Hi every one,
I have been using the SLEPc library (with fortran) for several years to solve large eigenvalues problems in parallel on distributed memory architectures.
Since I have started to use julia for a while (and really appreciate it!), I was missing this feature. So I’ve started to write a julia wrapper for this library : SlepcWrap.jl.
SLEPc can be considered as an “extension” to the well-known PETSc library; and heavily rely on it. There are several existing julia wrappers for the PETSc library, for instance : PETSc.jl and GridapPETSc.jl. However none of them were exactly fitting my needs so I have copied some parts of both packages to build my own wrapper : PetscWrap.jl. My concern was to have in hand a parallel and petsc-arch-flexible wrapper. Since I am new to the open-source community, I hope that I didn’t cause any trouble copying some code, I am open to change everything necessary to comply with licence issues.
For now, only the PETSc-SLEPc that I needed are wrapped, but don’t hesitate to contribute and/or post an issue to ask for a new feature.
Below, I have copied an example available in the repo (in the readme / docs) computing the eigenvalues of a 1D - Helmholtz problem (using finite-differences). It can be run using multiple cores : mpirun -n 4 julia helmholtz.jl:
In this example, we use the SLEPc to find the eigenvalues of the following Helmholtz equation:
u'' + \omega^2 u = 0 associated to Dirichlet boundary conditions on the domain [0,1]. Hence
the theoritical eigenvalues are \omega = k \pi with k \in \mathbb{Z}^*; and the associated
eigenvectors are u(x) = \sin(k \pi x).
A centered finite difference scheme is used for the spatial discretization.
The equation is written in matrix form Au = \alpha Bu where \alpha = \omega^2.
In this example, PETSc/SLEPc legacy method names are used. For more fancy names, check the next example.
Note that the way we achieve things in the document can be highly improved and the purpose of this example
is only demonstrate some method calls to give an overview.
Start by importing both PetscWrap, for the distributed matrices, and SlepcWrap for the eigenvalues.
using PetscWrap
using SlepcWrap
Number of mesh points and mesh step
n = 21
Δx = 1. / (n - 1)
Initialize SLEPc. Either without arguments, calling SlepcInitialize() or using “command-line” arguments.
To do so, either provide the arguments as one string, for instance
SlepcInitialize("-eps_max_it 100 -eps_tol 1e-5") or provide each argument in
separate strings : PetscInitialize(["-eps_max_it", "100", "-eps_tol", "1e-5").
Here we ask for the five closest eigenvalues to 0, using a non-zero pivot for the LU factorization and a
“shift-inverse” process.
SlepcInitialize("-eps_target 0 -eps_nev 5 -st_pc_factor_shift_type NONZERO -st_type sinvert")
Create the problem matrices, set sizes and apply “command-line” options. Note that we should
set the number of preallocated non-zeros to increase performance.
A = MatCreate()
B = MatCreate()
MatSetSizes(A, PETSC_DECIDE, PETSC_DECIDE, n, n)
MatSetSizes(B, PETSC_DECIDE, PETSC_DECIDE, n, n)
MatSetFromOptions(A)
MatSetFromOptions(B)
MatSetUp(A)
MatSetUp(B)
Get rows handled by the local processor
A_rstart, A_rend = MatGetOwnershipRange(A)
B_rstart, B_rend = MatGetOwnershipRange(B)
Fill matrix A with second order derivative central scheme
for i in A_rstart:A_rend
if(i == 1)
A[1, 1:2] = [-2., 1] / Δx^2
elseif (i == n)
A[n, n-1:n] = [1., -2.] / Δx^2
else
A[i, i-1:i+1] = [1., -2., 1.] / Δx^2
end
end
Fill matrix B with identity matrix
for i in B_rstart:B_rend
B[i,i] = -1.
end
Set boundary conditions : u(0) = 0 and u(1) = 0. Only the processor handling the corresponding rows are playing a role here.
(A_rstart == 1) && (A[1, 1:2] = [1. 0.] )
(B_rstart == 1) && (B[1, 1] = 0. )
(A_rend == n) && (A[n, n-1:n] = [0. 1.] )
(B_rend == n) && (B[n, n] = 0. )
Assemble the matrices
MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY)
MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY)
MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY)
MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY)
Now we set up the eigenvalue solver
eps = EPSCreate()
EPSSetOperators(eps, A, B)
EPSSetFromOptions(eps)
EPSSetUp(eps)
Then we solve
EPSSolve(eps)
And finally we can inspect the solution. Let’s first get the number of converged eigenvalues:
nconv = EPSGetConverged(eps)
Then we can get/display these eigenvalues (more precisely their square root, i.e \simeq \omega)
for ieig in 1:nconv
vpr, vpi = EPSGetEigenvalue(eps, ieig)
@show √(vpr), √(vpi)
end
We can also play with eigen vectors. First, create two Petsc vectors to allocate memory
vecr, veci = MatCreateVecs(A)
Then loop over the eigen pairs and retrieve eigenvectors
for ieig in 1:nconv
vpr, vpi, vecpr, vecpi = EPSGetEigenpair(eps, ieig, vecr, veci)
# At this point, you can call VecGetArray to obtain a Julia array (see PetscWrap examples).
# If you are on one processor, you can even plot the solution to check that you have a sinus
# solution. On multiple processors, this would require to "gather" the solution on one processor only.
end
Finally, let’s free the memory
MatDestroy(A)
MatDestroy(B)
EPSDestroy(eps)
And call finalize when you’re done
SlepcFinalize()
