Converting from sparse to dense to sparse again decreases density after constructing sparse matrix

Question

I am using scipy to generate a sparse finite difference matrix, constructing it initially from block matrices and then editing the diagonal to account for boundary conditions. The resulting sparse matrix is of the BSR type. I have found that if I convert the matrix to a dense matrix and then back to a sparse matrix using the scipy.sparse.BSR_matrix function, I am left with a sparser matrix than before. Here is the code I use to generate the matrix:

size = (4,4)

xDiff = np.zeros((size[0]+1,size[0]))
ix,jx = np.indices(xDiff.shape)
xDiff[ix==jx] = 1
xDiff[ix==jx+1] = -1

yDiff = np.zeros((size[1]+1,size[1]))
iy,jy = np.indices(yDiff.shape)
yDiff[iy==jy] = 1
yDiff[iy==jy+1] = -1

Ax = sp.sparse.dia_matrix(-np.matmul(np.transpose(xDiff),xDiff))
Ay = sp.sparse.dia_matrix(-np.matmul(np.transpose(yDiff),yDiff))

lap = sp.sparse.kron(sp.sparse.eye(size[1]),Ax) + sp.sparse.kron(Ay,sp.sparse.eye(size[0]))

#set up boundary conditions
BC_diag = np.array([2]+[1]*(size[0]-2)+[2]+([1]+[0]*(size[0]-2)+[1])*(size[1]-2)+[2]+[1]*(size[0]-2)+[2])

lap += sp.sparse.diags(BC_diag)

If I check the sparsity of this matrix I see the following:

lap
<16x16 sparse matrix of type '<class 'numpy.float64'>'
with 160 stored elements (blocksize = 4x4) in Block Sparse Row format>

However, if I convert it to a dense matrix and then back to the same sparse format I see a much sparser matrix:

sp.sparse.bsr_matrix(lap.todense())
<16x16 sparse matrix of type '<class 'numpy.float64'>'
with 64 stored elements (blocksize = 1x1) in Block Sparse Row format>

I suspect that the reason this is happening is because I constructed the matrix using the sparse.kron function but my question is if there is a way to arrive at the smaller sparse matrix without converting to dense first, for example if I end up wanting to simulate a very large domain.

Answer 1

BSR stores the data in dense blocks:

In [167]: lap.data.shape                                                        
Out[167]: (10, 4, 4)

In this case those blocks have quite a few zeros.

In [168]: lap1 = lap.tocsr() 
In [170]: lap1                                                                  
Out[170]: 
<16x16 sparse matrix of type '<class 'numpy.float64'>'
    with 160 stored elements in Compressed Sparse Row format>
In [171]: lap1.data                                                             
Out[171]: 
array([-2.,  1.,  0.,  0.,  1.,  0.,  0.,  0.,  1., -3.,  1.,  0.,  0.,
        1.,  0.,  0.,  0.,  1., -3.,  1.,  0.,  0.,  1.,  0.,  0.,  0.,
        1., -2.,  0.,  0.,  0.,  1.,  1.,  0.,  0.,  0., -3.,  1.,  0.,
        0.,  1.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  1., -4.,  1.,  0., 
        ...
        0.,  0.,  1., -2.])

In place cleanup:

In [172]: lap1.eliminate_zeros()                                                
In [173]: lap1                                                                  
Out[173]: 
<16x16 sparse matrix of type '<class 'numpy.float64'>'
    with 64 stored elements in Compressed Sparse Row format>

If I specify the csr format when using kron:

In [181]: lap2 = sparse.kron(np.eye(size[1]),Ax,format='csr') + sparse.kron(Ay,n
     ...: p.eye(size[0]), format='csr')                                         
In [182]: lap2                                                                  
Out[182]: 
<16x16 sparse matrix of type '<class 'numpy.float64'>'
    with 64 stored elements in Compressed Sparse Row format>