Hierarchical Particle Mesh: An FFT-accelerated Fast Multipole Method

Gnedin, Nickolay Y.

doi:10.3847/1538-4365/ab2d24

Cited by 7 publications

(8 citation statements)

References 22 publications

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“…Tree methods have for a long time been extraordinarily popular for evaluating the short range interactions also in hybrid tree-PM methods, as pioneered by Bagla (2002); Bagla and Ray (2003), or more recent FMM-PM (Gnedin 2019;Wang 2021;Springel et al 2021) approaches, thus supplementing an efficient method for periodic long-range interactions with an efficient method which is not limited to the uniform coarse resolution of FFT-based approaches (or also discrete jumps in resolution of AMR approaches). We discuss some technical aspects of these methods next.…”

Section: Hierarchical Tree Methodsmentioning

confidence: 99%

“…In order to avoid explicit Ewald summation, some recent methods employ hybrid FFT-FMM methods, where essentially a PM method is used to evaluate the periodic long range interactions as in tree-PM and the FMM method is used to increase the resolution beyond the PM mesh for short-range interactions (Gnedin 2019;Springel et al 2021).…”

Section: Fast-multipole Methodsmentioning

confidence: 99%

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Large-scale dark matter simulations

Angulo

Hahn

2022

Living Rev Comput Astrophys

111

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We review the field of collisionless numerical simulations for the large-scale structure of the Universe. We start by providing the main set of equations solved by these simulations and their connection with General Relativity. We then recap the relevant numerical approaches: discretization of the phase-space distribution (focusing on N-body but including alternatives, e.g., Lagrangian submanifold and Schrödinger–Poisson) and the respective techniques for their time evolution and force calculation (direct summation, mesh techniques, and hierarchical tree methods). We pay attention to the creation of initial conditions and the connection with Lagrangian Perturbation Theory. We then discuss the possible alternatives in terms of the micro-physical properties of dark matter (e.g., neutralinos, warm dark matter, QCD axions, Bose–Einstein condensates, and primordial black holes), and extensions to account for multiple fluids (baryons and neutrinos), primordial non-Gaussianity and modified gravity. We continue by discussing challenges involved in achieving highly accurate predictions. A key aspect of cosmological simulations is the connection to cosmological observables, we discuss various techniques in this regard: structure finding, galaxy formation and baryonic modelling, the creation of emulators and light-cones, and the role of machine learning. We finalise with a recount of state-of-the-art large-scale simulations and conclude with an outlook for the next decade.

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Section: Hierarchical Tree Methodsmentioning

confidence: 99%

Section: Fast-multipole Methodsmentioning

confidence: 99%

Large-scale dark matter simulations

Angulo

Hahn

2022

Living Rev Comput Astrophys

111

View full text Add to dashboard Cite

show abstract

“…Tree methods have for a long time been extraordinarily popular for evaluating the short range interactions also in hybrid tree-PM methods, as pioneered by Bagla (2002); Bagla and Ray (2003), or more recent FMM-PM (Gnedin 2019;Wang 2021;Springel et al 2020) approaches, thus supplementing an efficient method for periodic long-range interactions with an efficient method which is not limited to the uniform coarse resolution of FFT-based approaches (or also discrete jumps in resolution of AMR approaches). We discuss some technical aspects of these methods next.…”

Section: Hierarchical Tree Methodsmentioning

confidence: 99%

Large-scale dark matter simulations

Angulo,

Hahn

2021

Preprint

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We review the field of collisionless numerical simulations for the large-scale structure of the Universe. We start by providing the main set of equations solved by these simulations and their connection with General Relativity. We then recap the relevant numerical approaches: discretization of the phase-space distribution (focusing on N -body but including alternatives, e.g., Lagrangian submanifold and Schrödinger-Poisson) and the respective techniques for their time evolution and force calculation (Direct summation, mesh techniques, and hierarchical tree methods). We pay attention to the creation of initial conditions and the connection with Lagrangian Perturbation Theory. We then discuss the possible alternatives in terms of the micro-physical properties of dark matter (e.g., neutralinos, warm dark matter, QCD axions, Bose-Einstein condensates, and primordial black holes), and extensions to account for multiple fluids (baryons and neutrinos), primordial non-Gaussianity and modified gravity. We continue by discussing challenges involved in achieving highly accurate predictions. A key aspect of cosmological simulations is the connection to cosmological observables, we discuss various techniques in this regard: structure finding, galaxy formation and baryonic modelling, the creation of emulators and light-cones, and the role of machine learning. We finalise with a recount of state-of-the-art large-scale simulations and conclude with an outlook for the next decade.

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“…2). For this test the FMM-HPM method is implemented with the code described in Gnedin (2019). The oct-tree is at least 3 levels deep (with just 2 levels the PM and FMM-HPM results are identical by construction).…”

Section: Optimal Green Functionmentioning

confidence: 99%

Natively Periodic Fast Multipole Method: Approximating the Optimal Green Function

Gnedin

2020

Preprint

Self Cite

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The Fast Multipole Method (FMM) obeys periodic boundary conditions "natively" if it uses a periodic Green function for computing the multipole expansion in the interaction zone of each FMM oct-tree node. One can define the "optimal" Green function for such a method that results in the numerical solution that converges to the equivalent Particle-Mesh solution in the limit of sufficiently high order of multipoles. A discrete functional equation for the optimal Green function can be derived, but is not practically useful as methods for its solution are not known. Instead, this paper presents an approximation for the optimal Green function that is accurate to better than 10 −3 in L MAX norm and 10 −4 in L 2 norm for practically useful multipole counts. Such an approximately optimal Green function offers a practical way for implementing FMM with periodic boundary conditions "natively", without the need to compute lattice sums or to rely on hybrid FMM-PM approaches.

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Hierarchical Particle Mesh: An FFT-accelerated Fast Multipole Method

Cited by 7 publications

References 22 publications

Large-scale dark matter simulations

Large-scale dark matter simulations

Large-scale dark matter simulations

Natively Periodic Fast Multipole Method: Approximating the Optimal Green Function

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