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CorAL and the Future of Imaging

CorAL and the Future of Imaging. Outline: What is CorAL Tour of components: A test problem Imaging New bases.

jacqueline-talley
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CorAL and the Future of Imaging

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  1. CorAL and the Future of Imaging Outline: What is CorAL Tour of components: A test problem Imaging New bases This work was performed under the auspices of the U.S. Department of Energy by University of California, Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.

  2. The Correlation Algorithm Library Source & correlation data structures Various 1d Cartesian harmonics Spherical harmonics 3d histograms OSCAR formatted input Various model sources Imaging tools Fitting tools in development Kernels Oodles of wavefunctions Sample codes Status: It (sortof) runs, but we’re still cleaning it up Documentation in progress Code collaboration agreements available (see me later) Official GPL release “any time now” Platforms: Linux, Unix, MacOSX Written in C++ Requires Gnu Scientific Library What is CorAL?

  3. CorAL components • Core libraries • libcoral: core CorAL library • libcoralutils: utilitycodes CorAL uses • Binaries • CHUM: freeze-out point (emission function) generator • SHARK: precompute the kernels for CRAB and DIVER • CRAB: constructs correlations and sources from OSCAR data • DIVER: imaging code • plotting codes (bfplot, scplot, converts2c) • Various component tests

  4. Core Halo Ur-Model (CHUM)      Variation of the Core-Halo model of Nickerson, Csörgo˝, Kiang, Phys. Rev. C 57, 3251 (1998), etc. • Blast-wave like flow profile • Gaussian source with finite source lifetime •  production from source and resonance decay The emission function: f is fraction of’semitted directly from core, i.e.= f 2 in source.

  5. CHUM cont. • From exploding core, with Gaussian shape: • Use full 3-body decay kinematics and lifetime of =23 fm/c: • Blast-wave like flow profile: where , and

  6. CHUM source is non-Gaussian in 3d Set Rx=Ry=Rz=4 fm, f/o=10 fm/c, T=165 MeV, f=0.5 Difference in side, out and long directions mean tail in higher lm terms as well as the 00 term. The out-long tail due to lifetime of source.

  7. CoRrelation AfterBurner (CRAB) CRAB’s role has increased: it now computes sources and correlations from OSCAR data The Koonin-Pratt equation: Pair final state relative wave-function, q(r), defines the kernel: K(q,r)=|q(r)|2-1. Source function is related to emission function: Work in Bertsch-Pratt coordinates in pair Centre of Mass (CM) frame

  8. Breaking Problem into 1d Problems Where Expand in Ylm’s and Legendre polynomials: • l = 0 :Angle averaged correlation, get access toRinv • l= 1 :Access to Lednicky offset, i.e. who emitted first (unlike only) • l= 2 :Shape information, access to RO, RS, RL: C20RL C00-(C20±C22) RS, RO • l= 3 :Boomerang/triaxial deformation (unlike only) • l= 4 :Squares off shape Cartesian harmonics give analogous expressions

  9. Use CRAB to generate Slm(r)

  10. Use CRAB to generate Clm(q) Note: Coulomb turned off

  11. Demonstration InVERter (DIVER) imaging is an ill-posed problem Practical solution to linear inverse problem, minimize : Most probable source is: With covariance matrix: • kernel not square & may be singular • noisy data • error propagation Convert Koonin-Pratt equation to matrix form:

  12. Radial dependence of each term in terms of basis functions: Basis choice affects sensitivity to correlation. Characterize with l=0 kernel: Many bases to test: Orthogonal polynomials on interval (-1,1): Legendre polynomials Chebyshev polynomials Orthogonal polynomials on interval (0,∞): Laguerre polynomials Hermite polynomials Basis Splines Others possible: Spherical Bessel functions Coulomb wave functions What about radial basis? A crappy basis gives bad source with small uncertainty

  13. Basis Spline basis • Previous versions of CorAL (and HBTprogs) used Basis Splines: • Nb=0 is histogram, • Nb=1 is linear interpolation, • Nb=3 is equivalent to cubic interpolation. • Resolution controlled by knot placement • Knot placement controlled by • qscale parameter • Moving Nc-Nb-1 knots by hand • Local in r-space means non-local in q-space: • High q wiggles • Tails in r too sensitive to high q

  14. Optimizing Basis Spline basis tough Imaging for l=2 terms is problematic with Basis Spline basis: two length scales important in source (core & tail)

  15. Legendre polynomial basis • Orthonormal polynomial over fit range • Only 2 parameters: fit range and number of coefficients • Non-local in r-space, local in q-space • wiggles in q reduced • Chebyshev polynomials have very similar behavior

  16. Laguerre function basis • Polynomial times exponential weight • 0th term is exponential • Higher order terms encode deviation from exponential • Only can tune exponential weight radius, fit range, and number coefficients • May have trouble with non- exponential tails • Highly non-local in r-space so localized to low-q • Hermite basis has similar behavior

  17. First results are promising… S00(r) (fm-3) r (fm)

  18. Other terms in the imaged source

  19. CorAL developers and  testers DAVE BROWN JASON NEWBY MIKE HEFFNER RON SOLTZ NATHANIEL BROWN-PEREZ AKITOMO ENOKIZONO LI YANG PAWEL DANIELEWICZ SCOTT PRATT Lawrence Livermore National Laboratory

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