Added LSM and Updated CGLS to work with broadcasted arrays

[rohanbabbar04]

For #46

• Updated CGLS to work with broadcasted arrays
• Added a test to verify the above using MPIHStack
• Added LSM in tutorials section.

[mrava87]

The tutorial looks very good!

I have just a small comment regarding the tests for cgls. Since you test a case with x and y with partition.SCATTER and one with y with partition.BROADCAST, I thought it would make sense to also have the last combination (x with partition.SCATTER). I give it a try and it seems to work, so let's just add it :)

@pytest.mark.mpi(min_size=2)
@pytest.mark.parametrize(
    "par", [(par1), (par1j), (par2), (par2j), (par3), (par3j), (par4), (par4j)]
)
def test_cgls_broadcastmodel(par):
    A = (rank + 1) * np.ones((par["ny"], par["nx"])) + (rank + 2) * par[
        "imag"
    ] * np.ones((par["ny"], par["nx"]))
    Aop = MatrixMult(A, dtype=par["dtype"])
    VStack_MPI = MPIVStack(ops=[Aop, ])

    x = DistributedArray(global_shape= par['nx'], dtype=par['dtype'], partition=Partition.BROADCAST)
    x[:] = np.random.normal(1, 10, par['nx']) + par["imag"] * np.random.normal(10, 10, par['nx'])
    x_global = x.asarray()
    if par["x0"]:
        x0 = DistributedArray(global_shape=par['nx'], dtype=par['dtype'], partition=Partition.BROADCAST)
        x0[:] = np.random.normal(0, 10, par["nx"]) + par["imag"] * np.random.normal(
            10, 10, par["nx"]
        )
        x0_global = x0.asarray()
    else:
        x0 = None

    y = VStack_MPI @ x
    assert y.partition is Partition.SCATTER

    xinv = cgls(VStack_MPI, y, x0=x0, niter=par["nx"], tol=1e-5, show=True)[0]
    assert isinstance(xinv, DistributedArray)
    xinv_array = xinv.asarray()
    if rank == 0:
        ops = [MatrixMult((i + 1) * np.ones((par["ny"], par["nx"])) + (i + 2) * par[
            "imag"
        ] * np.ones((par["ny"], par["nx"])), dtype=par['dtype']) for i in range(size)]
        Vstack = VStack(ops=ops)
        if par["x0"]:
            x0 = x0_global
        else:
            x0 = None
        y1 = Vstack @ x_global
        xinv1 = pylops.cgls(Vstack, y1, x0=x0, niter=par["nx"], tol=1e-5, show=True)[0]
        assert_allclose(xinv_array, xinv1, rtol=1e-14)

[mrava87]

@rohanbabbar04 I noticed one more thing in the test_solver, I'll raise an issue and we can discuss there :)

[rohanbabbar04]

@rohanbabbar04 I noticed one more thing in the test_solver, I'll raise an issue and we can discuss there :)

Sure...