def test_100(self):
x, w = herme.hermegauss(100)
# test orthogonality. Note that the results need to be normalized,
# otherwise the huge values that can arise from fast growing
# functions like Laguerre can be very confusing.
v = herme.hermevander(x, 99)
vv = np.dot(v.T * w, v)
vd = 1 / np.sqrt(vv.diagonal())
vv = vd[:, None] * vv * vd
assert_almost_equal(vv, np.eye(100))
# check that the integral of 1 is correct
tgt = np.sqrt(2 * np.pi)
assert_almost_equal(w.sum(), tgt)
def test_100(self):
x, w = herme.hermegauss(100)
# test orthogonality. Note that the results need to be normalized,
# otherwise the huge values that can arise from fast growing
# functions like Laguerre can be very confusing.
v = herme.hermevander(x, 99)
vv = np.dot(v.T * w, v)
vd = 1/np.sqrt(vv.diagonal())
vv = vd[:,None] * vv * vd
assert_almost_equal(vv, np.eye(100))
# check that the integral of 1 is correct
tgt = np.sqrt(2*np.pi)
assert_almost_equal(w.sum(), tgt)
def eval_hermite(deg, x, analytic=True):
if analytic: # Use pre-computed polynomials
if deg == 0:
return 1.
elif deg == 1.:
return x
elif deg == 2:
return x*x - 1.
elif deg == 3:
return x*x*x - 3.*x
elif deg == 4:
return x*x*x*x - 6.*x*x + 3.
# If not returned, assume we should make the call to the numerical orthogonalizer
roots = hermite_e.hermegauss(deg)[0] if deg > 1 else np.array([0, ])
poly = np.poly1d(roots, r=True)
return poly(x)
def update_points(self, mean, scale, have_sqrt=False):
"""Updates the quadrature points given some initial point and scale.
Parameters
----------
mean : N x 1 numpy array
Point to represent by quadrature points.
scale : N x N numpy array
Covariance or square root of the covariance matrix of the given point.
have_sqrt : bool
Optional flag indicating if the square root of the matrix was
supplied. The default is False.
Returns
-------
None.
"""
def create_combos(points_per_ax, num_ax, tot_points):
combos = np.meshgrid(
*itertools.repeat(range(points_per_ax), num_ax))
return np.array(combos).reshape((num_ax, tot_points)).T
self.num_axes = mean.size
sqrt_cov = self._factor_scale_matrix(scale, have_sqrt)
self.points = np.nan * np.ones((self.num_points, self.num_axes))
self.weights = np.nan * np.ones(self.num_points)
# get standard values for 1 axis case, use "Probabilist's" Hermite polynomials
quad_points, weights = herm_e.hermegauss(self.points_per_axis)
ind_combos = create_combos(self.points_per_axis, self.num_axes,
self.num_points)
for ii, inds in enumerate(ind_combos):
point = quad_points[inds].reshape((inds.size, 1))
self.points[ii, :] = (mean + sqrt_cov @ point).ravel()
self.weights[ii] = np.prod(weights[inds])
self.weights = self.weights / np.sum(self.weights)
def unit_sigma_points(dim, degree=3):
"""
Unit Gauss-Hermite sigma-points of given degree.
Parameters
----------
dim : int
Dimension of the input random variable.
degree : int, optional
Degree of the integration rule.
Returns
-------
: (dim, num_points) ndarray
GH sigma-points.
"""
# 1D sigma-points (x) and weights (w)
x, w = hermegauss(degree)
# nD sigma-points by cartesian product
return cartesian([x] * dim).T # column/sigma-point
def weights(dim, degree=3):
"""
Gauss-Hermite quadrature weights.
Parameters
----------
dim : int
Dimension of the input random variable.
degree : int, optional
Degree of the integration rule.
Returns
-------
: (num_points, ) ndarray
GH quadrature weights of given degree.
"""
# 1D sigma-points (x) and weights (w)
x, w = hermegauss(degree)
# hermegauss() provides weights that cause posdef errors
w = factorial(degree) / (degree**2 * hermeval(x, [0] *
(degree - 1) + [1])**2)
return np.prod(cartesian([w] * dim), axis=1)
def inner3_herm(P,i,j,l):
#compute <\Phi_i\Phi_j\Phi_l>
#Set up Gauss-Hermite quadrature, weighting function is exp^{-x^2}
m=(P+1)**2
x, w=H.hermegauss(m)
inner=sum([product3_herm(i,j,l)(x[idx]) * w[idx] for idx in range(m)]) return inner/np.sqrt(2*np.pi) #because of the weight
def Compute_Q(proc_num, proc_max, mu1=0, mu2=0, sigma1=0.1, sigma2=0.1, gridx=50, gridy=50, p=1):
num_quad = 20
lambda1 = H.hermegauss(num_quad)[0]
lambda2 = H.hermegauss(num_quad)[0]
# Create the characteristic function class used to define the QoI
class AvgCharFunc(fn.UserExpression):
def __init__(self, region, **kwargs):
self.a = region[0]
self.b = region[1]
self.c = region[2]
self.d = region[3]
super().__init__(**kwargs)
def eval(self, v, x):
v[0] = 0
if (x[0] >= self.a) & (x[0] <= self.b) & (x[1] >= self.c) & (x[1] <= self.d):
v[0] = 1./( (self.b-self.a) * (self.d-self.c) )
return v
def value_shape(self):
return ()
def QoI_FEM(lam1,lam2,pointa,pointb,gridx,gridy,p):
aa = pointa[0]
bb = pointb[0]
cc = pointa[1]
dd = pointb[1]
mesh = fn.UnitSquareMesh(gridx, gridy)
V = fn.FunctionSpace(mesh, "Lagrange", p)
# Define diffusion tensor (here, just a scalar function) and parameters
A = fn.Expression((('exp(lam1)','a'),
('a','exp(lam2)')), a = fn.Constant(0.0), lam1 = lam1, lam2 = lam2, degree=3)
u_exact = fn.Expression("sin(lam1*pi*x[0])*cos(lam2*pi*x[1])", lam1 = lam1, lam2 = lam2, degree=2+p)
# Define the mix of Neumann and Dirichlet BCs
class LeftBoundary(fn.SubDomain):
def inside(self, x, on_boundary):
return (x[0] < fn.DOLFIN_EPS)
class RightBoundary(fn.SubDomain):
def inside(self, x, on_boundary):
return (x[0] > 1.0 - fn.DOLFIN_EPS)
class TopBoundary(fn.SubDomain):
def inside(self, x, on_boundary):
return (x[1] > 1.0 - fn.DOLFIN_EPS)
class BottomBoundary(fn.SubDomain):
def inside(self, x, on_boundary):
return (x[1] < fn.DOLFIN_EPS)
# Create a mesh function (mf) assigning an unsigned integer ('uint')
# to each edge (which is a "Facet" in 2D)
mf = fn.MeshFunction('size_t', mesh, 1)
mf.set_all(0) # initialize the function to be zero
# Setup the boundary classes that use Neumann boundary conditions
NTB = TopBoundary() # instatiate
NTB.mark(mf, 1) # set all values of the mf to be 1 on this boundary
NBB = BottomBoundary()
NBB.mark(mf, 2) # set all values of the mf to be 2 on this boundary
NRB = RightBoundary()
NRB.mark(mf, 3)
# Define Dirichlet boundary conditions
Gamma_0 = fn.DirichletBC(V, u_exact, LeftBoundary())
bcs = [Gamma_0]
# Define data necessary to approximate exact solution
f = ( fn.exp(lam1)*(lam1*fn.pi)**2 + fn.exp(lam2)*(lam2*fn.pi)**2 ) * u_exact
#g1:#pointing outward unit normal vector, pointing upaward (0,1)
g1 = fn.Expression("-exp(lam2)*lam2*pi*sin(lam1*pi*x[0])*sin(lam2*pi*x[1])", lam1=lam1, lam2=lam2, degree=2+p)
#g2:pointing downward (0,1)
g2 = fn.Expression("exp(lam2)*lam2*pi*sin(lam1*pi*x[0])*sin(lam2*pi*x[1])", lam1=lam1, lam2=lam2, degree=2+p)
g3 = fn.Expression("exp(lam1)*lam1*pi*cos(lam1*pi*x[0])*cos(lam2*pi*x[1])", lam1=lam1, lam2=lam2, degree=2+p)
fn.ds = fn.ds(subdomain_data=mf)
# Define variational problem
u = fn.TrialFunction(V)
v = fn.TestFunction(V)
a = fn.inner(A*fn.grad(u), fn.grad(v))*fn.dx
L = f*v*fn.dx + g1*v*fn.ds(1) + g2*v*fn.ds(2) + g3*v*fn.ds(3) #note the 1, 2 and 3 correspond to the mf
# Compute solution
u = fn.Function(V)
fn.solve(a == L, u, bcs)
psi = AvgCharFunc([aa, bb, cc, dd], degree=0)
Q = fn.assemble(fn.project(psi * u, V) * fn.dx)
return Q
Q_FEM = np.zeros(400)
num_Q_per_proc = 400//proc_max
if proc_num != proc_size -1:
for i in range(proc_num*num_Q_per_proc, (proc_num+1)*num_Q_per_proc):
Q_FEM[i] = QoI_FEM(mu1+sigma1*lambda1[i%num_quad],mu2+sigma2*lambda2[i//num_quad],[0.4,0.4],[0.6,0.6],gridx,gridy,p)
else:
for i in range(proc_num*num_Q_per_proc,400):
Q_FEM[i] = QoI_FEM(mu1+sigma1*lambda1[i%num_quad],mu2+sigma2*lambda2[i//num_quad],[0.4,0.4],[0.6,0.6],gridx,gridy,p)
filename = os.path.join(os.getcwd(), "Data", "Q_FEM_quad_") + str(proc_num) + ".mat"
data_dict = {'Q_FEM': Q_FEM}
sio.savemat(filename, data_dict)
return
def q(i,j):
x, w=H.hermegauss(20)
Q=sum([w[ldx]*sum([w[kdx] * Q_FEM_quad[ldx*20+kdx] * H.hermeval(x[kdx],Phi(i)) for kdx in range(20)])*H.hermeval(x[ldx],Phi(j)) for ldx in range(20)])
q= Q/(2*np.pi*factorial(i)*factorial(j))
return q
def quadrature(self, points):
xi, w = hermite_e.hermegauss(points)
w /= np.sqrt(2*np.pi)
return xi, w
def unit_sigma_points(dim, degree):
# 1D sigma-points (x) and weights (w)
x, w = hermegauss(degree)
# nD sigma-points by cartesian product
return cartesian([x] * dim).T # column/sigma-point
def unit_sigma_points(dim, degree):
# 1D sigma-points (x) and weights (w)
x, w = hermegauss(degree)
# nD sigma-points by cartesian product
return cartesian([x] * dim).T # column/sigma-point
def weights(dim, degree):
# 1D sigma-points (x) and weights (w)
x, w = hermegauss(degree)
# hermegauss() provides weights that cause posdef errors
w = factorial(degree) / (degree ** 2 * hermeval(x, [0] * (degree - 1) + [1]) ** 2)
return np.prod(cartesian([w] * dim), axis=1)