Orthogonal Basis Function Approximation of Particle Distributions In Numerical Simulations of Beams

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1 Orthogonal Basis Function Approximation of Particle Distributions In Numerical Simulations of Beams Balša Beam Physics and Astrophysics Group Northern Illinois University Jefferson Lab Beam Seminar April 3, 008

2 Motivation Studying dynamics of multi particle systems (charged particle beams, plasma, galaxies...) heavily relies on N body simulations It is important for N body codes to: be as efficient as possible, without compromising accuracy minimize numerical noise due to Nsimulation<< Nphysical account for multiscale dynamics for some applications: have a compact representation of history We present two orthonormal bases which, as a part of an N body code, address these requirements wavelet basis scaled Gauss Hermite basis

3 Outline of the Talk Algorithms for N body simulations Wavelet basis brief overview of wavelets wavelet based Poisson equation solver advantages applications Scaled Gauss Hermite basis mathematical formalism Poisson equation solver applications Discussion of further work 3

4 Algorithms for N body Simulations Direct summation: CPU cost scales as N Tree: direct summation nearby and statistical treatment farther away Particle In Cell (PIC): particles binned in cells (grid) system at t=t0 bin macroparticles to obtain particle distribution on a finite Nx Ny Nz grid N macroparticles Newton's equations advance particles by t system at t=t0+ t interpolate to find F for each particle F= solve the Poisson equation = on a finite Nx Ny Nz grid 4

5 Algorithms for N body Simulations Direct summation: CPU cost scales as N Tree: direct summation nearby and statistical treatment farther away Particle In Cell (PIC): particles binned in cells (grid) system at t=t0 bin macroparticles to obtain particle distribution on a finite Nx Ny Nz grid N macroparticles Newton's equations advance particles by t system at t=t0+ t interpolate to find F for each particle F= solve the Poisson equation = on a finite Nx Ny Nz grid use wavelets 5

6 Algorithms for N body Simulations Alternative N body algorithm: analytical function approximation analytical functions form a finite orthogonal basis N macroparticles, but no grid system at t=t0 expand particle distribution in a finite orthogonal basis (x,y,z)= ψ (x,y,z) N macroparticles ijk Newton's equations advance particles by t differentiate to get F for each particle F = system at t=t0+ t ijk ijk solve the Poisson equation = in the same basis (x,y,z)= ψ (x,y,z) ijk ijk ijk 6

7 Algorithms for N body Simulations Alternative N body algorithm: analytical function approximation analytical functions form a finite orthogonal basis N macroparticles, but no grid use scaled Gauss Hermite basis system at t=t0 expand particle distribution in a finite orthogonal basis (x,y,z)= ψ (x,y,z) N macroparticles ijk Newton's equations advance particles by t differentiate to get F for each particle F = system at t=t0+ t ijk ijk solve the Poisson equation = in the same basis (x,y,z)= ψ (x,y,z) ijk ijk ijk 7

8 Wavelets Wavelets: orthogonal basis composed of scaled and translated versions of the same localized mother wavelet ψ(x) and the scaling function ф(x): ψik (x)=k/ψ (kx i) f(x) =s00ф00 (x) + i k dikψik (x) Discrete Wavelet Transfrom (DFT) iteratively separates scales ~O(MN) operation, M size of the wavelet filter, N size of the signal Advantages: simultaneous localization in both space and frequency compact representation of data, enabling compression (FBI fingerprints) signal denoising: natural setting in which noise can be partially removed denoised simulation simulation with more macroparticles 8

9 Numerical Noise in PIC Simulations Any N body simulation will have numerical noise Sources of numerical noise in PIC simulations: graininess of the distribution function: Nsimulation<< Nphysical discreteness of the computational domain: and specified on a finite grid Each macroparticle is deposited onto a finite grid by either: Nearest Grid Point (NGP) dep. scheme Cloud In Cell (CIC) dep. scheme NGP CIC x marcoparticle location gridpoint location 9

10 Numerical Noise in PIC Simulations For NGP, at each gridpoint, particle dist. is Poissonian: n nj is the expected number in jth cell; P= n! n j e 1 n j For CIC, at each gridpoint, particle dist. is contracted Poissonian: n P= n! an j e 1 n integer an j D/ a= / 3 ~0.54 3D, 0.67 D, 0.8 1D Measure of error (noise) in depositing macroparticles onto a grid: = N grid 1 N grid Var qi i=1 NGP = Q total N N grid CIC = a Q total N N grid where qi= (Qtotal/N)ni, Qtotal total charge; Ngrid number of gridpoints (For more details see, Pogorelov & Bohn 007, PR STAB, 10, 03401) This error/noise estimate is crucial for optimal wavelet denoising IDEA: Solve the Poisson equation in such a way so as to minimize numerical noise USE WAVELETS 10

11 Numerical Noise in PIC Simulations In wavelet space: signal noise few large wavelet coefficients cij many small wavelet coefficients cij Denoising by wavelet thresholding: if cij < T, set to cij =0 (choose threshold T carefully!) A great deal of study has been devoted to estimating optimal T T = log N grid (σ was estimated earlier), Pogorelov & Bohn 007, PR STAB, 10,

12 Wavelet Denoising and Compression Whenever the discrete signal is analytically known, one can compute the Signal to Noise Ratio (SNR) which measures its quality SNR ~ Nppc ANALYTICAL Nppc: avg. # of particles per cell Nppc= 3 SNR =.0 Nppc = N/Ncells Nppc= 05 SNR =

13 Wavelet Denoising and Compression Whenever the discrete signal is analytically known, one can compute the Signal to Noise Ratio (SNR) which measures its quality SNR ~ Nppc Nppc: avg. # of particles per cell Nppc = N/Ncells D superimposed Gaussians on grid ANALYTICAL Nppc=3 SNR=.0 Nppc=05 SNR=

14 Wavelet Denoising and Compression Whenever the discrete signal is analytically known, one can compute the Signal to Noise Ratio (SNR) which measures its quality SNR ~ Nppc Nppc: avg. # of particles per cell Nppc = N/Ncells D superimposed Gaussians on grid ANALYTICAL Nppc=3 SNR=.0 Nppc=05 SNR=

15 Wavelet Denoising and Compression Whenever the discrete signal is analytically known, one can compute the Signal to Noise Ratio (SNR) which measures its quality SNR ~ Nppc Nppc: avg. # of particles per cell Nppc = N/Ncells D superimposed Gaussians on grid ANALYTICAL Nppc=3 SNR=.0 Nppc=05 SNR=

16 Wavelet Denoising and Compression Whenever the discrete signal is analytically known, one can compute the Signal to Noise Ratio (SNR) which measures its quality SNR ~ Nppc Nppc: avg. # of particles per cell Nppc = N/Ncells D superimposed Gaussians on grid ANALYTICAL Nppc=3 SNR=.0 Nppc=05 SNR=16.89 denoising by wavelet thresholding: if cij < T, set to 0 16

17 Wavelet Denoising and Compression Whenever the discrete signal is analytically known, one can compute the Signal to Noise Ratio (SNR) which measures its quality SNR ~ Nppc Nppc: avg. # of particles per cell Nppc = N/Ncells D superimposed Gaussians on grid COMPACT: only 0.1% of coeffs WAVELET THRESHOLDING ANALYTICAL Nppc=3 SNR=.0 Nppc=05 SNR=16.89 denoising by wavelet thresholding: if cij < T, set to 0 Advantages: DENOISED Nppc= 3 SNR = increase in SNR by c c more macroparticles (here c=8.3, c=69) compact storage in wavelet space (in this example 79/65536: 0.1%) 17

18 Wavelet Based Poisson Equation Solver Poisson equation in physical space u=f BCs: using Green's functions ubnd = g Preconditioned Conjugate Gradient in wavelet space A X B ε B DWT 1 solution U on the Nx Ny Nz grid transform discretized Poisson eq. to wavelet space discretize Poisson equation on a Nx Ny Nz grid LU=F DWT Lw Uw = Fw k (L) ~ O(Nx) wavelet threshold source Fw, operator Lw AX=B (PLwP) P 1Uw = PFw A compressed operator PLwP Precondition Laplacian Lw B denoised distribution PFw with diagonal preconditioner X denoised solution P 1Uw P k (Lw) ~ O(Nx) k (PLwP) ~ O(Nx) physical space wavelet space 18

19 Preconditioned Conjugate Gradient (PCG) i convergence rate depends on condition number k u u i k 1 u k ~ O(Nx) preconditioning (diagonal in wavelet space): good initial approximation: solution at previous time step k 1 k ~ O(Nx ) 19

20 Preconditioned Conjugate Gradient (PCG) i convergence rate depends on condition number k u u i k 1 u k ~ O(Nx) preconditioning (diagonal in wavelet space): good initial approximation: solution at previous time step k 1 k ~ O(Nx ) typical realistic beam simulation no preconditioning Ui=0 initial guess average over step run 75. 0

21 Preconditioned Conjugate Gradient (PCG) i convergence rate depends on condition number k u u i k 1 u k ~ O(Nx) preconditioning (diagonal in wavelet space): good initial approximation: solution at previous time step k 1 k ~ O(Nx ) typical realistic beam simulation average over step run no preconditioning Ui=0 initial guess 75. preconditioned Ui=0 initial guess

22 Preconditioned Conjugate Gradient (PCG) i convergence rate depends on condition number k u u i k 1 u k ~ O(Nx) preconditioning (diagonal in wavelet space): good initial approximation: solution at previous time step k 1 k ~ O(Nx ) typical realistic beam simulation no preconditioning Ui=0 initial guess 75. preconditioned Ui=0 initial guess 60.7 no preconditioning Ui=Ui 1 initial guess average over step run 4.8

23 Preconditioned Conjugate Gradient (PCG) i convergence rate depends on condition number k u u i k 1 u k ~ O(Nx) preconditioning (diagonal in wavelet space): good initial approximation: solution at previous time step k 1 k ~ O(Nx ) typical realistic beam simulation average over step run no preconditioning Ui=0 initial guess 75. preconditioned Ui=0 initial guess 60.7 no preconditioning Ui=Ui 1 initial guess 4.8 preconditioned Ui=Ui 1 initial guess.4 considerable computational speedup (, Pogorelov & Bohn 007) 3

24 Using PCG in Numerical Simulations Our goal: develop wavelet based Poisson solver which can easily be integrated into existing PIC codes First: test the PCG as a stand alone solver on examples from: beam dynamics galactic dynamics Second: insert the PCG Poisson solver into an existing PIC code (IMPACT T) IMPACT T and run realistic charged particle beam simulations (, Pogorelov & Bohn 007, PR STAB, 10, 03401) compare (conventional FFT based) IMPACT T Vs. IMPACT T with PCG: rms properties level of detail computational speed 4

25 Conventional IMPACT T vs. IMPACT T with PCG Code Comparison: rms Properties Fermilab/NICADD photoinjector grid 1 nc charge rms beam radius rms normalized transverse emittance rms bunch length rms normalized longitudinal emittance Good agreement to a few percent 5

26 Conventional IMPACT T vs. IMPACT T with PCG Code Comparison: Level of Detail & Speed IMPACT T transverse charge distribution for the Fermilab/NICADD photoinjector simulation very non axisymmetric beam grid, N=00000 very good agreement in detail initial distribution z = 1.0 m Speed comparison: IMPACT T w/ PCG ~ 10% faster than the conventional serial IMPACT T IMPACT T w/ PCG z =.0 m z = 4.0 m 6

27 Data Compression with PCG PCG provides excellent compression of data and operators in wavelet space Fermilab/NICADD photoinjector: Real Simulations particle distribution Laplacian operator grid, N=15 000, Nppc=4.58 ~ 3.5% coefficients retained on average grid, N= , Nppc=4.58 ~ 1.75% coefficients retained on average compact storage of beam's distribution history needed for CSR simulations compact storage of beam's potential needed for modeling halo formation 7

28 Ongoing Project: Improving the PCG Solver Currently, we (graduate student Ben Sprague and I) are working on a number of improvements to the wavelet based Poisson equation solver: change from fixed to adaptive grid simplify BC computation (currently a computational bottleneck) further exploit sparsity of operators and data sets use a non standard operator form to better separate scales use more sophisticated wavelet families (biorthogonal, lifted) explore other preconditioners parallelize and optimize Possible future applications of PCG solver: CSR simulations: computation of retarded potentials requires integration over history of the system compactly represented in wavelet space develop a new PIC code to simulate self gravitating systems 8

29 Scaled Gauss Hermite Basis 1 x ψ n x = n H n x e n! π Gauss Hermite orthonormal basis: (solution to quantum harmonic oscillator) Orthonormal: ψ l x ψ m x e x ψ m x dx=δ m dx=δ l m Basis functions: oscillatory exponentially decaying δlm, δm Kronecker delta Infinite expansion (D): 0, 1,, 4, 5 finite: f x, y = al m ψ l x ψ m y l =0 m=0 L M l=0 m=0 l =0 m=0 Scaled and translated version: L f M x y x, y = a l m ψ l x ψ m y α1 α l=0 m=0 1 1 α 1=, α = σx σy σ x x x f x, y dx dy y y { } { } σ y = x = x f x, y dx dy y y {} {} 9

30 Scaled Gauss Hermite Basis N Define collocation points: {γ j } j =0 roots of HL+1(x) { β k }km=0 roots of HM+1(y) L At collocation points: f γ j, β k = a l m ψ l γ j ψ m β k l=0 m =0 Take advantage of the relation for Hermite polynomials: n to obtain al m = H k x H k y k k! k =0 L C jk = = H n 1 x H n y H n x H n 1 y n 1 n! x y 1 C f γ j, β k ψ l γ j ψ m β k j=0 k=0 jk 0 l L, 0 m M M [ψ n γ j ] [ψ m β k ] l =0 γj γ j = x α1 βk β k = y α M L M 0 j L, 0 k M m=0 This formalism is general and can easily be extended to higher dimensions 30

31 Poisson Equation in Scaled Gauss Hermite Basis Poisson equation: x y x y x y ΔΦ x, y = [ x y ]Φ x, y = κf x, y α1 α α1 α α1 α x y Φ x, y = α1 α b l m ψ l x ψ m y l =0 m =0 where blm are given by the difference relation: α 1 l l 1 b n m α m m 1 b l m = κa l m with boundary coefficients: κ a l m, α m m 1 b n m= κ al m, α 1 l l 1 { l=0,1 m, m=0,1 l, } No need to invert the difference equation: compute boundary first, and then work inside computationally simple and efficient 31

32 Simulating Multiparticle Systems with Scaled Gauss Hermite Expansion N body realization of the discrete particle distribution: 1 f x, y = N N δ x x i δ y y i i=1 Expanding f(x,y) in scaled Gauss Hermite basis reduces to the following steps: L 1. tabulate the unchanging quantities: C jk = 1 N 1 y = N x = N xi i=1 N yi α1 = α = i=1 3. evaluate f γ j, β k at the nodes 4. compute coefficients [ψ l γ j ] [ψ m β k ] l =0 pl m j k =. compute x, y, α 1, α : al m = L [ [ N N N x i x i=1 N y i y i =1 1/ ] ] 1/ M m=0 ψ l γ j ψ m β k C jk γj x α1 β β k = k y α γ j = M p l m j k f γ j, β k j=0 k=0 3

33 Simulating Multiparticle Systems with Scaled Gauss Hermite Expansion N body realization of the discrete particle distribution: 1 f x, y = N N δ x x i δ y y i i=1 Expanding f(x,y) in scaled Gauss Hermite basis reduces to the following steps: L 1. tabulate the unchanging quantities: C jk = 1 N 1 y = N x = N xi i=1 N yi α1 = α = i=1 3. evaluate f γ j, β k at the nodes 4. compute coefficients [ψ l γ j ] [ψ m β k ] l =0 pl m j k =. compute x, y, α 1, α : al m = L [ [ N N N x i x i=1 N y i y i =1 1/ ] ] 1/ M M m=0 ψ l γ j ψ m β k C jk γj x α1 β β k = k y α γ j = function estimation from a discrete sample p l m j k f γ j, β k j=0 k=0 33

34 Simulating Multiparticle Systems with Scaled Gauss Hermite Expansion Evaluate f γ j, β k from a discrete sample (nonparametric density estimation) hx hy f γ l, β m = f γ l x, β m y d y d x shifted histogram estimator h x h y hx hy with window [ hx,hx] [ hy,hy] d y d x h x h y Optimal size of the window: 9 [ 1/ 5 hx f ' ' x d x h x ] N 1 /5 Integrated means square error (IMSE) IMSE = h opt = 1 /5 5 4 /5 1/ / 5 hx [ h x f ' ' x d x ] N 4/ 5 ~ N 4 / 5 Other, more sophisticated estimators are available: kernel, adaptive estimators, and others... 34

35 Applying Scaled Gauss Hermite Expansion Future application of scaled Gauss Hermite approximation: D CSR code of Bassi, Ellison, Heinemann and Warnock: particle distribution is sampled by N macroparticles distribution is approximated at each timestep with a cosine expansion beam self forces are computed from the analytic expansion Problems: unphysical wiggles in the tails of the distribution computational speed: each coefficient requires N cosine evaluations Problems resolved (?) with scaled Gauss Hermite: no wiggles basis functions are exponentially decaying computing coefficients scales more favorably and does not involve evaluation of any expensive function (addition & multiplication) 35

36 Applying Scaled Gauss Hermite Expansion Typical simulation: N=106 Wiggles in cosine expan. cosine Reduced by orders of mag. in scaled GH Nx=Ny=4 scaled GH Nx=Ny=10 fraction of negative volume Nx=Ny=16 18x18 grid 36

37 Applying Scaled Gauss Hermite Expansion Scaled Gauss Hermite expansion is computationally appreciably faster: cosine expansion: tcos ~ O(LMNpart) scaled Gauss Hermite: tsgh ~ O((L+M)Npart) + O(LM) ratio tcos/tsgh ~ O(L) + O(Npart/L) (assume L=M) 37

38 Applying Scaled Gauss Hermite Expansion Scaled Gauss Hermite expansion is computationally appreciably faster: cosine expansion: tcos ~ O(LMNpart) scaled Gauss Hermite: tsgh ~ O((L+M)Npart) + O(LM) ratio tcos/tsgh ~ O(L) + O(Npart/L) (assume L=M) ~0 times faster 38

39 Scaled Gauss Hermite Expansion: Loose Ends There are several issues with the scaled Gauss Hermite expansion that we are still exploring/resolving: convergence different estimators for evaluating f γ j, β k optimal number of basis functions (when is more less?) what do we lose by using an analytic expansion? avoid danger of smoothing over physical small scale structures adequate resolution: can this approach resolve physical small scale structure? When these issues are properly addressed, we will have another tool with which to attack CSR and integration over beam's history 39

40 Summary Designed an iterative wavelet based Poisson solver (PCG) wavelet compression and denoising achieves computational speedup preconditioning and sparsity of operators and data in wavelet space reduce CPU load integrated PCG into a PIC code (IMPACT T) for beam dynamics simulations current efforts: adaptive grid, parallelization, optimization future uses: probe usefulness of wavelet methodology in CSR simulations simulate self gravitating systems Developed a scaled Gauss Hermite approximation (still a prototype): efficient representation of particle distribution Poisson equation solved directly at a marginal cost current efforts: resolving issues of convergence, truncation of expansion future uses (?): in Bassi et al.'s D CSR code in Rui Li's D CSR code 40

41 Auxiliary Slides 41

42 Computing BCs: Current Implementation Green's function corresponding to a grounded rectangular pipe V=0 y computational grid EIGENFUNCTIONS sin l x pipe walls sin m y V=0 V=0 EIGENVALUES l =l / a V=0 x pipe walls a b 4 lm z = 0 0 x, y, z sin l x sin m y dx dy ab lm z z m=m / b lm = l m lm lm z = lm z Nx Ny lm x, y, z = l =1 m =1 z sin l x sin m y 4

43 Computing BCs: Adaptive Grid V=0 computational grid V=0 V=0 V=0 43

44 Computing BCs: Adaptive Grid V=0 computational grid V=0 V=0 V=0 44

45 Computing BCs: Adaptive Grid V=0 computational grid V=0 V=0 V=0 45

46 Preconditioner in Wavelet Space

47 Laplacian in Standard Form in Wavelet Space 47

48 Laplacian in Non Standard Form in Wavelet Space 48

49 Wavelet Decomposition The continuous wavelet transform of a function f (t) is s, = f t s, t dt 1 s, t = t s s t mother wavelet with scale and translation dimensions s and respectively 49

50 How Do Wavelets Work? Wavelet analysis (wavelet transform): S signal A approximation D detail Approximation apply low pass filter to Signal and down sample Detail apply high pass filter to Signal and down sample Wavelet synthesis (inverse wavelet transform): up sampling & filtering Complexity: 4MN, M the size of the wavelet, N number of cells Recall: FFT complexity 4N logn 50

51 Wavelets Wavelet transform separates scales superimposed gaussians FFT wavelet transform 51

52 Numerical Noise in PIC Simulations In wavelet space: signal noise few large wavelet coefficients cij many small wavelet coefficients cij Anscombe transformation Poissonian noise Denoising by wavelet thresholding: Gaussian noise if cij < T, set to cij =0 (choose threshold T carefully!) A great deal of study has been devoted to estimating optimal T Nppc=5 T = log N grid DF(T) (σ was estimated earlier) Ni=64 NGP Nppc=5 DF(T) CIC Ni=64 w/ Anscombe transformation w/out Anscombe transformation JLab Beam Terzi Seminar, April 008 & Bohn 007, PR STAB, 10, ć, Pogorelov

53 3D Plummer sphere on a 3x3x3 grid BCs: open in all directions (analytically specified) 53

54 3D 'fuzzy cigar' on a 3x3x3 grid BCs: grounded rectangular pipe (V=0 on the sides), open in z direction (Green's function solution) 54

55 D EXAMPLE: MICROBUNCHED DENSITY DISTRIBUTION (with U=0 starting guess) BCs: open in all directions (analytically specified) 55

56 IMPACT T Vs. IMPACT T with PCG: rms properties Fermilab/NICADD photoinjector (3 3 3 grid) 1 nc charge bunch compressor at z=5.65m from cathode rms beam radius rms normalized transverse emittance rms bunch length rms normalized longitudinal emittance 56

57 IMPACT T Vs. IMPACT T with PCG: level of detail IMPACT T transverse charge distribution for the Fermilab/NICADD photoinjector simulation 5 beamlets (masked photocathode) grid, N=00000 very good agreement in detail IMPACT T w/ PCG initial distribution z =.0 m z = 4.0 m z = 6.0 m 57

58 IMPACT T Vs. IMPACT T with PCG: level of detail Impact IMPACT T IMPACT Tresidue w/ PCG Impact with CPCG transverse charge distribution for the AES/JLab low charge photoinjector simulation very non axisymmetric beam grid, N=00000 very good agreement in detail initial distribution z = 0.5 m z = 1.9 m z =.5 m 58

59 IMPACT T Vs. IMPACT T with PCG: level of detail Impact IMPACT T IMPACT Tresidue w/ PCG Impact with CPCG transverse charge distribution for the AES/JLab low charge photoinjector simulation very non axisymmetric beam grid, N=00000 very good agreement in detail initial distribution z = 0.5 m z = 1.9 m z =.5 m 59

60 Scaled Gauss Hermite Basis 1 x ψ n x = n H n x e n! π Gauss Hermite orthonormal basis: (solution to quantum harmonic oscillator) Hermite polynomials Hn(x) relations: Orthonormal: ψ l x ψ m x e x δlm, δm Basis functions: oscillatory exponentially decaying 0, 1,, 4, 5 Infinite expansion (D): f x, y = al m ψ l x ψ m y Scaled and translated version: L M x y f x, y = a l m ψ l x ψ m y α1 α l=0 m=0 α 1= 1 1, α = σx σy σ x finite: l =0 m=0 Kronecker delta H n 1 x = x H n x n H n 1 x H n ' x = x H n 1 x ψ m x dx=δ m dx=δ l m L M l=0 m=0 l =0 m=0 x x f x, y dx dy y y { } { } σ y = x = y {} x f x, y dx dy y {} 60