def run_example(with_plots=True):
r"""
Example to demonstrate the use of the Sundials solver Kinsol
for the simple equation :math:`0 = 1 - y`
on return:
- :dfn:`alg_mod` problem instance
- :dfn:`alg_solver` solver instance
"""
#Define the res
def res(y):
return 1 - y
#Define an Assimulo problem
alg_mod = Algebraic_Problem(res, y0=0, name='Simple KINSOL Example')
#Define the KINSOL solver
alg_solver = KINSOL(alg_mod)
#Solve
y = alg_solver.solve()
#Basic test
nose.tools.assert_almost_equal(y, 1.0, 5)
return alg_mod, alg_solver
def run_example(with_plots=True):
r"""
Example to demonstrate the use of the Sundials solver Kinsol with
a user provided Jacobian.
on return:
- :dfn:`alg_mod` problem instance
- :dfn:`alg_solver` solver instance
"""
#Define the res
def res(y):
r1 = 2 * y[0] + 3 * y[1] - 6
r2 = 4 * y[0] + 9 * y[1] - 15
return N.array([r1, r2])
def jac(y):
return N.array([[2., 3.], [4., 9.]])
#Define an Assimulo problem
alg_mod = Algebraic_Problem(res,
y0=[0, 0],
jac=jac,
name='KINSOL example with Jac')
#Define the KINSOL solver
alg_solver = KINSOL(alg_mod)
#Sets the parameters
#Solve
y = alg_solver.solve()
#Basic test
nose.tools.assert_almost_equal(y[0], 1.5, 5)
nose.tools.assert_almost_equal(y[1], 1.0, 5)
return alg_mod, alg_solver
def run_example(with_plots=True):
r"""
Example to demonstrate the use of the Sundials solver Kinsol with
a user provided Jacobian and a preconditioner. The example is the
'Problem 4' taken from the book by Saad:
Iterative Methods for Sparse Linear Systems.
on return:
- :dfn:`alg_mod` problem instance
- :dfn:`alg_solver` solver instance
"""
#Read the original matrix
A_original = IO.mmread(os.path.join(file_path, "kinsol_ors_matrix.mtx"))
#Scale the original matrix
A = SPARSE.spdiags(1.0 / A_original.diagonal(), 0,
len(A_original.diagonal()), len(
A_original.diagonal())) * A_original
#Preconditioning by Symmetric Gauss Seidel
if True:
D = SPARSE.spdiags(A.diagonal(), 0, len(A_original.diagonal()),
len(A_original.diagonal()))
Dinv = SPARSE.spdiags(1.0 / A.diagonal(), 0,
len(A_original.diagonal()),
len(A_original.diagonal()))
E = -SPARSE.tril(A, k=-1)
F = -SPARSE.triu(A, k=1)
L = (D - E).dot(Dinv)
U = D - F
Prec = L.dot(U)
solvePrec = LINSP.factorized(Prec)
#Create the RHS
b = A.dot(N.ones((A.shape[0], 1)))
#Define the res
def res(x):
return A.dot(x.reshape(len(x), 1)) - b
#The Jacobian
def jac(x):
return A.todense()
#The Jacobian*Vector
def jacv(x, v):
return A.dot(v.reshape(len(v), 1))
def prec_setup(u, f, uscale, fscale):
pass
def prec_solve(r):
return solvePrec(r)
y0 = S.rand(A.shape[0])
#Define an Assimulo problem
alg_mod = Algebraic_Problem(res,
y0=y0,
jac=jac,
jacv=jacv,
name='ORS Example')
alg_mod_prec = Algebraic_Problem(res,
y0=y0,
jac=jac,
jacv=jacv,
prec_solve=prec_solve,
prec_setup=prec_setup,
name='ORS Example (Preconditioned)')
#Define the KINSOL solver
alg_solver = KINSOL(alg_mod)
alg_solver_prec = KINSOL(alg_mod_prec)
#Sets the parameters
def setup_param(solver):
solver.linear_solver = "spgmr"
solver.max_dim_krylov_subspace = 10
solver.ftol = LIN.norm(res(solver.y0)) * 1e-9
solver.max_iter = 300
solver.verbosity = 10
solver.globalization_strategy = "none"
setup_param(alg_solver)
setup_param(alg_solver_prec)
#Solve orignal system
y = alg_solver.solve()
#Solve Preconditionined system
y_prec = alg_solver_prec.solve()
print("Error , in y: ", LIN.norm(y - N.ones(len(y))))
print("Error (preconditioned), in y: ",
LIN.norm(y_prec - N.ones(len(y_prec))))
if with_plots:
P.figure(4)
P.semilogy(alg_solver.get_residual_norm_nonlinear_iterations(),
label="Original")
P.semilogy(alg_solver_prec.get_residual_norm_nonlinear_iterations(),
label='Preconditioned')
P.xlabel("Number of Iterations")
P.ylabel("Residual Norm")
P.title("Solution Progress")
P.legend()
P.grid()
P.figure(5)
P.plot(y, label="Original")
P.plot(y_prec, label="Preconditioned")
P.legend()
P.grid()
P.show()
#Basic test
for j in range(len(y)):
nose.tools.assert_almost_equal(y[j], 1.0, 4)
return [alg_mod, alg_mod_prec], [alg_solver, alg_solver_prec]