Treecode Algorithm Research Articles

A treecode algorithm for the Poisson equation in a general domain with unstructured grids

A treecode algorithm for the Poisson equation in a general domain with unstructured grids

Implementing the spherical grids and treecode algorithm for efficient parallelization of long-range electrostatic interactions

Implementing the spherical grids and treecode algorithm for efficient parallelization of long-range electrostatic interactions

A treecode algorithm based on tricubic interpolation

Treecode algorithms efficiently approximate N-body interactions in O(N) or O(NlogN). In order to treat general 3D kernels, recent developments employ polynomial interpolation to approximate the kernels. The polynomials are a tensor product of 1-dimensional polynomials. Here, we develop an O(NlogN) tricubic interpolation based treecode method for 3D kernels. The tricubic interpolation is inherently three-dimensional and as such does not employ a tensor product. The form allows for easy evaluation of the derivatives of the kernel, required in dynamical simulations, which is not the case for the tensor product approach. We develop both a particle-cluster and cluster-particle variants and present results for the Coulomb, screened Coulomb and the real space Ewald kernels. We also present results of an MD simulation of a Lennard-Jones liquid using the tricubic treecode.

Major merging of galaxies in multicomponent numerical models: mass loss and exchange

The process of collision of two multicomponent galaxies is considered in detail based on numerical simulations of the dynamics of gravitating gas, stars and dark mass. To solve the equations of motion of the gas component, we use the Smoothed Particle Hydrodynamics method. Modeling of collisionless components is based on the N-body model. The computations of gravitational forces are carried out using both the approximate hierarchical TreeCode algorithm and the direct method of summing the gravitational contribution from all particles, which provides an accurate solution. This approach allows testing various models and evaluating the resulting errors associated with the calculation of gravitational forces and a finite number of particles in each of the components. Both methods for calculating gravity are software implemented as parallel codes for Nvidia Tesla GPUs. The estimates of the lost mass and the efficiency of matter exchange between galaxies are discussed depending on the model parameters.

Cyclically parallelized treecode for fast computations of electrostatic interactions on molecular surfaces

We study the parallelization of a flexible order Cartesian treecode algorithm for evaluating electrostatic potentials of charged particle systems in which N particles are located on the molecular surfaces of biomolecules such as proteins. When the well-separated condition is satisfied, the treecode algorithm uses a far-field Taylor expansion to compute O(NlogN) particle–cluster interactions to replace the O(N2) particle–particle interactions. The algorithm is implemented using the Message Passing Interface (MPI) standard by creating identical tree structures in the memory of each task for concurrent computing. We design a cyclic order scheme to uniformly distribute spatially-closed target particles to all available tasks, which significantly improves parallel load balancing. We also investigate the parallel efficiency subject to treecode parameters such as Taylor expansion order p, maximum particles per leaf N0, and maximum acceptance criterion θ. This cyclically parallelized treecode can solve interactions among up to tens of millions of particles. However, if the problem size exceeds the memory limit of each task, a scalable domain decomposition (DD) parallelized treecode using an orthogonal recursive bisection (ORB) tree can be used instead In addition to efficiently computing the N-body problem of charged particles, our approach can potentially accelerate GMRES iterations for solving the boundary integral Poisson–Boltzmann equation.

A Fast Algorithm for Time-Dependent Radiative Transport Equation Based on Integral Formulation

In this work, we introduce a fast numerical algorithm to solve the time-dependent radiative transport equation (RTE). Our method uses the integral formulation of RTE and applies the treecode algorithm to reduce the computational complexity from O(M^{2+1/d} ) to O(M^{1+1/d} log M ), where M is the number of points in the physical domain. The error analysis is presented and numerical experiments are performed to validate our algorithm.

A Treecode Algorithm for 3D Stokeslets and Stresslets

The Stokeslet and stresslet kernels are commonly used in boundary element simulations and singularity methods for slow viscous flow. Evaluating the velocity induced by a collection of Stokeslets and stresslets by direct summation requires $O(N^2)$ operations, where $N$ is the system size. The present work develops a treecode algorithm for 3D Stokeslets and stresslets that reduces the cost to $O(N\log N)$. The particles are divided into a hierarchy of clusters, and well-separated particle-cluster interactions are computed by a far-field Cartesian Taylor approximation. The terms in the approximation are contracted to promote efficient computation. Serial and parallel results display the treecode performance for several test cases. In particular the method has relatively simple structure and low memory usage, and this enhances parallel efficiency for large systems.

Key generation for plain text in stream cipher via bi-objective evolutionary computing

Evolutionary algorithms are widely used to solve a wide variety of continuous, discrete and combinatorial optimization problems. Evolutionary multi-objective optimization problems seek Pareto front in order to negotiate the trade-off amongst various objective functions present in the problem. Much of the literature on cryptography focuses on making the inference problem harder, for securing the content. In this paper, we developed key generation algorithms using Non-Dominated Sorting Genetic Algorithm-II (NSGA-II) in the bi-objective optimization framework and Improved Modified Harmony Search + Differential Evolution (IMHS+DE), Differential Evolution (DE) and Improved Modified Harmony Search (IMHS), in the single objective optimization framework. For encoding the keystream thus generated as well as the plain text we employed the Mutated Huffman Tree Coding algorithm. In the next phase, we encrypted the encoded keystream as well as the encoded plain text in order to generate the cipher text. We then decrypted the cipher text using the encoded key stream followed by the decoding of the deciphered text using the code tables. Following the literature, we generated random texts of varying lengths and code table sizes in order to demonstrate the effectiveness of our proposed method. The proposed algorithms are compared with the extant methods. In the case of bi-objective optimization set up, we also plotted the empirical attainment function (EAF) surface to summarize the effectiveness of the NSGA-II based key generation algorithm. Of particular significance is the highest entropy value yielded by the NSGA-II based algorithm, which in turn indicates the strength of the key generated by the NSGA-II.

Biomolecular Simulations in a Continuum Ionic Solvent with Polarizable Force Fields, Using Python and GPUs

In molecular simulations, implicit-solvent models use continuum electrostatic theory to average the solvent degrees of freedom, drastically decreasing the amount of computations compared with explicit molecular dynamics. Mathematically, it can be formulated as a coupled system of the Poisson and Poisson-Boltzmann equations, interfaced by the biomolecule's solvent-excluded surface. Usually, these models use point-charge-based (classical) force fields to parameterize the biomolecule of interest, yet, more elaborate force fields that consider higher order multipoles and polarizability have emerged lately. Polarizable force fields generate more realistic charge distributions, but involve larger computations than a classical force field. One popular polarizable force field is AMOEBA, which uses point multipoles up to the quadrupole order to describe the charge distribution, and allows for the dipole component to polarize. In this work, we present an extension of the Poisson-Boltzmann solver PyGBe, which couples with the AMOEBA force field by solving iteratively for a self-consistent reaction potential. PyGBe is a boundary element method code, accelerated with a treecode algorithm and GPUs, that works from a Python interface. The boundary integral formulation treats the point multipoles analytically, which is an advantage over volumetric-based methods where very fine meshes are required to effectively place the point multipoles onto the numerical mesh. We will show results verifying our implementation in comparison with analytical solutions, available for spherical molecules, and computing the solvation energy of larger test proteins.

A treecode to simulate dust–plasma interactions**Substantial portions of this paper are adapted from chapters 4 and 5 of the first author’s recent PhD dissertation.

The interaction of a small object with surrounding plasma is an area of plasma-physics research with a multitude of applications. This paper introduces the plasma octree code pot, a microscopic simulator of a spheroidal dust grain in a plasma. pot uses the Barnes–Hut treecode algorithm to perform N-body simulations of electrons and ions in the vicinity of a chargeable spheroid, employing also the Boris particle-motion integrator and Hutchinson’s reinjection algorithm from SCEPTIC; a description of the implementation of all three algorithms is provided. We present results from pot simulations of the charging of spheres in magnetised plasmas, and of spheroids in unmagnetized plasmas. The results call into question the validity of using the Boltzmann relation in hybrid PIC codes.

A GPU accelerated Barnes–Hut tree code for FLASH4

We present a GPU accelerated CUDA-C implementation of the Barnes Hut (BH) tree code for calculating the gravitational potential on octree adaptive meshes. The tree code algorithm is implemented within the FLASH4 adaptive mesh refinement (AMR) code framework and therefore fully MPI parallel. We describe the algorithm and present test results that demonstrate its accuracy and performance in comparison to the algorithms available in the current FLASH4 version. We use a MacLaurin spheroid to test the accuracy of our new implementation and use spherical, collapsing cloud cores with effective AMR to carry out performance tests also in comparison with previous gravity solvers. Depending on the setup and the GPU/CPU ratio, we find a speedup for the gravity unit of at least a factor of 3 and up to 60 in comparison to the gravity solvers implemented in the FLASH4 code. We find an overall speedup factor for full simulations of at least factor 1.6 up to a factor of 10.

FrWF-Based LMBTC: Memory-Efficient Image Coding for Visual Sensors

After the successful development of JPEG2000, many state-of-the-art wavelet-based image coding algorithms have been developed. However, the traditional discrete wavelet transform (DWT) is implemented with memory intensive and time-consuming algorithms and, therefore, has very high system resource requirements. In particular, the very large requirement of memory poses a serious limitation for multimedia applications on memory-constrained portable devices, such as digital cameras and sensor nodes. In this paper, we propose a novel wavelet-based image coder with low memory requirements and low complexity that preserves the compression efficiency. Our encoder employs the fractional wavelet filter (FrWF) to calculate the DWT coefficients, which are quantized and encoded with a novel low memory block tree coding (LMBTC) algorithm. The LMBTC is a listless form of the wavelet block tree coding algorithm. Simulation results demonstrate that the proposed coder significantly reduces memory requirements and computational complexity and has competitive coding efficiency in comparison with other state-of-the-art coders. The FrWF combined with the LMBTC is, thus, a viable option for image communication over wireless sensor networks.

Enhanced Listless Block Tree Coding with Discrete Wavelet Transform for Image Compression

Set Partitioning in Hierarchal Trees (SPIHT) is an efficient method for compressing images under low bit rates. No List SPIHT (NLS) and Wavelet Based Block Tree Coding (WBTC) are two enhanced algorithms of SPIHT. The WBTC algorithm works on blocks instead of pixels in SPIHT. The size of root block in WBTC varies from one step to another. This reduces the memory requirement to a great extent. NLS uses markers instead of lists used for the storage of coefficients in SPIHT. The three lists used in SPIHT to manage the significant coefficients grow exponentially with each step as more number of coefficients is tracked. Due to this feature SPIHT algorithm requires a lot of memory management and hence it is complex for hardware implementation. But the 8 different markers used in NLS removes this drawback of original algorithm. Listless Block Tree Coding algorithm (LBTC) is evolved by combining the WBTC and NLS algorithms. In this algorithm image compression is performed on the block basis and the significant coefficients are tracked with the help of different markers. The LBTC algorithm when combined with Discrete Wavelet Transform (DWT) performs even well in the terms of Peak Signal to Noise Ratio (PSNR) and Mean Square Error (MSE). In this paper arithmetic encoding is applied on the LBTC-DWT algorithm which further enhances the compressed image quality in terms of PSNR and MSE though the time taken increases.

Efficient implementation of the Barnes-Hut octree algorithm for Monte Carlo simulations of charged systems

Computer simulation with Monte Carlo is an important tool to investigate the function and equilibrium properties of many systems with biological and soft matter materials solvable in solvents. The appropriate treatment of long-range electrostatic interaction is essential for these charged systems, but remains a challenging problem for large-scale simulations. We have developed an efficient Barnes-Hut treecode algorithm for electrostatic evaluation in Monte Carlo simulations of Coulomb many-body systems. The algorithm is based on a divide-and-conquer strategy and fast update of the octree data structure in each trial move through a local adjustment procedure. We test the accuracy of the tree algorithm, and use it to perform computer simulations of electric double layer near a spherical interface. It has been shown that the computational cost of the Monte Carlo method with treecode acceleration scales as $\log N$ in each move. For a typical system with ten thousand particles, by using the new algorithm, the speed has been improved by two orders of magnitude from the direct summation.

A Parallel Adaptive Treecode Algorithm for Evolution of Elastically Stressed Solids

AbstractThe evolution of precipitates in stressed solids is modeled by coupling a quasi-steady diffusion equation and a linear elasticity equation with dynamic boundary conditions. The governing equations are solved numerically using a boundary integral method (BIM). A critical step in applying BIM is to develop fast algorithms to reduce the arithmetic operation count of matrix-vector multiplications. In this paper, we develop a fast adaptive treecode algorithm for the diffusion and elasticity problems in two dimensions (2D). We present a novel source dividing strategy to parallelize the treecode. Numerical results show that the speedup factor is nearly perfect up to a moderate number of processors. This approach of parallelization can be readily implemented in other treecodes using either uniform or non-uniform point distribution. We demonstrate the effectiveness of the treecode by computing the long-time evolution of a complicated microstructure in elastic media, which would be extremely difficult with a direct summation method due to CPU time constraint. The treecode speeds up computations dramatically while fulfilling the stringent precision requirement dictated by the spectrally accurate BIM.

A Fast Treecode for Multiquadric Interpolation with Varying Shape Parameters

A treecode algorithm is presented for the fast evaluation of multiquadric radial basis function (RBF) approximations. The method is a dual approach to one presented by Krasny and Wang, which applies far-field expansions to clusters of RBF centers (source points). The new approach clusters evaluation points instead and is therefore easily able to cope with basis functions that have different multiquadric shape parameters. The new treecode is able to evaluate an approximation on N centers at M points in $O((N+M) \log M)$ time in the ideal case when evaluation points are uniformly distributed. When coupled with a two-level restricted additive Schwarz preconditioner for GMRES iterations, the treecode is well suited for use within an adaptive RBF iteration, previously described by Driscoll and Heryudono, as is demonstrated by experiments on test functions.

A sparse octree gravitational N-body code that runs entirely on the GPU processor

We present the implementation and performance of a new gravitational N-body tree-code that is specifically designed for the graphics processing unit (GPU).1The code is publicly available at: http://castle.strw.leidenuniv.nl/software.html.1 All parts of the tree-code algorithm are executed on the GPU. We present algorithms for parallel construction and traversing of sparse octrees. These algorithms are implemented in CUDA and tested on NVIDIA GPUs, but they are portable to OpenCL and can easily be used on many-core devices from other manufacturers. This portability is achieved by using general parallel-scan and sort methods. The gravitational tree-code outperforms tuned CPU code during the tree-construction and shows a performance improvement of more than a factor 20 overall, resulting in a processing rate of more than 2.8 million particles per second.

Comparisons of different codes for galacticN-body simulations

Context. N-body simulations are widely used for galactic dynamics studies. Several different algorithms have been developed and a number of authors have made their codes public. Aims. We wish to help potential users of these codes, particularly less experienced ones, to select the appropriate code by giving them relevant information. Methods. We consider different implementations of three different algorithms, namely the Barnes-Hut tree-code algorithm (implemented both in GADGET-2 and as a NEMO program), the fast multipole method (implemented in the FMB code – Fast Multipole with BLAS) and W. Dehnen’s algorithm (implemented in falcON – force algorithm with complexity O(N)). We compare extensively the timing and memory comparisons of these different implementations and discuss other advantages and disadvantages of these codes. Results. In terms of serial executions, falcON is clearly the fastest code, in many cases by a wide margin. In addition, it is much less sensitive to concentration than other codes, it conserves linear momentum, its computational costs depends roughly linearly on the number of particles, and it allows for manipulators. GADGET-2 is the second fastest and, being parallel, can outperform falcON if as ufficient number of processors is used, the number of which, in our tests is between three and ten. Its public version includes an SPH representation of the gas component. FMB is faster than the implementation of the Barnes-Hut algorithm in NEMO, as long as the particle distribution is not too concentrated, but remains considerably slower than falcON and GADGET-2. It has, nevertheless, the highest parallel efficiency and memory scalability. Conclusions. We recommend falcON for pure N-body simulations with a relatively restricted number of particles (up to a few million). In cases of a larger number of particles, a number of processors being available, or studies including gas and its physics, we recommend GADGET-2. FMB provides the highest parallel efficiency and memory scalability, but is considerably slower in execution time and is ill-adapted to the concentrated distributions encountered in galactic work.

Adaptive panel representation for oblique collision of two vortex rings

In this article, we use a hierarchical panel method for representing vortex sheet surface motion in 3D flow to investigate the oblique collision of two vortex rings. The particles representing the sheet are advected by a regularized Biot–Savart integral with smoothed Rosenhead–Moore kernel. The particle velocities are evaluated by an adaptive treecode algorithm based on Taylor expansions in Cartesian coordinates. The method allowed us to consider late stages of a vortex rings collision, producing a funnel region. Vorticity iso-surfaces evolution is also investigated.

Illuminating hot Jupiters in caustic crossing

In recent years a large number of Hot Jupiters orbiting in a very close orbit around the parent stars have been explored with the transit and doppler effect methods. Here in this work we study the gravitational microlensing effect of a binary lens on a parent star with a Hot Jupiter revolving around it. Caustic crossing of the planet makes enhancements on the light curve of the parent star in which the signature of the planet can be detected by high precision photometric observations. We use the inverse ray shooting method with tree code algorithm to generate the combined light curve of the parent star and the planet. In order to investigate the probability of observing the planet signal, we do a Monte-Carlo simulation and obtain the observational optical depth of $\tau \sim 10^{-8}$. We show that about ten years observations of Galactic Bulge with a network of telescopes will enable us detecting about ten Hot Jupiter with this method. Finally we show that the observation of the microlensing event in infra-red band will increase the probability for detection of the exo-planets.