def hodge_decomposition(omega):
"""
For a given p-cochain \omega there is a unique decomposition
\omega = d(\alpha) + \delta(\beta) (+) h
for p-1 cochain \alpha, p+1 cochain \beta, and harmonic p-cochain h.
This function returns (non-unique) representatives \beta, \gamma, and h
which satisfy the equation above.
Example:
#decompose a random 1-cochain
sc = SimplicialComplex(...)
omega = sc.get_cochain(1)
omega.[:] = rand(*omega.shape)
(alpha,beta,h) = hodge_decomposition(omega)
"""
sc = omega.complex
p = omega.k
alpha = sc.get_cochain(p - 1)
beta = sc.get_cochain(p + 1)
# Solve for alpha
A = delta(d(sc.get_cochain_basis(p - 1))).v
b = delta(omega).v
alpha.v = cg(A, b, tol=1e-8)[0]
# Solve for beta
A = d(delta(sc.get_cochain_basis(p + 1))).v
b = d(omega).v
beta.v = cg(A, b, tol=1e-8)[0]
# Solve for h
h = omega - d(alpha) - delta(beta)
return (alpha, beta, h)
def hodge_decomposition(omega):
"""
For a given p-cochain \omega there is a unique decomposition
\omega = d(\alpha) + \delta(\beta) (+) h
for p-1 cochain \alpha, p+1 cochain \beta, and harmonic p-cochain h.
This function returns (non-unique) representatives \beta, \gamma, and h
which satisfy the equation above.
Example:
#decompose a random 1-cochain
sc = SimplicialComplex(...)
omega = sc.get_cochain(1)
omega.[:] = rand(*omega.shape)
(alpha,beta,h) = hodge_decomposition(omega)
"""
sc = omega.complex
p = omega.k
alpha = sc.get_cochain(p - 1)
beta = sc.get_cochain(p + 1)
# Solve for alpha
A = delta(d(sc.get_cochain_basis(p - 1))).v
b = delta(omega).v
alpha.v = cg( A, b, tol=1e-8 )[0]
# Solve for beta
A = d(delta(sc.get_cochain_basis(p + 1))).v
b = d(omega).v
beta.v = cg( A, b, tol=1e-8 )[0]
# Solve for h
h = omega - d(alpha) - delta(beta)
return (alpha,beta,h)
def plotjson(fn):
"""
plotjson: make plots from json output of fiedler.py
fn: the filename of the json file
"""
fo=open(fn)
data=json.load(fo)
fo.close()
if "adj" in data:
(A,adj,Npts) = fiedler.adj_mat(data["adj"])
#scew symetricise
A = (A.T - A)/2
A=A.tocoo()
pos=A.data>0
skew = numpy.column_stack((A.row[pos],A.col[pos],A.data[pos])).tolist()
# #method from hodge decomposition driver.py
# sc = simplicial_complex(([[el] for el in range(0,A.shape[0])],numpy.column_stack((A.row[pos],A.col[pos])).tolist()))
# omega = sc.get_cochain(1)
# omega.v[:] = A.data[pos]
# p = omega.k
# alpha = sc.get_cochain(p - 1)
# #beta = sc.get_cochain(p + 1)
# # Solve for alpha
# A2 = delta(d(sc.get_cochain_basis(p - 1))).v
# b = delta(omega).v
# rank=cg( A2, b, tol=1e-8 )[0]
# method from ranking driver.py
asc = abstract_simplicial_complex([numpy.column_stack((A.row[pos],A.col[pos])).tolist()])
B1 = asc.chain_complex()[1] # boundary matrix
rank = lsqr(B1.T, A.data[pos])[0] # solve least squares problem
sc = simplicial_complex(([[el] for el in range(0,A.shape[0])],numpy.column_stack((A.row[pos],A.col[pos])).tolist()))
omega = sc.get_cochain(1)
omega.v[:] = A.data[pos]
p = omega.k
alpha = sc.get_cochain(p - 1)
alpha.v = rank
v = A.data[pos]-d(alpha).v
cyclic_adj_list=numpy.column_stack((A.row[pos],A.col[pos],v)).tolist()
div_adj_list=numpy.column_stack((A.row[pos],A.col[pos],d(alpha).v)).tolist()
data["hodge"]=list(rank)
data["hodgerank"]=list(numpy.argsort(numpy.argsort(rank)))
fo = open(fn,"w")
json.dump(data,fo, indent=2)
fo.close()
fn=fn+".abstract"
#fiedler.doPlots(numpy.array(data["f1"]),numpy.array(data["f2"]),numpy.array(data["d"]),cyclic_adj_list,fn+".decomp.cyclic.",widths=[6],vsdeg=False,nByi=data["nByi"],directed=True)
#fiedler.doPlots(numpy.array(data["f1"]),numpy.array(data["f2"]),numpy.array(data["d"]),div_adj_list,fn+".decomp.acyclic.",widths=[6],vsdeg=False,nByi=data["nByi"],directed=True)
#fiedler.doPlots(numpy.array(data["f1"]),numpy.array(data["f2"]),numpy.array(data["d"]),data["adj"],fn+".decomp.acyclic.over.all.",widths=[6],vsdeg=False,nByi=data["nByi"],adj_list2=div_adj_list,directed=True)
#fiedler.doPlots(numpy.array(data["f1"]),-1*numpy.array(rank),numpy.array(data["d"]),cyclic_adj_list,fn+".decomp.harmonic.v.grad.",widths=[6],heights=[2],vsdeg=False,nByi=data["nByi"],directed=True)
#fiedler.doPlots(numpy.array(data["f1"]),-1*numpy.array(rank),numpy.array(data["d"]),skew,fn+".decomp.skew.v.grad.",widths=[6],heights=[2],vsdeg=False,nByi=data["nByi"],directed=True)
#fiedler.doPlots(numpy.array(data["f1"]),-1*numpy.array(rank),numpy.array(data["d"]),data["adj"],fn+".decomp.acyclic.over.all.v.grad.",widths=[6],heights=[2],vsdeg=False,nByi=data["nByi"],adj_list2=div_adj_list,directed=True)
fiedler.doPlots(numpy.array(data["f1"]),-1*numpy.array(rank),numpy.array(data["d"]),data["adj"],fn+".all.v.grad.",widths=[24],heights=[6],vsdeg=False,nByi=data["nByi"],directed=False)
