220
5 Formulation and Implementation of the Gamow Shell Model
matrix elements of the look-up procedure are not contiguous in general. Therefore,
the search process induces time lags absent from the full storage scheme.
Let us quantify the memory gain of the partial storage scheme compared to
the full storage scheme. In the full storage scheme, one stores the index of Slater
determinants |SD f , which is an integer of 4 bytes, and the Hamiltonian matrix
element, which is a complex number of 16 bytes. This adds up to 20 bytes. In the
partial storage scheme, one still has to store the index of Slater determinants |SD f
but, instead of the Hamiltonian matrix element, one stores an integer of 4 bytes,
from which the index of the two-body matrix element in an array and associated
phase can be recovered. Therefore, one stores 8 bytes in the partial storage scheme,
as compared to the 20 bytes of the full storage scheme, hence obtaining the gain
factor of 2.5. Consequently, the partial storage scheme is a compromise between
full storage and on-the-fly schemes. Indeed, the partial storage scheme is faster than
the on-the-fly scheme, as the Hamiltonian matrix is not reconstructed for every ˆ
H
times vector operation. However, the partial storage scheme still leads to significant
memory requirements, albeit not as large as with the full storage method.
5.8.3 2D Partitioning Parallelization Scheme of the Hamiltonian
A 2D partitioning scheme has been implemented in Gamow shell model [28].
It allows in particular to distribute the Gamow shell model vectors among all
nodes, so that memory usage is optimized for vectors as well. The symmetry
of the Hamltonian ˆ
H is included in this scheme. Moreover, the overlapping
between message passing interface communications and calculations has been
introduced therein. In this method, using a hybrid message passing interface/open
multiprocessing scheme, a single thread takes care of message passing interface
data transfer, while all the others are dedicated to matrix-vector multiplication. As
message passing interface communications become of the same order of magnitude
as matrix-vector multiplication when Gamow shell model dimensions increase, this
scheme allows to save a significant amount of time.
In the 2D partitioning scheme, ˆ
H is divided in n d (n d + 1)/2 squares, equal
to N, where both |SD i and |SD f Slater determinants scale with the number of
message passing interface nodes. The 1p-1h part of ˆ
H is inexpensive in terms
of both time and memory and hence will not be considered. Let us concentrate
firstly on the neutron 2p-2h part of ˆ
H . For each Slater determinant |SD i present
in the column, one loops over all intermediate configurations C int and Slater
determinants |SD int (see Sect. 5.8.2). One then obtains the SD int |a δ a γ |SD i
phase and associated one-body states. Using the same procedure on |SD int , one
generates SD f |a †
α a
†
β |SD int phase and associated one-body states, so that the twobody phase SD f |a †
α a
†
β a δ a γ |SD i is obtained with its one-body states.
5 Formulation and Implementation of the Gamow Shell Model
matrix elements of the look-up procedure are not contiguous in general. Therefore,
the search process induces time lags absent from the full storage scheme.
Let us quantify the memory gain of the partial storage scheme compared to
the full storage scheme. In the full storage scheme, one stores the index of Slater
determinants |SD f , which is an integer of 4 bytes, and the Hamiltonian matrix
element, which is a complex number of 16 bytes. This adds up to 20 bytes. In the
partial storage scheme, one still has to store the index of Slater determinants |SD f
but, instead of the Hamiltonian matrix element, one stores an integer of 4 bytes,
from which the index of the two-body matrix element in an array and associated
phase can be recovered. Therefore, one stores 8 bytes in the partial storage scheme,
as compared to the 20 bytes of the full storage scheme, hence obtaining the gain
factor of 2.5. Consequently, the partial storage scheme is a compromise between
full storage and on-the-fly schemes. Indeed, the partial storage scheme is faster than
the on-the-fly scheme, as the Hamiltonian matrix is not reconstructed for every ˆ
H
times vector operation. However, the partial storage scheme still leads to significant
memory requirements, albeit not as large as with the full storage method.
5.8.3 2D Partitioning Parallelization Scheme of the Hamiltonian
A 2D partitioning scheme has been implemented in Gamow shell model [28].
It allows in particular to distribute the Gamow shell model vectors among all
nodes, so that memory usage is optimized for vectors as well. The symmetry
of the Hamltonian ˆ
H is included in this scheme. Moreover, the overlapping
between message passing interface communications and calculations has been
introduced therein. In this method, using a hybrid message passing interface/open
multiprocessing scheme, a single thread takes care of message passing interface
data transfer, while all the others are dedicated to matrix-vector multiplication. As
message passing interface communications become of the same order of magnitude
as matrix-vector multiplication when Gamow shell model dimensions increase, this
scheme allows to save a significant amount of time.
In the 2D partitioning scheme, ˆ
H is divided in n d (n d + 1)/2 squares, equal
to N, where both |SD i and |SD f Slater determinants scale with the number of
message passing interface nodes. The 1p-1h part of ˆ
H is inexpensive in terms
of both time and memory and hence will not be considered. Let us concentrate
firstly on the neutron 2p-2h part of ˆ
H . For each Slater determinant |SD i present
in the column, one loops over all intermediate configurations C int and Slater
determinants |SD int (see Sect. 5.8.2). One then obtains the SD int |a δ a γ |SD i
phase and associated one-body states. Using the same procedure on |SD int , one
generates SD f |a †
α a
†
β |SD int phase and associated one-body states, so that the twobody phase SD f |a †
α a
†
β a δ a γ |SD i is obtained with its one-body states.