Kernel Independent Fast Multipole Method Research Articles

On accuracy of translations by kernel independent fast multipole methods

The paper concerns with accurate evaluation of local fields in problems, like those of fracture mechanics, for which extreme, rather than mean values, are of prime significance. We focus on large-scale problems, solved iteratively by a boundary element method (BEM) with employing the kernel independent fast multipole method (KI-FMM) for decreasing the complexity of matrix-to-vector multiplications. We aim to study the accuracy of individual translations far-fields for the major types of kernels (weakly singular, singular and hypersingular) entering the boundary integral equations (BIE) of harmonic and elasticity problems in 2D and 3D. It is revealed, analytically and numerically, that (i) the accuracy of individual translations of fields, generated by sources involving the normal at a source point, is notably less than that by sources independent on the normal; (ii) the accuracy is notably less for a hypersingular BIE, then for a singular BIE; and (iii) under the same complexity, the accuracy of translations, performed by using smooth (circular in 2D, spherical in 3D) equivalent surfaces (ES), is notably better than that when using non-smooth (square in 2D, cubic in 3D) ES. These conclusions yield recommendations for minimizing the complexity without loss of the accuracy of translated fields.

Kernel-Independent Fast Multipole Boundary Element Solver for Coupled Conduction–Radiation Heat Transfer Problem

The purpose of this paper is to present a kernel-independent fast multipole accelerated boundary element method for the simulation of a system of coupled conduction–radiation heat transfer in three-dimensional participating media. Radiative transfer integral equations are considered as radiation heat transfer model in participating media. For the nonlinear energy equation, Newton–Generalized Minimal Residual (GMRES) iterative scheme is adopted. The loose coupling iterative technique is applied to this strongly coupled system. Then, the separate solutions are both obtained by the kernel-independent fast multipole boundary element method based on the collocation scheme. Numerical examples show that the present fast algorithm is effective and efficient.

Kernel aggregated fast multipole method

Many different simulation methods for Stokes flow problems involve a common computationally intense task -- the summation of a kernel function over $O(N^2)$ pairs of points. One popular technique is the Kernel Independent Fast Multipole Method (KIFMM), which constructs a spatial adaptive octree for all points and places a small number of equivalent multipole and local equivalent points around each octree box, and completes the kernel sum with $O(N)$ cost, using these equivalent points. Simpler kernels can be used between these equivalent points to improve the efficiency of KIFMM. Here we present further extensions and applications to this idea, to enable efficient summations and flexible boundary conditions for various kernels. We call our method the Kernel Aggregated Fast Multipole Method (KAFMM), because it uses different kernel functions at different stages of octree traversal. We have implemented our method as an open-source software library STKFMM based on the high performance library PVFMM, with support for Laplace kernels, the Stokeslet, regularized Stokeslet, Rotne-Prager-Yamakawa (RPY) tensor, and the Stokes double-layer and traction operators. Open and periodic boundary conditions are supported for all kernels, and the no-slip wall boundary condition is supported for the Stokeslet and RPY tensor. The package is designed to be ready-to-use as well as being readily extensible to additional kernels.

Accurate and efficient calculation of photoionization in streamer discharges using fast multipole method

This paper focuses on the three-dimensional simulation of the photoionization in streamer discharges, and provides a general framework to efficiently and accurately calculate the photoionization model using the integral form. The simulation is based on the kernel-independent fast multipole method (FMM). The accuracy of this method is studied quantitatively for different domains and various pressures in comparison with other existing models based on partial differential equations (PDEs). The comparison indicates the numerical error of the FMM is much smaller than those of other PDE-based methods, with the reference solution given by direct numerical integration. Such accuracy can be achieved with affordable computational cost, and its performance in both efficiency and accuracy is quite stable for different domains and pressures. Meanwhile, the simulation accelerated by the FMM exhibits good scalability using up to 1280 cores, which shows its capability of three-dimensional simulations using parallel (distributed) computing. The difference of the proposed method and other efficient approximations are also studied in a three-dimensional dynamic problem where two streamers interact.

Solution Optimization of RBF Interpolation for Implicit Modeling of Orebody

To interpolate large drillhole data sets efficiently in implicit modeling of orebody, we optimize the solution of the RBF equation based on the two-level domain decomposition method. The solution performance is improved by balancing the two goals of the convergence rate and the iteration efficiency. The solution method converts the large domain into small subdomains which can be solved directly. The optimum division of subdomains can improve both the convergence of iteration and the efficiency of a single iteration, whereas the subdivision of a trivial subdomain leads the iteration failed to converge. For the efficiency of a single iteration, the kernel independent fast multipole method is used to calculate the matrix-vector product efficiently. Moreover, the initial solution of iteration and center reduction strategy are optimized for the dynamic updating of implicit surface. Combined with the preconditioned Krylov subspace method, the optimized solution method was implemented. The experimental results of several drillhole data sets show that the optimized solution method converges rapidly within a small number of iterations.

A fast directional boundary element method for wideband multi-domain elastodynamic analysis

A fast directional boundary element method for wideband multi-domain elastodynamic analysis is developed. The directional low rank property of the elastodynamic kernel is proved, which serves as the theoretical basis of the fast directional algorithm. The boundary integral evaluation is accelerated domain by domain. For each domain, the octree is divided into low frequency and high frequency regimes according to its S-wave number. The computation in the low frequency regime is the same with the kernel independent fast multipole method, while the high frequency regime is computed efficiently by applying the parabolic low rank property of the kernels on directional wedges. The matrices on only one directional wedge for each level have to be calculated, since that for other directions can be obtained by coordinate frame rotations. Thus the harmonic responses for any frequencies can be computed efficiently. Numerical results show that the computational complexity for wideband multi-domain elastodynamic problems is successfully brought down to O(NlogαN). Its applications to viscoelastic materials and inverse problems are also validated.

Complex dynamics of long, flexible fibers in shear

The macroscopic properties of polymeric fluids are inherited from the material properties of the fibers embedded in the solvent. The behavior of such passive fibers in flow has been of interest in a wide range of systems, including cellular mechanics, nutrient acquisition by diatom chains in the ocean, and industrial applications such as paper manufacturing. The rotational dynamics and shape evolution of fibers in shear depends upon the slenderness of the fiber and the non-dimensional “elasto-viscous” number that measures the ratio of the fluid’s viscous forces to the fiber’s elastic forces. For a small elasto-viscous number, the nearly-rigid fiber rotates in the shear, but when the elasto-viscous number reaches a threshold, buckling occurs. For even larger elasto-viscous numbers, there is a transition to a “snaking behavior” where the fiber remains aligned with the shear axis, but its ends curl in, in opposite directions. These experimentally-observed behaviors have recently been characterized computationally using slender-body theory and immersed boundary computations. However, classical experiments with nylon fibers and recent experiments with actin filaments have demonstrated that for even larger elasto-viscous numbers, multiple buckling sites and coiling can occur. Using a regularized Stokeslet framework coupled with a kernel independent fast multipole method, we present simulations that capture these complex fiber dynamics.

A Dual-Level Method of Fundamental Solutions in Conjunction with Kernel-Independent Fast Multipole Method for Large-Scale Isotropic Heat Conduction Problems

A Dual-Level Method of Fundamental Solutions in Conjunction with Kernel-Independent Fast Multipole Method for Large-Scale Isotropic Heat Conduction Problems

Fast algorithms for large dense matrices with applications to biofluids

Numerical simulation of biofluids entails solving equations of fluid-structure interactions. At zero Reynolds number, solvers such as the Method of Regularized Stokeslets (MRS) give rise to large and dense matrices in practical applications where the number of structures immersed in the fluid is large. Building on previous work for an unbounded fluid domain, we first extend the Kernel-Independent Fast Multipole Method (KIFMM) for MRS to compute the matrix-vector products for the fluid flow induced by point forces above a stationary wall. In this case, the use of a regularized image system introduces additional terms to the solution which cause the matrix-vector multiplication to be quite challenging. In addition, we study the case where a linear system needs to be solved for the unknown forces that structures with known velocities exert on the fluid. Our main contribution is proposing several preconditioning techniques for the matrices associated with a few variants of MRS, including the case where a force-free, torque-free condition is imposed. They take advantage of the data-sparsity of FMM matrices as well as properties of Krylov subspaces. Our approach is memory efficient, capable of handling non-uniformly distributed structures and applicable to all FMM matrices. It enables efficient computation of the flow field surrounding a large group of dynamic micro-structures; in particular, we study the effects of fluid mixing caused by the periodic beating of a dense carpet of lung cilia.

A fast multipole algorithm for radiative heat transfer in 3D semitransparent media

This paper presents a fast multipole accelerated boundary element method for simulating the radiative heat transfer in 3D semitransparent media. The participating media is absorbing, emitting and isotropically scattering. The proposed implementation adopts the boundary element method based on the collocation scheme to discretize the radiative integral transfer equations. The resulting algebraic system is solved iteratively with GMRES (generalised minimum residual method). A kernel-independent fast multipole method to accelerate the matrix-vector product in each iteration. It is found from three numerical examples that the computational efficiency of the developed fast algorithm is much better than the conventional boundary element method.

Universal image systems for non-periodic and periodic Stokes flows above a no-slip wall

It is well-known that by placing judiciously chosen image point forces and doublets to the Stokeslet above a flat wall, the no-slip boundary condition can be conveniently imposed on the wall Blake (1971) [8]. However, to further impose periodic boundary conditions on directions parallel to the wall usually involves tedious derivations because single or double periodicity in Stokes flow may require the periodic unit to have no net force, which is not satisfied by the well-known image system. In this work we present a force-neutral image system. This neutrality allows us to represent the Stokes image system in a universal formulation for non-periodic, singly periodic and doubly periodic geometries. This formulation enables the black-box style usage of fast kernel summation methods. We demonstrate the efficiency and accuracy of this new image method with the periodic kernel independent fast multipole method in both non-periodic and periodic geometries. We then extend this new image system to other widely used Stokes fundamental solutions, including the Laplacian of the Stokeslet and the Rotne–Prager–Yamakawa tensor.

A modified dual-level algorithm for large-scale three-dimensional Laplace and Helmholtz equation

A modified dual-level algorithm is proposed in the article. By the help of the dual level structure, the fully-populated interpolation matrix on the fine level is transformed to a local supported sparse matrix to solve the highly ill-conditioning and excessive storage requirement resulting from fully-populated interpolation matrix. The kernel-independent fast multipole method is adopted to expediting the solving process of the linear equations on the coarse level. Numerical experiments up to 2-million fine-level nodes have successfully been achieved. It is noted that the proposed algorithm merely needs to place 2–3 coarse-level nodes in each wavelength per direction to obtain the reasonable solution, which almost down to the minimum requirement allowed by the Shannon’s sampling theorem. In the real human head model example, it is observed that the proposed algorithm can simulate well computationally very challenging exterior high-frequency harmonic acoustic wave propagation up to 20,000 Hz.

Flexibly imposing periodicity in kernel independent FMM: A multipole-to-local operator approach

An important but missing component in the application of the kernel independent fast multipole method (KIFMM) is the capability for flexibly and efficiently imposing singly, doubly, and triply periodic boundary conditions. In most popular packages such periodicities are imposed with the hierarchical repetition of periodic boxes, which may give an incorrect answer due to the conditional convergence of some kernel sums. Here we present an efficient method to properly impose periodic boundary conditions using a near-far splitting scheme. The near-field contribution is directly calculated with the KIFMM method, while the far-field contribution is calculated with a multipole-to-local (M2L) operator which is independent of the source and target point distribution. The M2L operator is constructed with the far-field portion of the kernel function to generate the far-field contribution with the downward equivalent source points in KIFMM. This method guarantees the sum of the near-field & far-field converge pointwise to results satisfying periodicity and compatibility conditions. The computational cost of the far-field calculation observes the same O(N) complexity as FMM and is designed to be small by reusing the data computed by KIFMM for the near-field. The far-field calculations require no additional control parameters, and observes the same theoretical error bound as KIFMM. We present accuracy and timing test results for the Laplace kernel in singly periodic domains and the Stokes velocity kernel in doubly and triply periodic domains.

Scalable topology optimization with the kernel-independent fast multipole method

The paper presents a new method for shape and topology optimization based on an efficient and scalable boundary integral formulation for elasticity. To optimize topology, our approach uses iterative extraction of isosurfaces of a topological derivative. The numerical solution of the elasticity boundary value problem at every iteration is performed with the boundary element formulation and the kernel-independent fast multipole method. Providing excellent single node performance and scalable parallelization, our method is among the fastest optimization tools available today. The performance of our approach is studied on few illustrative examples, including the optimization of engineered constructions for the minimum compliance and the optimization of the microstructure of a metamaterial for the desired macroscopic tensor of elasticity.

On solving continuum-mechanics problems by fast multipole methods

It is shown analytically and numerically that the kernel-independent fast multipole method provides the accuracy and computational complexity of the analytical method if circular (spherical in 3D) equivalent surfaces are used.

Parallel optimization with boundary elements and kernel independent fast multipole method

Parallel optimization with boundary elements and kernel independent fast multipole method

Kernel-independent fast multipole method within the framework of regularized Stokeslets

The method of regularized Stokeslets (MRS) uses a radially symmetric blob function of infinite support to smooth point forces and allows for evaluation of the resulting flow field. This is a common method to study swimmers at zero Reynolds number where the Stokeslet is the fundamental solution corresponding to the kernel of the single layer potential. Simulating the collective motion of N micro-swimmers using the MRS results in at least N2 pair-wise interactions. Efficient simulation of a large number of swimmers in free space is observed with the implementation of the kernel-independent fast multipole method (FMM) for radial basis functions. We illustrate the complexity of the algorithm on a simple test case where we study regularized point forces, showing that the method is of order N. Additionally, we explore accuracy in time for the MRS where the swimmers are modeled as Kirchhoff rods and the kernel-independent FMM is compared to the direct calculation using the standard MRS. Optimal hydrodynamic efficiency is also explored for different configurations of swimmers.

Algorithm 967

The solution of a constant-coefficient elliptic Partial Differential Equation (PDE) can be computed using an integral transform: A convolution with the fundamental solution of the PDE, also known as a volume potential. We present a Fast Multipole Method (FMM) for computing volume potentials and use them to construct spatially adaptive solvers for the Poisson, Stokes, and low-frequency Helmholtz problems. Conventional N-body methods apply to discrete particle interactions. With volume potentials, one replaces the sums with volume integrals. Particle N-body methods can be used to accelerate such integrals. but it is more efficient to develop a special FMM. In this article, we discuss the efficient implementation of such an FMM. We use high-order piecewise Chebyshev polynomials and an octree data structure to represent the input and output fields and enable spectrally accurate approximation of the near-field and the Kernel Independent FMM (KIFMM) for the far-field approximation. For distributed-memory parallelism, we use space-filling curves, locally essential trees, and a hypercube-like communication scheme developed previously in our group. We present new near and far interaction traversals that optimize cache usage. Also, unlike particle N-body codes, we need a 2:1 balanced tree to allow for precomputations. We present a fast scheme for 2:1 balancing. Finally, we use vectorization, including the AVX instruction set on the Intel Sandy Bridge architecture to get better than 50% of peak floating-point performance. We use task parallelism to employ the Xeon Phi on the Stampede platform at the Texas Advanced Computing Center (TACC). We achieve about 600 gflop /s of double-precision performance on a single node. Our largest run on Stampede took 3.5s on 16K cores for a problem with 18 e +9 unknowns for a highly nonuniform particle distribution (corresponding to an effective resolution exceeding 3 e +23 unknowns since we used 23 levels in our octree).

An O(N) and parallel approach to integral problems by a kernel-independent fast multipole method: Application to polarization and magnetization of interacting particles

Large classes of materials systems in physics and engineering are governed by magnetic and electrostatic interactions. Continuum or mesoscale descriptions of such systems can be cast in terms of integral equations, whose direct computational evaluation requires O(N2) operations, where N is the number of unknowns. Such a scaling, which arises from the many-body nature of the relevant Green’s function, has precluded wide-spread adoption of integral methods for solution of large-scale scientific and engineering problems. In this work, a parallel computational approach is presented that relies on using scalable open source libraries and utilizes a kernel-independent Fast Multipole Method (FMM) to evaluate the integrals in O(N) operations, with O(N) memory cost, thereby substantially improving the scalability and efficiency of computational integral methods. We demonstrate the accuracy, efficiency, and scalability of our approach in the context of two examples. In the first, we solve a boundary value problem for a ferroelectric/ferromagnetic volume in free space. In the second, we solve an electrostatic problem involving polarizable dielectric bodies in an unbounded dielectric medium. The results from these test cases show that our proposed parallel approach, which is built on a kernel-independent FMM, can enable highly efficient and accurate simulations and allow for considerable flexibility in a broad range of applications.

A kernel-independent fast multipole BEM for large-scale elastodynamic analysis

Purpose – The purpose of this paper is to develop an easy-to-implement and accurate fast boundary element method (BEM) for solving large-scale elastodynamic problems in frequency and time domains. Design/methodology/approach – A newly developed kernel-independent fast multipole method (KIFMM) is applied to accelerating the evaluation of displacements, strains and stresses in frequency domain elastodynamic BEM analysis, in which the far-field interactions are evaluated efficiently utilizing equivalent densities and check potentials. Although there are six boundary integrals with unique kernel functions, by using the elastic theory, the authors managed to accelerate these six boundary integrals by KIFMM with the same kind of equivalent densities and check potentials. The boundary integral equations are discretized by Nyström method with curved quadratic elements. The method is further used to conduct the time-domain analysis by using the frequency-domain approach. Findings – Numerical results show that by the fast BEM, high accuracy can be achieved and the computational complexity is brought down to linear. The performance of the present method is further demonstrated by large-scale simulations with more than two millions of unknowns in the frequency domain and one million of unknowns in the time domain. Besides, the method is applied to the topological derivatives for solving elastodynamic inverse problems. Originality/value – An efficient KIFMM is implemented in the acceleration of the elastodynamic BEM. Combining with the Nyström discretization based on quadratic elements and the frequency-domain approach, an accurate and highly efficient fast BEM is achieved for large-scale elastodynamic frequency domain analysis and time-domain analysis.