Nessun titolo diapositiva

1
New (iterative) methods for solving the nuclear
eigenvalue problem Pisa 05
2
An importance sampling algorithm for large scale
shell model calculations F. Andreozzi N. Lo
Iudice A. Porrino.
3
Currently adopted methods

• Stochastic methodology Monte Carlo (C.W. Johnson
  et al. PRL 92), suitable for ground state. Minus
  sign problem.
• Direct Diagonalization Lanczos (see G.H. Golub
  and C.F. Van Loan Matrix Computations 96).
  Critical point Sizes of the matrix.
• In between Quantum MC (M. Honma et al. PRL 95).
  MC to select the relevant basis states. Problems
  Redundancy, symmetries broken by the stochastic
  procedure.

4

• Diagonalization algorithm
• (A. Andreozzi, A. Porrino, and N.Lo Iudice J.
  Phys. A 02)Iterative generation of an
  eigenspace
• A ? Symmetric matrix representing a
  self-adjoint operator in an orthonormal basis
• 1 gt , 2 gt , , Ngt
• A ? aij lt i  j gt
• Lowest eigenvalue and eigenvector

5
a11 a12 a13 a14 ..
a1N a21 a22
a23 a24 .. a2N a31 a32 a33
a34 .. a3N a41 a42 a43
a44 .. a4N ..
aN1 .. aNN
6
  ?1 a11 f1 gt 1 gt basis 1 gt,
2 gt
Diagonalize the matrix
 
?2 , f2 gt
k1(2) 1 gt k2(2) 2 gt
7
Updated basis f2 gt, 3 gt Compute
b3 lt f2 Â 3 gt
k1(2) a13 k2(2) a23

Diagonalize the matrix
 


?3 , f3 gt S ki(3) i gt



i 1, 3
8
Updated basis fN-1 gt, N gt Compute bN
lt fN-1 Â N gt
Diagonalize
the matrix  
?N ?
E(1) , fN gt ?(1) gt ? ki(N) i
gt



i 1,
N End first iteration loop
9
First step of the second iteration
Def. f 1(2) gt ?(1) gt ?1(2)
E(1)
Compute b1 lt f1(2) Â 1 gt
the states f1(2) gt, 1gt are not
linearly independent
Generalized eigenvalue problem  
det ( - ?
) 0

10
E(1) , ?(1) E(2) , ?(2)

THEOREM If the sequence E(i) converges ,
then E(i) E (eigenvalue of
the matrix A) ?(i) ?
(eigenvector of the matrix A)
11
Simultaneous determination of v eigensolutions
The structure of the algorithm unchanged
12
?1 0 . 0 b11 .... b1j
0 ?2 . 0
b21 b2j . .
0 0 .. ?v bv1 .
bvj b11 .... bv1 a11 ..
a1j b12 bv2 a21 .. a2j
.
b1j . bvj aj1 .... ajj
13

• Easy implementation
• Variational foundation
• Robust
• Convergence to the extremal eigenvalues
• Numerically stable and ghost-free solutions
• Orthogonality of the computed eigenvectors
• Fast O( N2) operations

c
14
IMPORTANCE SAMPLING ? gt S ci
i gt i
1,Nlocalization property ? only m ( N )
ci important diagonalization algorithm
gives quite accurate solutions already in the
first approximation loop
15
Sampling procedure (F. Andreozzi, N. L.
A. Porrino, J. Phys. G 03)given e aij ?
? v diag (?i) (i,j 1, , v)
for j v1 , N diagonalize Aj
bj b1j , , bvj
?v bj bjT ajj
16

• select the v lowest eigenvalues
• ?1 , , ?v if S i 1,v ?i -
  ?i gt e accept the j th state
  
  end loop
• requires ? N ? ( v 1)3 operations

17
Importance sampling reduces by a factor
N / Nsampled the number of operations The
effectiveness of the reduction depends on the
localization properties of the wave
function Increase of the localization
through the use of a correlated basis ?
model space partitioning
18
Numerical Applications

• Semimagic nuclei 108Sn
• NZ 48Cr
• Ngt Z 133Xe

19

108Sn
1h11/2 3s1/2
2d3/2 1g7/2 2d5/2
Realistic effective interaction deduced from
Bonn A potential . Jp 2 N 17467
20
(No Transcript)
21
(No Transcript)
22
(No Transcript)
23
(No Transcript)
24
(No Transcript)
25
scaling with n (number of sampled states)
? (n) a b (N/n) exp(-c N/n)
e (n) (d/n2) exp(-c N/n) it allows for
high precision extrapolation n ? N
26
Heuristic argument

• consistency
• e(n) ? d? / dn
• from the sampling condition (one target
  state) ?? Sj ??j Sj bj2 / ( ajj - ?)
  

27
In the convergence region

• ajj - ? ? ann - ? ? n
• bj2 lt?j-1 H jgt2 (?j-1 ?i ci
  igt)
• small and random for j lt n
  
• zero for jgt
  n
• bj2 ? exp (- j/n)

28

• ?? ? b (N/n) exp(-c N/n)
• e(n) ? d? / dn
• ? (d/n2) exp(-c N/n)

29
Conclusions

• The algorithm is simple, robust and has a
  variational foundation
• Once endowed with the importance sampling,
• a) it keeps the extent of space truncation under
  strict control
• b) it allows for extrapolation to exact
  eigensolutions
• It is very promising for heavy nuclei
• It may be applied to systems others than nuclei
  (molecules, metal clusters, quantum dots etc.)

30
Nuclear Eigenvalue problem in a microscopic
multiphonon spaceIterative equation of motion
method

• Naples(Andreozzi, Lo Iudice, Porrino)
• Prague (Knapp, Kvasil)
• collaboration

31
Preliminary remarks

• Standard Shell model is exact and complete within
  a given model space.
• Often the model space is spanned by ?N 0 h?
• Thus it does not include the high-energy
  configurations building up the collective states.
• TDA or RPA act in a more restricted but more
  selective space (p-h or 2qp configurations up to
  n h?) and therefore are in general more suitable
  for collective excitations. They, however, do not
  account for anharmonic effects.
• A multiphonon space is needed for describing
  anharmonicity
• Problem with multiphonon space
• Antisymmetry
  Redundancy
• Proposed way out Equation of motion method

32
Eigenvalue problem Formulation

• Goal Solving
• H a gt Ea a gt
• in a multiphonon space spanned by
• 0 gt, 1, ?1 gt, 2, ?2 gt, . n, ?n gt
• where
• n, ?n gt ?1 ?2 ..?n gt
• ?i gt Sph C
  ph(?i ) Bph 0gt
• Sph Cph(?i ) ap
  ah 0gt

33

• lt n ?n H, Bph n-1 ?n-1 gt
• S? lt n ?n H n ? gt lt n ? Bph
  n-1 ?n-1 gt -
• S? lt n ?n Bph n-1 ? gt lt n-1 ? H
  n-1 ?n-1 gt
• lt n ?n H n ? gt E?n d?n?
• lt n-1 ? H n-1 ?n-1 gt E?n-1 d?n-1?

Amplitude

• lt n ?n H, Bph n-1 ?n-1 gt
• (E?n - E?n-1) lt n ?n Bph n-1 ?n-1
  gt.

34

• The commutator yields
• lt n ?n H, Bph n-1 ?n-1 gt
• (ep- eh) lt n ?n Bph n-1 ?n-1 gt
• Sph Gp h h p lt n ?n Bph n-1 ?n-1
  gt
• S Gpppp lt n ?n Bph Bpp
  n-1 ?n-1 gt
• S
  Gphhp lt n ?n Bph Bhh) n-1
  ?n-1 gt
• S Gph ph lt n ?n Bph Bpp
  n-1 ?n-1 gt )
• S
  Ghhhh lt n ?n Bph Bhh) n-1
  ?n-1 gt

Amplitudes
35
Linearization
Amplitudes
lt n ?n Bph Bpp n-1 ?n-1 gt
lt n ?n Bph Bhh) n-1 ?n-1
gt
lt n ?n Bph n-1 ?gt lt
n-1 ?Bpp n-1 ?n-1 gt
Î S ? n-1 ? gtlt n-1 ?
lt n ?n Bph n-1 ?gt lt
n-1 ?Bhh n-1 ?n-1 gt
Î S ? n-1 ? gtlt n-1 ?
36
Eigenvalue Equation

• ?(n) X (n)
  E?n X (n)
• Where
• X(n)?n-1 ph lt n ?n Bph n-1 ?n-1 gt
• (?(n))ph,ph(?n-1?n-1) A ph,phd?n-1?n-1
  
• Hpp(?n-1?n-1)dhh
• - Hhh(?n-1?n-1)dpp
• Aph,ph (ep eh E?n-1) dph,ph - Gphph
• Hpp(?n-1?n-1) Sh1h2 Gph1ph2 R h1h2
  (?n-1?n-1)
• - ½ Sp1p2 Gpp1p2p R p1p2 (?n-1?n-1)
• Hhh(?n-1?n-1) Sp1p2 Ghp1hp2 R p1p2
  (?n-1?n-1)
• - ½ Sh1h2 Ghh1h2h R h1h2 (?n-1?n-1)
• Rab(?n-1?n-1) lt n-1 ?n-1 Bab n-1
  ?n-1 gt

37
Redundancy

• The states
• Bph n-1 ?n-1 gt
• form an overcomplete linearly
• dependent set.
• Is there a way out? Yes

38

• Let us perform the expansion in the redundant
  basis
• n ?ngt S ?n-1ph Cph (?n ?n-1) Bph
  n-1?n-1 gt
• We obtain
• X (n)?n-1 ph lt n ?n Bph n-1 ?n-1 gt
• S ?n-1ph Cph(?n ?n-1) Dphph(?n-1
  ?n-1)
• where
• Dphph (?n-1 ?n-1)
• lt n-1 ?n-1 B ph
  Bph n-1 ?n-1 gt

39

• In matrix form
• X D C
• Therefore
• ? X E X
• (?D)C H C E DC
• This Eq. is ill defined with respect to inversion
  (D is singular)

40
The way out Choleski method

• Choleski selects a basis of linear independent
  states
• Bph n-1 ?n-1 gt
• spanning the physical subspace of the
• right dimension
• Ng lt N
• Using this basis, we compute a non singular
• matrix D and get
• (D-1?D)C EC

41

• Eq.
• (D-1?D)C E C
• yields Ng exact eigensolutions for the
• n-phonon subspace.
• We can now move to the (n1)-phonon subspace.
• We only need to know X(n) and R(n).
• X(n) is given by
• X D C

42

• R(n) is given by the recursive relations
• Rpp(?n?n) lt n ?n Bpp n ?n gt
• S ?n-1h Cph (?n ?n-1) X(?n)?n-1 ph
• S ?n-1?n-1p1h Cph (?n ?n-1) X(?n)?n-1 p1h
• Rpp(?n-1?n-1)
• Rhh(?n?n)
• S ?n-1p Cph (?n ?n-1) X(?n)?n-1 ph
• S ?n-1?n-1ph1 Cph1 (?n ?n-1) X(?n)?n-1 ph1
• Rhh(?n-1?n-1)

43
Outcome of iteration the Hamiltonian matrix

• E?0 H ?0 ?1 H ?0 ?2
  0 0
• E?1 0 . . .0 H ?1 ?2
  H ?1 ?3 0
• E?10.0 H ?1 ?2 H
  ?1 ?3 0
• E ?1 0.0 H ?1 ?2
  H ?1 ?3 0
• ..
• E?2 0.. 0
  H ?2 ?3 H ?2 ?4
• E?2
  0........ 0 H ?2 ?3H ?2 ?4
• 
  E?2 0..0 H ?2 ?3H ?2 ?4
• 
  ........................
• 
  E?3 0 ..0 H ?3 ?4
• 
  

44
The off diagonal terms

• are also computed by iteration
• lt n-1 ?n-1 H n ?n gt S (ph)k
  C(?n)(ph)k
• lt n-1 ?n-1 H, B(ph)k n-1 ?n-1k gt
  S l X(ph)l (?n-1 ?n-2) lt n-2 ?n-1l H n-1
  ?nk gt
• lt n-2 ?n-2 H n ?n gt S (ph)k
  C(?n)(ph)k
• lt n-2 ?n-2 H, B(ph)k n-1 ?n-1k gt
  
• X(ph)l (?n-2 ?n-3) lt n-3 ?n-3l H n-1
  ?n-1k gt

45
The Hamiltonian matrix

• E?0 H ?0 ?1 H ?0 ?2
  0 0
• E?1 0 . . .0 H ?1 ?2
  H ?1 ?3 0
• E?10.0 H ?1 ?2 H
  ?1 ?3 0
• E ?1 0.0 H ?1 ?2
  H ?1 ?3 0
• ..
• E?2 0.. 0
  H ?2 ?3 H ?2 ?4
• E?2
  0........ 0 H ?2 ?3H ?2 ?4
• 
  E?2 0..0 H ?2 ?3H ?2 ?4
• 
  ........................
• 
  E?3 0 ..0 H ?3 ?4
• 
  

46
Properties of H

• It is composed of central diagonal blocks
• Each block corresponds to a given n-phonon
  subspace
• A given n-block is coupled only to (n?1)- and
  (n?2)-blocks
• Partitioning Importance
  sampling
• Severe truncation

47
Status of art Program tests successfully
completed

• A 16
• Phonon space
• p-configurations ? d
• h-configurations ? s p-1
• Hamiltonian BonnA

48
Choleski effect Jp 0 T 1

• Two-phonon space
• 122
  26
• Three phonon space
• 3142
  329

49
Choleski effect Jp 3- T 1

• Two-phonon space
• 252
  62
• Three phonon space
• 14956
  1438

50
Future program

• Immediate applications
• Coupled scheme p-h.
• Detailed study of
• anharmonic effects on giant resonances
• Peculiar collective modes
• i. ISGDR (squeezed mode), which requires up to
  3h? p-h configurations
• ii. Twist mode (orbital M2 mode)
• iii. Double GDR

51
Future program

• In parallel
• Eq. of M. in uncoupled (spherical and deformed)
  scheme
• From p-h to qp to treat open shell nuclei as a
  cheap alternative to large scale shell model

52
More ambitious goal

• Combine SM (iterative algorithm) with Eq. of M.
  to enlarge the SM space and study i.e. intruders
• It si possible since the Eq. of M. formalism
  holds for any vacuum state.
• It can actually be used as alternative to SM in
  several cases (closed subshells)

53
(No Transcript)
54
In many cases the information of interest is
restricted to a few (usually low-lying) states
whose accurate description presumably requires
only a limited subset of the basis states
Identification of the relevant components implies
the knowledge of the wave function ? Adaptive
diagonalization algorithm
55
(No Transcript)
56
Bh ltfi(h-1) Â j gt i 1, , v

j 1, , p
à is a principal submatrix of A
Ã
57
(No Transcript)
58
a more efficient way through the similarity
transform A O -1 A
O
O ? v-dimensional row vector
Iv 0 ? 1
59
(No Transcript)
60
transformed matrix A wj - (?
bj) ? - ? ?v ajj ? bjT decoupling
condition wj 0 ? ajj - ? bj
eigenvalue of A
?vbj ? ? bj wj
ajj - ? bj
61
(No Transcript)
62
Decoupling condition can be recast in form of a
dispersion relation ? bj - Si1,v
bij2 / (ajj-?i - ? bj) ? (? bj)min
? ?max interlacing property of the
eigenvalues ? S i 1,v ?i -
?i (? bj)min gt e
63

• Choleski decomposition
• Any real non singular symmetric matrix can be
  written as
• D L LT
• Where L ?lij is a lower triangular matrix and
  LT its transpose
• DetD (DetL)2 l112 l222 ...lii2..

64

• The elements of L are recursively defined as
• l211 d11
• l11 lj1 dj1 j2,.,n
• l2ii dii Sk1,i-1 l2ik
• lii lji dji Sk1,i-1 lik ljk

65

• The decomposition goes on until
• lrr 0
• DetL 0 DetD
  0
• ?r gt is linearly dependent
• and is to be discarded

66

• Ordering
• A linearly independent basis may yield an
  overlap matrix ill defined with respect to
  inversion.
• To avoid this we arrange the basis in decreasing
  order
• ?ii ?jj ? j gt I
• This is automatically achieved
• if we choose at each step

67

• the vector yielding the maximum value of
• dii Sk1,i-1 l2ik
• 
  
• 
  
• dii Sk1,i-1 l2ik djj Sk1,j-1 l2jk
  ? j gt i
• ?ii ?jj ? j gt i

68
(No Transcript)
69
decompose accordingly the Hamiltonian
H H1 H2
H12 solve the eigenvalue equations
Hi ai Nigt Eai ai Ni gt replace
the standard shell model basis with
a N gt a1 N1 a2 N2 gt ? Orthonormal
correlated basis