can be computed to any specified accuracy, at an additional cost of the order of the free-space FMM. Several technical details and efficient algorithms for auxiliary computations are provided, as are numerical comparisons.

physics.comp-ph↗

arXiv2014

Recursive computation of spherical harmonic rotation coefficients of large degree

Computation of the spherical harmonic rotation coefficients or elements of Wigner's d-matrix is important in a number of quantum mechanics and mathematical physics applications. Particularly, this is important for the Fast Multipole Methods in three dimensions for the Helmholtz, Laplace and related equations, if rotation-based decomposition of translation operators are used. In these and related problems related to representation of functions on a sphere via spherical harmonic expansions computation of the rotation coefficients of large degree $n$ (of the order of thousands and more) may be necessary. Existing algorithms for their computation, based on recursions, are usually unstable, and do not extend to $n$. We develop a new recursion and study its behavior for large degrees, via computational and asymptotic analyses. Stability of this recursion was studied based on a novel application of the Courant-Friedrichs-Lewy condition and the von Neumann method for stability of finite-difference schemes for solution of PDEs. A recursive algorithm of minimal complexity $O\left(n^{2}\right)$ for degree $n$ and FFT-based algorithms of complexity $O\left(n^{2}\log n\right) $ suitable for computation of rotation coefficients of large degrees are proposed, studied numerically, and cross-validated. It is shown that the latter algorithm can be used for $n\lesssim 10^{3}$ in double precision, while the former algorithm was tested for large $n$ (up to $10^{4}$ in our experiments) and demonstrated better performance and accuracy compared to the FFT-based algorithm.

math.NA↗

arXiv2013

Performance of a GPU-based Direct Summation Algorithm for Computation of Small Angle Scattering Profile

Small Angle Scattering (SAS) of X-rays or neutrons is an experimental technique that provides valuable structural information for biological macromolecules under physiological conditions and with no limitation on the molecular size. In order to refine molecular structure against experimental SAS data, ab initio prediction of the scattering profile must be recomputed hundreds of thousands of times, which involves the computation of the sinc kernel over all pairs of atoms in a molecule. The quadratic computational complexity of predicting the SAS profile limits the size of the molecules and and has been a major impediment for integration of SAS data into structure refinement protocols. In order to significantly speed up prediction of the SAS profile we present a general purpose graphical processing unit (GPU) algorithm, written in OpenCL, for the summation of the sinc kernel (Debye summation) over all pairs of atoms. This program is an order of magnitude faster than a parallel CPU algorithm, and faster than an FMM-like approximation method for certain input domains. We show that our algorithm is currently the fastest method for performing SAS computation for small and medium size molecules (around 50000 atoms or less). This algorithm is critical for quick and accurate SAS profile computation of elongated structures, such as DNA, RNA, and sparsely spaced pseudo-atom molecules.

q-bio.BM↗

arXiv2013

Parallel Algorithms for Constructing Data Structures for Fast Multipole Methods

We present efficient algorithms to build data structures and the lists needed for fast multipole methods. The algorithms are capable of being efficiently implemented on both serial, data parallel GPU and on distributed architectures. With these algorithms it is possible to map the FMM efficiently on to the GPU or distributed heterogeneous CPU-GPU systems. Further, in dynamic problems, as the distribution of the particles change, the reduced cost of building the data structures improves performance. Using these algorithms, we demonstrate example high fidelity simulations with large problem sizes by using FMM on both single and multiple heterogeneous computing facilities equipped with multi-core CPU and many-core GPUs.

cs.MS↗

arXiv2012

Efficient FMM accelerated vortex methods in three dimensions via the Lamb-Helmholtz decomposition

Vortex element methods are often used to efficiently simulate incompressible flows using Lagrangian techniques. Use of the FMM (Fast Multipole Method) allows considerable speed up of both velocity evaluation and vorticity evolution terms in these methods. Both equations require field evaluation of constrained (divergence free) vector valued quantities (velocity, vorticity) and cross terms from these. These are usually evaluated by performing several FMM accelerated sums of scalar harmonic functions. We present a formulation of the vortex methods based on the Lamb-Helmholtz decomposition of the velocity in terms of two scalar potentials. In its original form, this decomposition is not invariant with respect to translation, violating a key requirement for the FMM. One of the key contributions of this paper is a theory for translation for this representation. The translation theory is developed by introducing "conversion" operators, which enable the representation to be restored in an arbitrary reference frame. Using this form, extremely efficient vortex element computations can be made, which need evaluation of just two scalar harmonic FMM sums for evaluating the velocity and vorticity evolution terms. Details of the decomposition, translation and conversion formulae, and sample numerical results are presented.

physics.comp-ph↗