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The most time-consuming part of the static codes is BdG matrix diagonalization. It is important how the BdG matrix is decomposed among the computing process in order to efficiently utilize `st-wslda` codes. The decomposition depends on the dimensionality version of the code. The most time-consuming part of the static codes is BdG matrix diagonalization. It is important to know how the BdG matrix is decomposed in the computing process to utilize `st-wslda` codes efficiently. The decomposition depends on the dimensionality version of the code.
# st-wslda-3d # st-wslda-3d
## Block-cyclic decomposition of BdG matrix ## Block-cyclic decomposition of BdG matrix
The code diagonalizes one matrix per iteration of size `matrix_size=2*NX*NY*NZ`. The matrix is decomposed between MPI process in block-cyclic (bc) fashion. To understand the idea of bc decomposition let us suppose that our (artificial) BdG matrix has size 9x9 (in practice this cannot happen since the size will be always an even number). The code will be executed on `np=6` processes: The code diagonalizes one matrix per iteration of size `matrix_size=2*NX*NY*NZ`. The matrix is decomposed between MPI processes in a block-cyclic (bc) fashion. To understand the idea of bc decomposition, let us suppose that our (artificial) BdG matrix has size 9x9 (in practice, this cannot happen since the size will always be an even number). The code will be executed on `np=6` processes:
```bash ```bash
mpirun -np 6 ./st-wslda-3d input.txt mpirun -np 6 ./st-wslda-3d input.txt
``` ```
...@@ -19,20 +19,20 @@ nb 2 ...@@ -19,20 +19,20 @@ nb 2
``` ```
Then the matrix will be distributed in the following fashion: Then the matrix will be distributed in the following fashion:
![bc-decomp](uploads/2cffebe8b09bbb6fef13c909065b3899/bc-decomp.png) ![bc-decomp](uploads/2cffebe8b09bbb6fef13c909065b3899/bc-decomp.png)
where by colors we indicate matrix elements that are handled by different MPI processes (6 different colors): whereby colors indicate matrix elements that are handled by different MPI processes (6 different colors):
![bc-decomp-2](uploads/d5d9bf64c421faefecd6967fde32dd66/bc-decomp-2.png) ![bc-decomp-2](uploads/d5d9bf64c421faefecd6967fde32dd66/bc-decomp-2.png)
To learn more about bc decomposition see [here](http://wlazlowski.fizyka.pw.edu.pl/pdfs/dydaktyka/NTO/meeting-8.pdf). To learn more about bc decomposition, see [here](http://wlazlowski.fizyka.pw.edu.pl/pdfs/dydaktyka/NTO/meeting-8.pdf).
## Selecting p, q, mb and nb ## Selecting p, q, mb and nb
By construction following constraint must be satisfied: `np=p*q`. When selecting decomposition parameters user should take into account: By construction, the following constraint must be satisfied: `np=p*q`. When selecting decomposition parameters, the user should take into account:
* Typically, best performance is achieved for `p=q`, thus it is recommended to select `p` and `q` to be as close as possible. If `p` and `q` are commented out then the algorithm will select automatically their values to satisfy this requirement. * Typically, the best performance is achieved for `p=q`; thus, it is recommended to select `p` and `q` to be as close as possible. If `p` and `q` are commented out, then the algorithm will automatically select their values to satisfy this requirement.
*Note*: Typically, the constraint `p=q` cannot be satisfied. Then we empirically found that setting with `p<q` gives a better performance than settings with `p>q`. For example, if we run code with `np=24` than we have two options `(p,q)=(4,6)` or `(p,q)=(6,4)`. According to our expertise we expect that setting `(p,q)=(4,6)` will provide better performance of the computation process. *Note*: Typically, the constraint `p=q` cannot be satisfied. Then we empirically found that setting with `p<q` gives a better performance than settings with `p>q`. For example, if we run code with `np=24` than we have two options `(p,q)=(4,6)` or `(p,q)=(6,4)`. According to our expertise, we expect that setting `(p,q)=(4,6)` will provide better performance of the computation process.
* By construction bock sizes must satisfy the following constraints: `mb<=matrix_size/p` and `nb<=matrix_size/q`. In case of decomposition codes (like diagonalization) settings block size to their maximal values does not provide good performance. We find that the best performance is obtained if `mb` and `nb` are much smaller than their maximal allowed values and at the same time number of matrix elements in the block `mb*nb` is significant (of the order of hundred or higher). Empirically we find that typically good performance is obtained for block sizes to be 16, 32, 64 (powers of 2). We recommend the user to try with these values, and based on results decide if further increase or decrease is profitable. * By construction, block sizes must satisfy the following constraints: `mb<=matrix_size/p` and `nb<=matrix_size/q`. In the case of decomposition codes (like diagonalization), settings block sizes to their maximal values do not perform well. We find that the best performance is obtained if `mb` and `nb` are much smaller than their maximal allowed values and at the same time number of matrix elements in the block `mb*nb` is significant (of the order of a hundred or higher). Empirically we find that typically good performance is obtained for block sizes to be 16, 32, 64 (powers of 2). We recommend the user try these values and based on the results, decide if further increase or decrease is profitable.
* If `p=q=1` then the parallelization is not applied. It corresponds to the single CPU version of the code. The code should work for these settings as well (do not expect that you will be able to solve large problems then). * If `p=q=1`, then the parallelization is not applied. It corresponds to the single CPU version of the code. The code should work for these settings as well (do not expect that you will be able to solve large problems then).
# st-wslda-2d # st-wslda-2d
In this variant the code assumes that the quasi-particle wave functions have form: In this variant, the code assumes that the quasi-particle wave functions have the form:
```math ```math
\psi(x,y,z)=\varphi(x,y)\frac{1}{\sqrt{L_z}}e^{ik_z z} \psi(x,y,z)=\varphi(x,y)\frac{1}{\sqrt{L_z}}e^{ik_z z}
``` ```
...@@ -40,11 +40,11 @@ where ...@@ -40,11 +40,11 @@ where
```math ```math
k_z = 0, \pm 1 \frac{2\pi}{L_z}, \pm 2 \frac{2\pi}{L_z}, \ldots , +(N_z-1) \frac{2\pi}{L_z} k_z = 0, \pm 1 \frac{2\pi}{L_z}, \pm 2 \frac{2\pi}{L_z}, \ldots , +(N_z-1) \frac{2\pi}{L_z}
``` ```
and $`L_z = NZ*DZ`$ is the box length along z-direction. From physical point of view, it means that we impose translation symmetry along z-direction. Under this assumption BdG matrix acquires block-diagonal form: and $`L_z = NZ*DZ`$ is the box length along the z-direction. From the physical point of view, it means that we impose translation symmetry along the z-direction. Under this assumption, BdG matrix acquires block-diagonal form:
![HBdG-2d](uploads/9659540e2c865b7a6736e55cec10aa50/HBdG-2d.png) ![HBdG-2d](uploads/9659540e2c865b7a6736e55cec10aa50/HBdG-2d.png)
and diagonalization of the matrix is equivalent to diagonalizations of submatrices $`H(k_{z,i})`$, each of them of size `matrix_size=2*NX*NY`. Moreover, the translation symmetry imposes that $`H(k_{z})=H(-k_{z})`$ and in practice it is sufficient to diagonalize only submatrices for positive $`k_z`$, which takes `NZ/2` values. Submatrices can be diagonalized simultaneously. and diagonalization of the matrix is equivalent to diagonalizations of submatrices $`H(k_{z,i})`$, each of them of size `matrix_size=2*NX*NY`. Moreover, the translation symmetry imposes that $`H(k_{z})=H(-k_{z})`$ and in practice it is sufficient to diagonalize only submatrices for positive $`k_z`$, which takes `NZ/2` values. Submatrices can be diagonalized simultaneously.
To demonstrate parallelization scheme in 2D case, let us consider following lattice: To demonstrate the parallelization scheme in 2D case, let us consider the following lattice:
```c ```c
#define NX 8 #define NX 8
#define NY 10 #define NY 10
...@@ -60,7 +60,7 @@ and we execute code with `np=24` processes: ...@@ -60,7 +60,7 @@ and we execute code with `np=24` processes:
```bash ```bash
mpirun -np 24 ./st-wslda-3d input.txt mpirun -np 24 ./st-wslda-3d input.txt
``` ```
For these settings, the single iteration requires `NZ/2=6` diagonalizations. The total set of processes will be dived into subgroups, each of size `p*q=6`. Thus the number of subgroups will be `24/6=4`. Each submatrix will be decomposed in block-cyclic fashion among `p*q` processes as in 3D case. For these settings, the single iteration requires `NZ/2=6` diagonalizations. The total set of processes will be divided into subgroups, each of size `p*q=6`. Thus the number of subgroups will be `24/6=4`. Each submatrix will be decomposed in the block-cyclic fashion among `p*q` processes as in 3D case.
This information is provided in the code output: This information is provided in the code output:
``` ```
# CODE: ST-WSLDA-2D # CODE: ST-WSLDA-2D
...@@ -81,7 +81,7 @@ This information is provided in the code output: ...@@ -81,7 +81,7 @@ This information is provided in the code output:
# CREATING CBLACS GRIDs OF SIZE (pzheev): [2 x 3] # CREATING CBLACS GRIDs OF SIZE (pzheev): [2 x 3]
``` ```
Note that here hamiltonian size means the size of submatrix `160=2*8*10`. Note that here hamiltonian size means the size of submatrix `160=2*8*10`.
The computation process for single iteration is presented schematically in the figure below: The computation process for a single iteration is presented schematically in the figure below:
![pscheme-2d](uploads/4792eecc243913bfd2e18d0fef09c90e/pscheme-2d.png) ![pscheme-2d](uploads/4792eecc243913bfd2e18d0fef09c90e/pscheme-2d.png)
and it is reflected in the code output: and it is reflected in the code output:
...@@ -102,9 +102,9 @@ and it is reflected in the code output: ...@@ -102,9 +102,9 @@ and it is reflected in the code output:
# Number of nwf in [-ecut,+ecut] to be extracted is: 976 (50.8% of total number of states) # Number of nwf in [-ecut,+ecut] to be extracted is: 976 (50.8% of total number of states)
``` ```
## Selecting p, q, and np ## Selecting p, q, and np
By construction following constraint must be satisfied: `np/(p*q) = i` where `i` is an integer number (number of subgroups) and `i<=NZ/2`. If `p` and `q` are specified in the input file, the code will select them in a such way to maximize the number of simultaneous diagonalizations. Concerning matrix distribution, all requirements as described for 3D case apply here as well. By construction, the following constraint must be satisfied: `np/(p*q) = i` where `i` is an integer number (number of subgroups) and `i<=NZ/2`. If `p` and `q` are specified in the input file, the code will select them in such a way as to maximize the number of simultaneous diagonalizations. Concerning matrix distribution, all requirements as described for 3D case apply here as well.
# st-wslda-1d # st-wslda-1d
The diagonalization scheme for 1D code is the same as for 2D code, wich modification that now submatrices depend on two wave-vectors $`H(k_{y,i}, k_{z,j})`$. The diagonalization scheme for 1D code is the same as for 2D code, which modification that now submatrices depend on two wave-vectors $`H(k_{y,i}, k_{z,j})`$.
In 1D case typically we need to diagonalize a large number of small matrices of size `matrix_size=2*NX`. For small sizes, of the order of thousand, typically the best performance is obtained for `p=q=1`, i.e. no block-cyclic distribution of the submatrices. In 1D case, typically, we need to diagonalize a large number of small matrices of size `matrix_size=2*NX`. For small sizes of the order of thousand, typically, the best performance is obtained for `p=q=1`, i.e. no block-cyclic distribution of the submatrices.
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