Things I don't understand about Vlasov-Poisson numerics

Transcription

1 Things I don't understand about Vlasov-Poisson numerics

2 1. Modeling stellar systems 2. Interpolation in simulations of cold dark matter 3. Orbit integration in N-body systems 4. How accurately should we integrate?

3 1. Modeling stellar systems

4 1. Modeling stellar systems the problem: model a stellar system such as an E0 galaxy the most we can hope to determine is ν(r,vlos), the surface brightness at projected position R in a unit interval of line-ofsight velocity vlos given a spherical, stationary, non-rotating stellar system with known ν(r,vlos), find the gravitational potential Φ(r) and the distribution function (df) f(r,vr,v ) Jeans s theorem tells us that the df can only depend on energy E = v 2 /2 + Φ(r) and angular momentum L=rv, that is, f(r,vr,v ) = f(e,l)

5 given a spherical, stationary, non-rotating stellar system with known ν(r,vlos), find the gravitational potential Φ(r) and the df f(e,l) why this is difficult: given Φ(r) it s straightforward to determine f(e,l) but in general you can t determine a function of two variables and a function of one variable [ν(r,vlos) and Φ(r)] from a single function of two variables f(e,l)

6 given a spherical, stationary, non-rotating stellar system with known ν(r,vlos), find the gravitational potential Φ(r) and the df f(e,l) in general you can t determine a function of two variables and a function of one variable [ν(r,vlos) and Φ(r)] from a single function of two variables f(e,l) Dejonghe & Merritt (1992): assume Φ(r) and determine f(e,l) if Φ(r) is wrong, it s likely to give f(e,l) < 0 in some places, which is unphysical therefore Φ(r) is determined through a nonlinear, implicit, integral inequality other difficulties: the integral equation for f(e,l) given Φ(r) is ill-conditioned so noise is amplified determination of ν(r,vlos) is subject to systematic errors because of template mismatches and continuum subtraction

7 given a spherical, stationary, non-rotating stellar system with known ν(r,vlos), find the gravitational potential Φ(r) and the df f(e,l) how this is done in practice: the gold standard is Schwarzschild s (1978) method: assume some potential Φ(r) divide phase space into bins of volume Δwj = ΔxjΔvj integrate N orbits from a wide variety of initial conditions and track the fraction of time orbit i spends in phase-space bin j, fij assign luminosity λi to orbit i and compute the expected surface brightness νλ(r,vlos). Calculate χ 2 = [ν(rk,vlos,k) - νλ(rk,vlos,k)] 2 and adjust the luminosities λi to minimize χ 2 df is then f(xj,vj)= i λifij/δwj repeat for other potentials and find the potential that minimizes χ 2

8 given a spherical, stationary, non-rotating stellar system with known ν(r,vlos), find the gravitational potential Φ(r) and the df f(e,l) how this is done in practice: the gold standard is Schwarzschild s (1978) method problems with Schwarzschild s method: the problem is under-determined and ill-conditioned so you must regularize, e.g., find maximum-entropy solution, apply smoothness constraints to the df, etc. the number of degrees of freedom is impossible to estimate the method is noisy because orbits have square-root singularities at their turning points in estimating the parameters of the potential one should marginalize over the df (the λi) but all existing methods just find the maximumlikelihood λi (Magorrian 2007)

9 given a spherical, stationary, non-rotating stellar system with known ν(r,vlos), find the gravitational potential Φ(r) and the df f(e,l) how this is done in practice: the gold standard is Schwarzschild s (1978) method there are many problems with Schwarzschild s method despite all these concerns, Schwarzschild s method, and simpler methods validated by Schwarzschild s method, is the basis for almost all we know about the dynamics of hot stellar systems existence and masses of black holes in the centers of galaxies distribution of dark matter in elliptical galaxies mass-to-light ratios of elliptical galaxies, tilt of the fundamental plane cores vs. cusps in dwarf spheroidal galaxies

10 N test particles orbit a point mass M in a steady state. We have three-dimensional position and velocity vectors for all N particles. What is the best estimate of M? 1. Virial theorem: A simple toy problem drawbacks: can t be generalized to potentials depending on several parameters inefficient use of data if there is a range of radii no error estimates

11 A simple toy problem 2. Projected mass estimator: drawbacks: can t be generalized to case where only line-of-sight velocities are available, without introducing dependence on eccentricity distribution can t be generalized to potentials depending on several parameters

12 A simple toy problem 3. Orbital roulette (Beloborodov & Levin 2004) drawbacks: can t be generalized to case where only line-of-sight velocities are available fails for zero eccentricity orbits result depends on the choice of statistical test

13 4. What would Bayes do? A simple toy problem

14 A simple toy problem

15 4. What would Bayes do? A simple toy problem

16 A simple toy problem Test on 1000 realizations of a 10- particle system; uniform distribution in e 2 and central mass M=1

17 A simple toy problem Test on 1000 realizations of a 10- particle system with central mass M=1 uniform in e 2

18 now apply the same method to a general integrable potential...

19

20 Bayesian approach to determining potential parameters fails, because there is a unique orbit consistent with every star so all potentials have the same likelihood

21 Bayesian approach to estimating potential parameters the Bayesian approach works well for power-law potentials Φ(r) = cr -k if it is applied to variables with the same dimension as c, i.e. variables of the form v 2 r k and vr 2 r k for a general integrable potential and a non-smooth distribution function f(j), a Bayesian approach to determining the gravitational potential fails bigger observational errors give better determinations of the potential parameters! (Magorrian 2013) the results from Bayesian analyses depend on arbitrary assumptions about the smoothness of the distribution function

22 Bayesian approach to estimating potential parameters the Bayesian approach works well for power-law potentials Φ(r) = cr -k if it is applied to variables with the same dimension as c, what i.e. variables are we of the really form doing v 2 r k and when vr 2 r k we estimate gravitational for a general integrable parameters potential of and galaxies a non-smooth using distribution function f(j), a Bayesian approach to determining the Schwarzschild s method? gravitational potential fails bigger observational errors give better determinations of the potential parameters! (Magorrian 2013) the results from Bayesian analyses depend on arbitrary assumptions about the smoothness of the distribution function

23 2. N-body simulations of cold dark matter

24 standard cosmological model involves cold dark matter: at high redshift, dark matter occupies a 3-dimensional manifold or sheet in 6-dimensional phase space, with near-zero comoving velocity as structure develops this sheet is stretched and folded but never torn, and remains 3-dimensional N-body simulations follow a Monte Carlo sampling of the mass distribution in this sheet particles in a simulation are vertices in a 3-dimensional mesh that describes the sheet how can we best characterize the properties of the sheet, i.e. how should we interpolate between particles? Schulz et al. (2013)

25 N-body simulations follow a Monte Carlo sampling of the mass distribution in this sheet particles in a simulation are vertices in a 3-dimensional mesh that describes the sheet how can we best characterize the properties of the sheet, i.e. how should we interpolate between particles? Simplest interpolation schemes use only position information: traditionally these were developed to generate gravitational potential from Poisson s equation. Two approaches: replace each particle with a smooth density kernel (as in softening and SPH) divide space into a grid and assign mass of particle to nearby grid points (NGP, CIC, etc.) Voroni and Delaunay tesselations (Bernardeau & van de Weygaert 1996) k nearest neighbors recursive binary splitting (Ascasibar & Binney 2005) More sophisticated schemes use both position and velocity information (interpolation in phase space)

26 Phase-space interpolation schemes for cold dark matter interpolate onto a grid in phase space extremely memory-intensive: grid with N elements per dimension needs N 6 grid points Yoshikawa et al. (2013): direct solution of Vlasov-Poisson equations using 646 phase-space grid (run on 64 nodes, 32 GB RAM per node, 180 s per timestep) kernel density estimation needs a metric for phase space Delaunay tesselation (Arad et al. 2004) expensive in time and memory k nearest neighbors needs a metric recursive binary splitting (Maciejewski et al. 2009)

27 Phase-space interpolation schemes for cold dark matter interpolate onto a grid in phase space kernel density estimation Delaunay tesselation k nearest neighbors recursive binary splitting Lagrangian tesselation (Abel et al. 2012, Shandarin et al. 2012) tesselate the phase space at t=0 and use an N-body simulation to follow the nodes of the tesselation

28 Phase-space interpolation schemes for cold dark matter particle-based Lagrangian tesselation Shandarin et al. (2012)

29 Phase-space interpolation schemes for cold dark matter Abel et al. (2012) grid-based density estimate Lagrangian tesselation density estimate

30 Phase-space interpolation schemes for cold dark matter interpolate onto a grid in phase space kernel density estimation Delaunay tesselation k nearest neighbors recursive binary splitting Lagrangian tesselation (Shandarin et al. 2012) geodesic deviation (Vogelsberger et al. 2008)

31 Phase-space interpolation schemes for cold dark matter

32 Phase-space interpolation schemes for cold dark matter interpolate onto a grid in phase space kernel density estimation Delaunay tesselation k nearest neighbors recursive binary splitting Lagrangian tesselation (Shandarin et al. 2012) geodesic deviation (Vogelsberger et al. 2008) symplectic interpolation

33 Symplectic interpolation

34 Symplectic interpolation assume we tesselate 3-space with tetrahedra and use linear or quadratic interpolation within each tetrahedron: linear interpolation quadratic interpolation number of available interpolation constants number of constraints from symplectic condition 3 30 number of constraints from the 4 vertices of the tetrahedron number of constraints from adjacent tetrahedra -?

35 Symplectic interpolation assume we tesselate 3-space with tetrahedra and use linear or quadratic interpolation within each tetrahedron: is symplectic interpolation a useful tool for improving the resolution linear interpolation and power quadratic of large interpolation cosmological N-body simulations? number of available interpolation constants number of constraints from symplectic condition 3 30 number of constraints from the 4 vertices of the tetrahedron number of constraints from adjacent tetrahedra -?

36 3. Orbit integration in N-body systems

37 Orbit integration in N-body systems

38 Orbit integration in N-body systems standard approach is Leapfrog or Verlet method advantages of leapfrog time-reversible symplectic no memory needed for intermediate steps disadvantages of leapfrog only second-order accurate not suitable for collisional simulations not symplectic if variable timestep Consider an N-body system with stars of mass m and radius R. Typical velocity is v 2 =GNm/R. Crossing or dynamical time is tcr = R/v. Typical inter-particle separation is r = R/N 1/3. Leapfrog fails if the timestep h exceeds time required to cross r, h > r/v ~ tcr/n 1/3 as N increases and the potential becomes smoother, the maximum timestep must shrink

39 Impulse integrator with Hanno Rein (IAS and Toronto)

40 Impulse-approximation integrator symplectic second-order accurate (like leapfrog) no memory needed for intermediate steps no restriction to timesteps less than inter-particle travel time disadvantages: algebraic formulas are more complicated, so slower to evaluate implicit with Hanno Rein

41 rel. energy error rel. energy error e-05 1e-06 1e-07 1e-08 1e e-05 1e-06 1e-07 1e-08 1e-09 N = 100,000 Impulse-approximation integrator time [crossing times] N = 10,000,000 2nd order impulse integrator leapfrog integratorr time [crossing times] 2nd order impulse integrator leapfrog integratorr with Hanno Rein ape velocity neareast star N fixed particles in a uniform sphere, plus one test particle on circular orbit 1000 energy of test particle should be conserved 100 errors in leapfrog get bigger as N increases errors in impulse integrator get smaller as N increases N = 10,000,000 2nd order impulse integrator leapfrog integrator Test particle velocity

42 rel. energy error rel. energy error e-05 1e-06 1e-07 1e-08 1e e-05 1e-06 1e-07 1e-08 1e-09 N = 100,000 Impulse-approximation integrator time [crossing times] N = 10,000,000 2nd order impulse integrator leapfrog integratorr time [crossing times] 2nd order impulse integrator leapfrog integratorr with Hanno Rein ape velocity neareast star why does the error saturate? N = 10,000,000 why does it saturate at the same timestep for both methods? 1000 what determines the saturation level? 100 2nd order impulse integrator leapfrog integrator Test particle velocity

43 rel. energy error rel. energy error e-05 1e-06 1e-07 1e-08 1e e-05 1e-06 1e-07 1e-08 1e-09 N = 100,000 Impulse-approximation integrator time [crossing times] 2nd order impulse integrator leapfrog integratorr N = 10,000, can the impulse integrator be developed into a useful tool for N-body integrations? time [crossing times] 2nd order impulse integrator leapfrog integratorr with Hanno Rein ape velocity neareast star why does the error saturate? N = 10,000,000 why does it saturate at the same timestep for both methods? 1000 what determines the saturation level? 100 2nd order impulse integrator leapfrog integrator Test particle velocity

44 4. How do we know if an N-body integration is correct?

45 How do we know if an N-body integration is correct? small errors in an N-body integration grow exponentially (Miller 1964, 1971; Hayli 1970) e-folding or Liapunov time ~ crossing time tcr (Heggie 1988, Goodman et al. 1992). Not related to the relaxation time and independent of N therefore detailed final positions and velocities of particles in a simulation are not reproducible this has not deterred subsequent investigations from being carried out in an optimistic spirit (Aarseth & Lecar 1975) a pessimist might ask, why should we believe the results?

46 How do we know if an N-body integration is correct? small errors in an N-body integration grow exponentially (Miller 1964, 1971; Hayli 1970) e-folding or Liapunov time ~ crossing time tcr (Heggie 1988, Goodman et al. 1992). Not related to the relaxation time and independent of N therefore detailed final positions and velocities of particles in a simulation are not reproducible this has not deterred subsequent investigations from being carried out in an optimistic spirit (Aarseth & Lecar 1975) a pessimist might ask, why should we believe the results? forward error analysis: measures the divergence of the numerical solution from the exact solution with the same initial conditions backward error analysis: asks if the numerical solution is the same as the exact solution for a slightly different physical system

47 When should we believe the results of an N-body simulation? never because of the growth of computational errors N-body calculations in stellar dynamics are of limited usefulness (Miller 1964) - forward error analysis always, so long as the integrator is symplectic the argument is that then we are following a Hamiltonian system that is close to the real one - backward error analysis when the integration is time-reversible this is easy if it uses a time-reversible integrator and roundoff is negligible if it conserves energy and angular momentum this is very crude if the result looks OK implies we should use bigger and bigger timesteps until something goes grossly wrong if the result is reproduced in a statistical sample of simulations maybe they give consistent results only because they are all equally inaccurate (Heggie 1991) if the result looks the same when the integration is repeated with twice the particles, half the timestep, etc.

48 Shadowing (Quinlan & Tremaine 1992) forward error analysis: measures the divergence of the numerical solution from the exact solution with the same initial conditions backward error analysis: asks if the numerical solution is the same as the exact solution for a slightly different physical system shadowing: asks if the numerical solution is the same as an exact solution with slightly different initial conditions

49 Shadowing (Quinlan & Tremaine 1992) shadowing: asks if the numerical solution is the same as an exact solution with slightly different initial conditions a shadow orbit is an exact solution that stays within a small distance δ of the numerical solution over the integration time 0 < t < T existence of shadow orbits is not a rigorous proof that the N-body integration is realistic since the shadow orbits may have peculiar initial conditions finding shadow orbits is numerically challenging. Best approach is to use shooting: start the shadow in the contracting manifold of phase space at t=0 and integrate forward; start another one in the expanding manifold at t=t and integrate backward, then match in the middle works better in chaotic dynamical systems than integrable systems: it s easier to shadow ẍ = x than ẍ = const typically the existence of a shadow orbit terminates abruptly at a glitch ; these are real, not numerical artifacts

50 t=20.1 shadow exact t=3.25 numerical start, t=0 N=100 Plummer model, 99 particles fixed no softening numerical orbit followed with a standard Aarsethtype N-body integrator shadow distance in position δ = typical shadow times T ~ 10 and get longer if accuracy of numerical orbit is improved or softening is added Quinlan & Tremaine (1992

51 Hayes (2003): shadowed a wide variety of N-body integrations using leapfrog non-symplectic integrators are shdowed less well glitches appear to be a Poisson process the half-life for shadowing is crossing time softening timestep system size shadow duration is insensitive to N softening is needed to obtain long-lasting shadow orbits with any practical integration algorithm shadowing for many crossing times requires that particles travel a small fraction of the softening length in one timestep

52 Hayes (2003): shadowed a wide variety of N-body integrations using leapfrog non-symplectic integrators are shdowed less well glitches appear to be a Poisson process the half-life for shadowing is crossing time softening timestep system size how badly are we allowed to integrate? shadow duration is insensitive to N softening is needed to obtain long-lasting shadow orbits with any practical integration algorithm shadowing for many crossing times requires that particles travel a small fraction of the softening length in one timestep