def test_sylv_schur_V_E(n, r, m):
np.random.seed(0)
A, E = diff_conv_1d_fem(n, 1, 1)
B = np.random.randn(n, m)
Ar = np.random.randn(r, r) - r * np.eye(r)
Er = np.random.randn(r, r)
Er = (Er + Er.T) / 2
Er += r * np.eye(r)
Br = np.random.randn(r, m)
Aop = NumpyMatrixOperator(A)
Eop = NumpyMatrixOperator(E)
Bop = NumpyMatrixOperator(B)
Arop = NumpyMatrixOperator(Ar)
Erop = NumpyMatrixOperator(Er)
Brop = NumpyMatrixOperator(Br)
Vva = solve_sylv_schur(Aop, Arop, E=Eop, Er=Erop, B=Bop, Br=Brop)
V = Vva.to_numpy().T
AVErT = A.dot(V.dot(Er.T))
EVArT = E.dot(V.dot(Ar.T))
BBrT = B.dot(Br.T)
assert fro_norm(AVErT + EVArT + BBrT) / fro_norm(BBrT) < 1e-10
def test_sylv_schur_W_E(n, r, p):
np.random.seed(0)
A, E = diff_conv_1d_fem(n, 1, 1)
C = np.random.randn(p, n)
Ar = np.random.randn(r, r) - r * np.eye(r)
Er = np.random.randn(r, r)
Er = (Er + Er.T) / 2
Er += r * np.eye(r)
Cr = np.random.randn(p, r)
Aop = NumpyMatrixOperator(A)
Eop = NumpyMatrixOperator(E)
Cop = NumpyMatrixOperator(C)
Arop = NumpyMatrixOperator(Ar)
Erop = NumpyMatrixOperator(Er)
Crop = NumpyMatrixOperator(Cr)
Wva = solve_sylv_schur(Aop, Arop, E=Eop, Er=Erop, C=Cop, Cr=Crop)
W = Wva.to_numpy().T
ATWEr = A.T.dot(W.dot(Er))
ETWAr = E.T.dot(W.dot(Ar))
CTCr = C.T.dot(Cr)
assert fro_norm(ATWEr + ETWAr + CTCr) / fro_norm(CTCr) < 1e-10
def _projection_matrices(self, rom, projection):
if self.fom.parametric:
fom = self.fom.with_(**{
op: getattr(self.fom, op).assemble(mu=self.mu)
for op in ['A', 'B', 'C', 'D', 'E']
},
parameter_space=None)
else:
fom = self.fom
self.V, self.W = solve_sylv_schur(fom.A,
rom.A,
E=fom.E,
Er=rom.E,
B=fom.B,
Br=rom.B,
C=fom.C,
Cr=rom.C)
if projection == 'orth':
self.V = gram_schmidt(self.V, atol=0, rtol=0)
self.W = gram_schmidt(self.W, atol=0, rtol=0)
elif projection == 'biorth':
self.V, self.W = gram_schmidt_biorth(self.V, self.W, product=fom.E)
self._pg_reductor = LTIPGReductor(fom, self.W, self.V,
projection == 'biorth')
def test_sylv_schur_V_E(n, r, m):
np.random.seed(0)
A, E = diff_conv_1d_fem(n, 1, 1)
B = np.random.randn(n, m)
Ar = np.random.randn(r, r) - r * np.eye(r)
Er = np.random.randn(r, r)
Er = (Er + Er.T) / 2
Er += r * np.eye(r)
Br = np.random.randn(r, m)
Aop = NumpyMatrixOperator(A)
Eop = NumpyMatrixOperator(E)
Bop = NumpyMatrixOperator(B)
Arop = NumpyMatrixOperator(Ar)
Erop = NumpyMatrixOperator(Er)
Brop = NumpyMatrixOperator(Br)
Vva = solve_sylv_schur(Aop, Arop, E=Eop, Er=Erop, B=Bop, Br=Brop)
V = Vva.to_numpy().T
AVErT = A.dot(V.dot(Er.T))
EVArT = E.dot(V.dot(Ar.T))
BBrT = B.dot(Br.T)
assert fro_norm(AVErT + EVArT + BBrT) / fro_norm(BBrT) < 1e-10
def test_sylv_schur_W_E(n, r, p):
np.random.seed(0)
A, E = diff_conv_1d_fem(n, 1, 1)
C = np.random.randn(p, n)
Ar = np.random.randn(r, r) - r * np.eye(r)
Er = np.random.randn(r, r)
Er = (Er + Er.T) / 2
Er += r * np.eye(r)
Cr = np.random.randn(p, r)
Aop = NumpyMatrixOperator(A)
Eop = NumpyMatrixOperator(E)
Cop = NumpyMatrixOperator(C)
Arop = NumpyMatrixOperator(Ar)
Erop = NumpyMatrixOperator(Er)
Crop = NumpyMatrixOperator(Cr)
Wva = solve_sylv_schur(Aop, Arop, E=Eop, Er=Erop, C=Cop, Cr=Crop)
W = Wva.to_numpy().T
ATWEr = A.T.dot(W.dot(Er))
ETWAr = E.T.dot(W.dot(Ar))
CTCr = C.T.dot(Cr)
assert fro_norm(ATWEr + ETWAr + CTCr) / fro_norm(CTCr) < 1e-10
def _projection_matrices(self, rom, projection):
fom = self.fom
self.V, self.W = solve_sylv_schur(fom.A, rom.A,
E=fom.E, Er=rom.E,
B=fom.B, Br=rom.B,
C=fom.C, Cr=rom.C)
if projection == 'orth':
self.V = gram_schmidt(self.V, atol=0, rtol=0)
self.W = gram_schmidt(self.W, atol=0, rtol=0)
elif projection == 'biorth':
self.V, self.W = gram_schmidt_biorth(self.V, self.W, product=fom.E)
self._pg_reductor = LTIPGReductor(fom, self.W, self.V, projection == 'biorth')
def _projection_matrices(self, rom, projection):
fom = self.fom
self.V, self.W = solve_sylv_schur(fom.A,
rom.A,
E=fom.E,
Er=rom.E,
B=fom.B,
Br=rom.B,
C=fom.C,
Cr=rom.C)
if projection == 'orth':
self.V = gram_schmidt(self.V, atol=0, rtol=0)
self.W = gram_schmidt(self.W, atol=0, rtol=0)
elif projection == 'biorth':
self.V, self.W = gram_schmidt_biorth(self.V, self.W, product=fom.E)
self._pg_reductor = LTIPGReductor(fom, self.W, self.V,
projection == 'biorth')
def _set_V_W_reductor(self, rom, projection):
fom = (
self.fom.with_(
**{op: getattr(self.fom, op).assemble(mu=self.mu)
for op in ['A', 'B', 'C', 'D', 'E']}
)
if self.fom.parametric
else self.fom
)
self.V, self.W = solve_sylv_schur(fom.A, rom.A,
E=fom.E, Er=rom.E,
B=fom.B, Br=rom.B,
C=fom.C, Cr=rom.C)
if projection == 'orth':
gram_schmidt(self.V, atol=0, rtol=0, copy=False)
gram_schmidt(self.W, atol=0, rtol=0, copy=False)
elif projection == 'biorth':
gram_schmidt_biorth(self.V, self.W, product=fom.E, copy=False)
self._pg_reductor = LTIPGReductor(fom, self.W, self.V,
projection == 'biorth')
def test_sylv_schur_V(n, r, m):
np.random.seed(0)
A = diff_conv_1d_fd(n, 1, 1)
B = np.random.randn(n, m)
Ar = np.random.randn(r, r) - r * np.eye(r)
Br = np.random.randn(r, m)
Aop = NumpyMatrixOperator(A)
Bop = NumpyMatrixOperator(B)
Arop = NumpyMatrixOperator(Ar)
Brop = NumpyMatrixOperator(Br)
Vva = solve_sylv_schur(Aop, Arop, B=Bop, Br=Brop)
V = Vva.to_numpy().T
AV = A.dot(V)
VArT = V.dot(Ar.T)
BBrT = B.dot(Br.T)
assert fro_norm(AV + VArT + BBrT) / fro_norm(BBrT) < 1e-10
def test_sylv_schur_W(n, r, p):
np.random.seed(0)
A = diff_conv_1d_fd(n, 1, 1)
C = np.random.randn(p, n)
Ar = np.random.randn(r, r) - r * np.eye(r)
Cr = np.random.randn(p, r)
Aop = NumpyMatrixOperator(A)
Cop = NumpyMatrixOperator(C)
Arop = NumpyMatrixOperator(Ar)
Crop = NumpyMatrixOperator(Cr)
Wva = solve_sylv_schur(Aop, Arop, C=Cop, Cr=Crop)
W = Wva.to_numpy().T
ATW = A.T.dot(W)
WAr = W.dot(Ar)
CTCr = C.T.dot(Cr)
assert fro_norm(ATW + WAr + CTCr) / fro_norm(CTCr) < 1e-10
def test_sylv_schur_V(n, r, m):
np.random.seed(0)
A = diff_conv_1d_fd(n, 1, 1)
B = np.random.randn(n, m)
Ar = np.random.randn(r, r) - r * np.eye(r)
Br = np.random.randn(r, m)
Aop = NumpyMatrixOperator(A)
Bop = NumpyMatrixOperator(B)
Arop = NumpyMatrixOperator(Ar)
Brop = NumpyMatrixOperator(Br)
Vva = solve_sylv_schur(Aop, Arop, B=Bop, Br=Brop)
V = Vva.to_numpy().T
AV = A.dot(V)
VArT = V.dot(Ar.T)
BBrT = B.dot(Br.T)
assert fro_norm(AV + VArT + BBrT) / fro_norm(BBrT) < 1e-10
def test_sylv_schur_W(n, r, p):
np.random.seed(0)
A = diff_conv_1d_fd(n, 1, 1)
C = np.random.randn(p, n)
Ar = np.random.randn(r, r) - r * np.eye(r)
Cr = np.random.randn(p, r)
Aop = NumpyMatrixOperator(A)
Cop = NumpyMatrixOperator(C)
Arop = NumpyMatrixOperator(Ar)
Crop = NumpyMatrixOperator(Cr)
Wva = solve_sylv_schur(Aop, Arop, C=Cop, Cr=Crop)
W = Wva.to_numpy().T
ATW = A.T.dot(W)
WAr = W.dot(Ar)
CTCr = C.T.dot(Cr)
assert fro_norm(ATW + WAr + CTCr) / fro_norm(CTCr) < 1e-10