Density Estimation Techniques for Density Estimation Techniques for Charged Particle Beams Charged Particle Beams Bal Bal š š a Terzić a Terzić Beam Physics and Astrophysics Group, NICADD Beam Physics and Astrophysics Group, NICADD Northern Illinois University, USA Northern Illinois University, USA Work done in collaboration with Work done in collaboration with Gabriele Bassi Gabriele Bassi Cockcroft Institute Cockcroft Institute Cockcroft Institute Seminar Cockcroft Institute Seminar February 26, 2009 February 26, 2009
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Density Estimation Techniques for Density Estimation Techniques for Charged Particle Beams Charged Particle Beams
BalBalšša Terzića TerzićBeam Physics and Astrophysics Group, NICADDBeam Physics and Astrophysics Group, NICADD
Northern Illinois University, USANorthern Illinois University, USA
Work done in collaboration with Work done in collaboration with Gabriele BassiGabriele Bassi
Cockcroft InstituteCockcroft Institute
Cockcroft Institute SeminarCockcroft Institute SeminarFebruary 26, 2009February 26, 2009
MotivationMotivation• When a charged particle beam traverses a curved trajectory, beam's selfinteraction leads to coherent synchrotron radiation (CSR)
• Beam's selfinteraction has adverse effects:
• Appreciable emittance degradation• Microbunching instability which degrades beam quality (major concern for FELs, which require bright electron beams)
• Proper numerical modeling of CSR requires:
• Pointtopoint methods: solving the microscopic Maxwell's equation (LienardWiechert potentials) • Mean field methods: solving VlasovMaxwell equation (finite difference, finite element, Green's function, retarded potentials)
• Possible simplifications to 3D CSR modeling:
• 1D line approximation (IMPACT, ELEGANT): probably too simplistic • 2D approximation: codes by Li 1998, Bassi et al. 2006
Comparison: The New Methods Vs. The OldComparison: The New Methods Vs. The OldEfficiencyEfficiency
Execution time Vs. N
cosine expansion: Ncx=40, N
cz=100
grid resolution: Nx=128, N
z=1024
(Ngrid
=131072)
t deposition
tMC cos
~ O1
N cx N cz
● Both alternative methods for density estimation presented here are about N
cxN
cz times faster than Monte Carlo cosine expansion
● In the present implementation of Bassi's code: t(density estimation) : t(field computation) = 40% : 60%, so making the density estimation ~103 faster by using these new methods speeds up the code by ~ 40%
Comparison: The New Methods Vs. The OldComparison: The New Methods Vs. The OldEfficiencyEfficiency
cosine expansion: Ncx=40, N
cz=100
grid resolution: Nx=128, N
z=1024
Execution time: The New Methods Vs. The Old
t/tMC cos
= 1/(NcxN
cz)
● We will present a detailed and systematic comparison of the new gridbased methods and the Monte Carlo cosine expansion (PAC 2009)
● With these alternative methods for representing beam density, we have optimized only one part of the simulation (density estimation)
● A more timeconsuming part of the algorithm is a field calculation.We will focus on optimizing the field calculation by exploiting the advantages afforded by wavelet formulation:
– Compact representation(<0.5% of grid coefficients needed)
– Sparsity of operators (already designed a waveletbased Poisson equation solver)
– Denoised representation: more accurate
Discussion of Further WorkDiscussion of Further Work
Sparsity Vs. N1%
0.1%
● We presented two new, gridbased techniques for density estimation in beams simulations as an alternative to Monte Carlo cosine expansion:
– Truncated FCT technique:
● Orders of magnitude faster; just as accurate (equivalent to MC cosine)– Waveletdenoised technique:
● Orders of magnitude faster and appreciably more accurate
● The simulation times are significantly reduced
● Further optimization is needed on the field calculation bottleneck
– Take advantage of wavelets (sparsity, denoising)
● Compare with Li's 2D CSR code for consistency
● Closing in on the big goal: having an accurate, efficient and trustworthy code which properly accounts for beam selfinteraction due to the CSR
SummarySummary
Auxiliary SlidesAuxiliary Slides
Wavelet DecompositionWavelet Decomposition
The continuous wavelet transform of a function f (t) is
mother wavelet with scale and translation dimensions s and respectively
s ,=∫−∞
∞
f t s , t dt
s ,t =1
s t−
s t
● Approximation – apply lowpass filter to Signal and downsample
● Detail – apply highpass filter to Signal and downsample