SlepcWrap.jl

SlepcWrap.jl is a parallel Julia wrapper for the (awesome) SLEPc library. As described on their main page, "SLEPc is a software library for the solution of large scale sparse eigenvalue problems on parallel computers. It is an extension of PETSc and can be used for linear eigenvalue problems in either standard or generalized form, with real or complex arithmetic.".

Note that as SLEPc is an extension of PETSc, SlepcWrap.jl is an extension of PetscWrap.jl.

The project is far from covering all SLEPc methods, but adding a new wrapper is very quick and easy.

How to install it

You must have installed the SLEPc library (and necessarily the PETSc library as wall) on your computer and set the two following environment variables : SLEPC_DIR and PETSC_ARCH.

At run time, PetscWrap.jl looks for the libslepc.so using these environment variables and "load" the library.

To install the package, use the Julia package manager:

pkg> add SlepcWrap

Contribute

Any contribution(s) and/or remark(s) are welcome! If you need a function that is not wrapped yet but you don't think you are capable of contributing, post an issue with a minimum working example.

SLEPc compat.

This version of PetscWrap.jl has been tested with slepc-3.13.1. Complex numbers are supported.

How to use it

SLEPc methods wrappers share the same name as their C equivalent : for instance EPSCreate or EPSGetEigenvalue. Furthermore, an optional "higher level" API, referred to as "fancy", is exposed : for instance create_eps or get_eig). Note that this second way of manipulating SLEPc will evolve according the package's author needs while the first one will try to follow SLEPc official API.

You will find examples of use by building the documentation: julia SlepcWrap.jl/docs/make.jl. Here is one of the examples:

Helmholtz equation

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\pix). 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.

To run this example, simplfy excute mpirun -n your_favourite_integer julia helmholtz_FD.jl

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-1
    if (i == 0)
        MatSetValues(A, [0], [0, 1], [-2., 1] / Δx^2, INSERT_VALUES)
    elseif (i == n-1)
        MatSetValues(A, [n-1], [n-2, n-1], [1., -2.] / Δx^2, INSERT_VALUES)
    else
        MatSetValues(A, [i], i-1:i+1, [1., -2., 1.] / Δx^2, INSERT_VALUES)
    end
end

Fill matrix B with identity matrix

for i in B_rstart:B_rend-1
    MatSetValue(B, i, i, -1., INSERT_VALUES)
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 == 0) && MatSetValues(A, [0], [0,1], [1., 0.], INSERT_VALUES)
(B_rstart == 0) && MatSetValue(B, 0, 0, 0., INSERT_VALUES)

(A_rend == n) && MatSetValues(A, [n-1], [n-2,n-1], [0., 1.], INSERT_VALUES)
(B_rend == n) && MatSetValue(B, n-1, n-1, 0., INSERT_VALUES)

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 0:nconv - 1
    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 0:nconv-1
    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()