def test_noisy_runge():
n = 100
np.random.seed(3)
x0 = 2 * np.random.rand(n) - 1.0
y0 = 1 / (1 + 25 * x0**2)
y0 += 1.0e-1 * (2 * np.random.rand(*x0.shape) - 1)
a = -1.5
b = +1.5
plt.plot(x0, y0, "xk")
x = np.linspace(a, b, 201)
# plt.plot(x, 1 / (1 + 25 * x ** 2), "-", color="0.8", label="1 / (1 + 25 * x**2)")
lmbda = 0.2
u = smoothfit.fit1d(x0,
y0,
a,
b,
200,
degree=1,
lmbda=lmbda,
variant="dolfin")
x = np.linspace(a, b, 201)
vals = [u(xx) for xx in x]
plt.plot(x, vals, "-")
# plt.title(f"lmbda = {lmbda:.1e}")
plt.xlim(a, b)
plt.ylim(-0.2, 1.2)
plt.grid()
plt.gca().set_aspect("equal")
# plt.show()
plt.savefig("runge-noise-02.svg", bbox_inches="tight", transparent=True)
def lambda_effect():
n = 100
numpy.random.seed(1)
x0 = 2 * numpy.random.rand(n) - 1.0
y0 = 1 / (1 + 25 * x0 ** 2) + 1.0e-1 * (2 * numpy.random.rand(*x0.shape) - 1)
a = -1.5
b = +1.5
for k, lmbda in enumerate(numpy.logspace(-3, 1, num=41)):
plt.plot(x0, y0, "xk")
x = numpy.linspace(a, b, 201)
# plt.plot(x, 1 / (1 + 25 * x ** 2), "-", color="0.8", label="1 / (1 + 25 * x**2)")
u = smoothfit.fit1d(x0, y0, a, b, 1000, degree=1, lmbda=lmbda)
x = numpy.linspace(a, b, 201)
vals = [u(xx) for xx in x]
plt.plot(x, vals, "-", color="#d62728")
plt.title(f"lmbda = {lmbda:.1e}")
plt.xlim(a, b)
plt.ylim(-0.2, 1.2)
# plt.show()
plt.savefig(
f"smoothfit-lambda-{k:02d}.png", bbox_inches="tight", transparent=True
)
plt.close()
return
def test_noisy_runge():
n = 100
rng = np.random.default_rng(123)
x0 = 2 * rng.random(n) - 1.0
y0 = 1 / (1 + 25 * x0**2)
y0 += 1.0e-1 * (2 * np.random.rand(*x0.shape) - 1)
a = -1.5
b = +1.5
plt.plot(x0, y0, "xk")
# x = np.linspace(a, b, 201)
# plt.plot(x, 1 / (1 + 25 * x ** 2), "-", color="0.8", label="1 / (1 + 25 * x**2)")
lmbda = 0.2
basis, u = smoothfit.fit1d(x0, y0, a, b, 200, degree=1, lmbda=lmbda)
plt.plot(basis.mesh.p[0], u[basis.nodal_dofs[0]], "-", label="smooth fit")
# plt.title(f"lmbda = {lmbda:.1e}")
plt.xlim(a, b)
plt.ylim(-0.2, 1.2)
plt.grid()
plt.gca().set_aspect("equal")
# plt.show()
# plt.savefig("runge-noise-02.svg", bbox_inches="tight", transparent=True)
plt.savefig("runge-noise-02.png", bbox_inches="tight", transparent=True)
def test_1d():
n = 20
# x0 = numpy.linspace(-1.0, 1.0, n)
numpy.random.seed(123)
x0 = numpy.random.rand(n) * 2 - 1
# y0 = x0.copy()
# y0 = 1 / (1 + 25*x0**2) + 5.0e-2 * (2*numpy.random.rand(n)-1)
# y0 = numpy.exp(x0) + 1.0e-1 * (2*numpy.random.rand(n) - 1)
# y0 = x0 + 1.0e-1 * (2*numpy.random.rand(n) - 1)
# y0 = x0**3 + 1.0e-1 * (2*numpy.random.rand(n) - 1)
y0 = numpy.sin(1 * numpy.pi *
x0) # + 1.0e-1 * (2*numpy.random.rand(n) - 1)
# y0 = 1 / (x0 + 2) + 1.0e-2 * (2*numpy.random.rand(n) - 1)
a = -1.5
b = +1.5
u = smoothfit.fit1d(x0, y0, a, b, 10, degree=2, eps=1.0e-1)
# ref = 1.5552074468182238
# print(assemble(u*u * dx))
# assert abs(assemble(u*u * dx) - ref) < 1.0e-2 * ref
# x = u.function_space().mesh().coordinates()
x = numpy.linspace(a, b, 201)
vals = [u(xx) for xx in x]
plt.plot(x0, y0, 'xk', label='data')
plt.plot(x, vals, '-', color='k', alpha=0.3, label='smooth fit')
plt.xlim(a, b)
plt.legend()
plt.show()
return
def update(lmbda):
basis, coeffs = smoothfit.fit1d(x0, y0, a, b, 500, degree=1, lmbda=lmbda)
fit.set_xdata(basis.mesh.p[0])
fit.set_ydata(coeffs[basis.nodal_dofs[0]])
plt.title(f"lambda = {lmbda:.3e}")
plt.ylim(-0.1, 1.1)
plt.xlim(a, b)
plt.legend()
def test_samples():
# From <https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm#Example>
x0 = np.array([0.038, 0.194, 0.425, 0.626, 1.253, 2.500, 3.740])
y0 = np.array([0.050, 0.127, 0.094, 0.2122, 0.2729, 0.2665, 0.3317])
basis, u = smoothfit.fit1d(x0, y0, 0, 4, 1000, degree=1, lmbda=1.0)
# plot the function
plt.plot(basis.mesh.p[0], u[basis.nodal_dofs[0]], "-", label="smooth fit")
plt.plot(x0, y0, "xk")
plt.xlim(0, 4)
plt.ylim(0)
plt.grid()
# plt.show()
plt.savefig("smoothfit-samples.png", bbox_inches="tight", transparent=True)
def test_1d(solver, show=False):
n = 20
# x0 = np.linspace(-1.0, 1.0, n)
rng = np.random.default_rng(2)
x0 = rng.random(n) * 2 - 1
# y0 = x0.copy()
# y0 = 1 / (1 + 25*x0**2) + 5.0e-2 * (2*np.random.rand(n)-1)
# y0 = np.exp(x0) + 1.0e-1 * (2*np.random.rand(n) - 1)
# y0 = x0 + 1.0e-1 * (2*np.random.rand(n) - 1)
# y0 = x0**3 + 1.0e-1 * (2*np.random.rand(n) - 1)
y0 = np.sin(1 * np.pi * x0) # + 1.0e-1 * (2*np.random.rand(n) - 1)
# y0 = 1 / (x0 + 2) + 1.0e-2 * (2*np.random.rand(n) - 1)
a = -1.5
b = +1.5
lmbda = 1.0e-2
basis, coeffs = smoothfit.fit1d(x0,
y0,
a,
b,
64,
lmbda,
degree=1,
solver=solver)
if solver == "Nelder-Mead":
ref = 14.27840047789019
else:
ref = 36.311654900989524
assert abs(np.dot(coeffs, coeffs) - ref) < 1.0e-10 * ref
if show:
plt.plot(basis.mesh.p[0],
coeffs[basis.nodal_dofs[0]],
"-",
label="smooth fit")
plt.plot(x0, y0, "xk", label="samples")
x = np.linspace(a, b, 101)
plt.plot(x, np.sin(np.pi * x), "-", color="0.8", label="original")
plt.xlim(a, b)
plt.legend()
plt.show()
return
def update(k):
# k sample points
xs = np.linspace(-1.0, 1.0, k)
ys = f(xs)
samples.set_xdata(xs)
samples.set_ydata(ys)
# plt.plot(xs, ys, "xk")
basis, coeffs = smoothfit.fit1d(xs, ys, a, b, 200, degree=1, lmbda=1.0e-6)
fit.set_xdata(basis.mesh.p[0])
fit.set_ydata(coeffs[basis.nodal_dofs[0]])
plt.title(f"{k} sample points")
plt.ylim(-0.1, 1.1)
plt.xlim(a, b)
plt.legend()
def test_1d(solver, show=False):
n = 20
# x0 = np.linspace(-1.0, 1.0, n)
np.random.seed(2)
x0 = np.random.rand(n) * 2 - 1
# y0 = x0.copy()
# y0 = 1 / (1 + 25*x0**2) + 5.0e-2 * (2*np.random.rand(n)-1)
# y0 = np.exp(x0) + 1.0e-1 * (2*np.random.rand(n) - 1)
# y0 = x0 + 1.0e-1 * (2*np.random.rand(n) - 1)
# y0 = x0**3 + 1.0e-1 * (2*np.random.rand(n) - 1)
y0 = np.sin(1 * np.pi * x0) # + 1.0e-1 * (2*np.random.rand(n) - 1)
# y0 = 1 / (x0 + 2) + 1.0e-2 * (2*np.random.rand(n) - 1)
a = -1.5
b = +1.5
lmbda = 1.0e-2
u = smoothfit.fit1d(x0,
y0,
a,
b,
64,
lmbda,
degree=1,
solver=solver,
variant="dolfin")
if solver == "Nelder-Mead":
ref = 0.659_143_389_243_738_2
else:
ref = 1.417_961_801_095_804_4
val = assemble(u * u * dx)
assert abs(val - ref) < 1.0e-10 * ref
if show:
# x = u.function_space().mesh().coordinates()
x = np.linspace(a, b, 201)
vals = [u(xx) for xx in x]
plt.plot(x, vals, "-", label="smooth fit")
plt.plot(x0, y0, "xk", label="samples")
plt.plot(x, np.sin(np.pi * x), "-", color="0.8", label="original")
plt.xlim(a, b)
plt.legend()
plt.show()
def test_samples():
# From <https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm#Example>
x0 = numpy.array([0.038, 0.194, 0.425, 0.626, 1.253, 2.500, 3.740])
y0 = numpy.array([0.050, 0.127, 0.094, 0.2122, 0.2729, 0.2665, 0.3317])
u = smoothfit.fit1d(x0, y0, 0, 4, 1000, degree=1, lmbda=1.0)
# plot the function
x = numpy.linspace(0, 4, 201)
vals = [u(xx) for xx in x]
plt.plot(x, vals, "-", label="smooth fit")
plt.plot(x0, y0, "xk")
plt.xlim(0, 4)
plt.ylim(0)
plt.grid()
# plt.show()
plt.savefig("smoothfit-samples.svg", bbox_inches="tight", transparent=True)
return