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  • 1.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Saffar Shamshirgar, Davoud
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Fast Ewald summation for free-space Stokes potentials2017In: Research in the Mathematical Sciences, ISSN 2197-9847, Vol. 4, no 1Article in journal (Refereed)
    Abstract [en]

    We present a spectrally accurate method for the rapid evaluation of free-space Stokes potentials, i.e., sums involving a large number of free space Green’s functions. We consider sums involving stokeslets, stresslets and rotlets that appear in boundary integral methods and potential methods for solving Stokes equations. The method combines the framework of the Spectral Ewald method for periodic problems (Lindbo and Tornberg in J Comput Phys 229(23):8994–9010, 2010. doi: 10.1016/j.jcp.2010.08.026 ), with a very recent approach to solving the free-space harmonic and biharmonic equations using fast Fourier transforms (FFTs) on a uniform grid (Vico et al. in J Comput Phys 323:191–203, 2016. doi: 10.1016/j.jcp.2016.07.028 ). Convolution with a truncated Gaussian function is used to place point sources on a grid. With precomputation of a scalar grid quantity that does not depend on these sources, the amount of oversampling of the grids with Gaussians can be kept at a factor of two, the minimum for aperiodic convolutions by FFTs. The resulting algorithm has a computational complexity of $$O(N \log N)$$ O ( N log N ) for problems with N sources and targets. Comparison is made with a fast multipole method to show that the performance of the new method is competitive.

  • 2.
    Saffar Shamshirgar, Davood
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Fast methods for electrostatic calculations in molecular dynamics simulations2018Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis deals with fast and efficient methods for electrostatic calculations with application in molecular dynamics simulations. The electrostatic calculations are often the most expensive part of MD simulations of charged particles. Therefore, fast and efficient algorithms are required to accelerate these calculations. In this thesis, two types of methods have been considered: FFT-based methods and fast multipole methods (FMM).

    The major part of this thesis deals with fast N.log(N) and spectrally accurate methods for accelerating the computation of pairwise interactions with arbitrary periodicity. These methods are based on the Ewald decomposition and have been previously introduced for triply and doubly periodic problems under the name of Spectral Ewald (SE) method. We extend the method for problems with singly periodic boundary conditions, in which one of three dimensions is periodic. By introducing an adaptive fast Fourier transform, we reduce the cost of upsampling in the non periodic directions and show that the total cost of computation is comparable with the triply periodic counterpart. Using an FFT-based technique for solving free-space harmonic problems, we are able to unify the treatment of zero and nonzero Fourier modes for the doubly and singly periodic problems. Applying the same technique, we extend the SE method for cases with free-space boundary conditions, i.e. without any periodicity.

    This thesis is also concerned with the fast multipole method (FMM) for electrostatic calculations. The FMM is very efficient for parallel processing but it introduces irregularities in the electrostatic potential and force, which can cause an energy drift in MD simulations. In this part of the thesis we introduce a regularized version of the FMM, useful for MD simulations, which approximately conserves energy over a long time period and even for low accuracy requirements. The method introduces a smooth transition over the boundary of boxes in the FMM tree and therefore it removes the discontinuity at the error level inherent in the FMM.

  • 3.
    Saffar Shamshirgar, Davood
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Hess, Berk
    KTH, Centres, Science for Life Laboratory, SciLifeLab.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    A comparison of the Spectral Ewald and Smooth Particle Mesh Ewald methods in GROMACSManuscript (preprint) (Other academic)
    Abstract [en]

    The smooth particle mesh Ewald (SPME) method is an FFT based methodfor the fast evaluation of electrostatic interactions under periodic boundaryconditions. A highly optimized implementation of this method is availablein GROMACS, a widely used software for molecular dynamics simulations.In this article, we compare a more recent method from the same family ofmethods, the spectral Ewald (SE) method, to the SPME method in termsof performance and efficiency. We consider serial and parallel implementa-tions of both methods for single and multiple core computations on a desktopmachine as well as the Beskow supercomputer at KTH Royal Institute ofTechnology. The implementation of the SE method has been well optimized,however not yet comparable to the level of the SPME implementation thathas been improved upon for many years. We show that the SE method isvery efficient whenever used to achieve high accuracy and that it already atthis level of optimization can be competitive for low accuracy demands.

  • 4.
    Saffar Shamshirgar, Davood
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Hess, Berk
    KTH, Centres, Science for Life Laboratory, SciLifeLab.
    Yokota, Rio
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Regularized FMM for MD simulationsManuscript (preprint) (Other academic)
    Abstract [en]

    A regularized fast multipole method (FMM) which approximately conserves the total energy in Molecular dynamics (MD) simulations is presented. The new algorithm introduces a regularization which eliminates the discontinuity inherent in the FMM. This allows us to use FMM in simulations as a substitute for widely used FFT based methods. For a system of N particles, the computational complexity of the resulting method is still of order N though with a larger constant compared to the plain FMM. Numerical examples are provided to confirm that the new algorithm improves the accuracy and approximately conserves the long term total energy.

  • 5.
    Saffar Shamshirgar, Davood
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    A fast multipole method for evaluating exponential integral type kernelsManuscript (preprint) (Other academic)
    Abstract [en]

    We present a fast multipole method for evaluation of sums with exponential  integral type kernels. These sums appear while solving free space Poisson problems in two dimensions and in the derivation of 1d-periodic Ewald sums. The presented method uses recurrence relations to derive multipole expansions for computing interactions between particles and far clusters.

  • 6.
    Saffar Shamshirgar, Davood
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Fast Ewald summation for electrostatic potentials with arbitrary periodicityManuscript (preprint) (Other academic)
    Abstract [en]

    A unified treatment for fast and spectrally accurate evaluation of electrostatic potentials subject to periodic boundary conditions in any or none of the three space dimensions is presented. Ewald decomposition is used to split the problem into a real space and a Fourier space part, and the FFT based Spectral Ewald (SE) method is used to accelerate the computation of the latter. A key component in the unified treatment is an FFT based solution technique for the free-space Poisson problem in three, two or one dimensions, depending on the number of non-periodic directions. The cost of calculations is furthermore reduced by employing an adaptive FFT for the doubly and singly periodic cases, allowing for different local upsampling rates. The SE method will always be most efficient for the triply periodic case as the cost for computing FFTs will be the smallest, whereas the computational cost for the rest of the algorithm is essentially independent of the periodicity. We show that the cost of removing periodic boundary conditions from one or two directions out of three will only marginally increase the total run time. Our comparisons also show that the computational cost of the SE method for the free-space case is typically about four times more expensive as compared to the triply periodic case.

    The Gaussian window function previously used in the SE method, is here compared to an approximation of the Kaiser-Bessel window function, recently introduced. With a carefully tuned shape parameter that is selected based on an error estimate for this new window function, runtimes for the SE method can be further reduced.

  • 7.
    Saffar Shamshirgar, Davoud
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    The Spectral Ewald method for singly periodic domains2017In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 347, p. 341-366Article in journal (Refereed)
    Abstract [en]

    We present a fast and spectrally accurate method for efficient computation of the three dimensional Coulomb potential with periodicity in one direction. The algorithm is FFT-based and uses the so-called Ewald decomposition, which is naturally most efficient for the triply periodic case. In this paper, we show how to extend the triply periodic Spectral Ewald method to the singly periodic case, such that the cost of computing the singly periodic potential is only marginally larger than the cost of computing the potential for the corresponding triply periodic system. In the Fourier space contribution of the Ewald decomposition, a Fourier series is obtained in the periodic direction with a Fourier integral over the non-periodic directions for each discrete wave number. We show that upsampling to resolve the integral is only needed for modes with small wave numbers. For the zero wave number, this Fourier integral has a singularity. For this mode, we effectively need to solve a free-space Poisson equation in two dimensions. A very recent idea by Vico et al. makes it possible to use FFTs to solve this problem, allowing us to unify the treatment of all modes. An adaptive 3D FFT can be established to apply different upsampling rates locally. The computational cost for other parts of the algorithm is essentially unchanged as compared to the triply periodic case, in total yielding only a small increase in both computational cost and memory usage for this singly periodic case.

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