def CalculateSolution(Pe):
# Create the TFC Class:
N = 200
m = 190
nC = 2
tfc = utfc(N, nC, m, basis='LeP', x0=0., xf=1.)
x = tfc.x
# Get the Chebyshev polynomials
H = tfc.H
H0 = H(np.array([0.]))
Hf = H(np.array([1.]))
# Create the constraint expression and its derivatives
y = lambda x, xi: np.dot(H(x), xi) + (1. - x) * (1. - np.dot(H0, xi)
) - x * np.dot(Hf, xi)
yd = egrad(y)
ydd = egrad(yd)
L = lambda xi: ydd(x, xi) - Pe * yd(x, xi)
# Calculate the solution
zXi = np.zeros(H(x).shape[1])
xi, it = NLLS(zXi, L)
# Create the test set:
N = 1000
xTest = np.linspace(0., 1., N)
err = np.abs(y(xTest, xi) - soln(xTest, Pe))
return np.max(err), np.mean(err)
def BVP_tfc(N, m, basis, iterMax, tol):
## Unpack Paramters: *********************************************************
x0 = 0.
xf = np.pi
## Initial Conditions: *******************************************************
y0 = 0.
yf = 0.
nC = 2 # number of constraints
## Determine call tfc class needs to be 1 for ELMs
if basis == 'CP' or basis == 'LeP':
c = 2. / (xf - x0)
elif basis == 'FS':
c = 2. * np.pi / (xf - x0)
else:
c = 1. / (xf - x0)
## Compute true solution
ytrue = lambda x: np.exp(-x) * np.sin(x)
err = onp.ones_like(m) * np.nan
res = onp.ones_like(m) * np.nan
## GET CHEBYSHEV VALUES: *********************************************
tfc = utfc(N, nC, int(m), basis=basis, x0=x0, xf=xf)
x = tfc.x
H = tfc.H
H0 = H(tfc.x[0])
Hf = H(tfc.x[-1])
## DEFINE THE ASSUMED SOLUTION: *************************************
phi1 = lambda x: (np.pi - x) / np.pi
phi2 = lambda x: x / np.pi
f = lambda x: np.exp(-2. * x) * np.sin(x) * (np.cos(x) - np.sin(
x)) - 2. * np.exp(-x) * np.cos(x)
y = lambda x, xi: np.dot(H(x), xi) + phi1(x) * (y0 - np.dot(
H0, xi)) + phi2(x) * (yf - np.dot(Hf, xi))
yp = egrad(y)
ypp = egrad(yp)
## DEFINE LOSS AND JACOB ********************************************
L = lambda xi: ypp(x, xi) + y(x, xi) * yp(x, xi) - f(x)
## SOLVE THE SYSTEM *************************************************
xi = onp.zeros(H(x).shape[1])
xi, _, time = NLLS(xi, L, timer=True, maxIter=iterMax)
## COMPUTE ERROR AND RESIDUAL ***************************************
err = onp.linalg.norm(y(x, xi) - ytrue(x))
res = onp.linalg.norm(L(xi))
return err, res, time
def IVP2BVP(N, m, gamma, basis, iterMax, tol):
## Unpack Paramters: *********************************************************
x0 = -1.
xf = 1.
## Initial Conditions: *******************************************************
y0 = -2.
y0p = -2.
yf = 2.
nC = 2 # number of constraints
## Determine call tfc class needs to be 1 for ELMs
if basis == 'CP' or basis == 'LeP':
c = 2./ (xf - x0)
elif basis == 'FS':
c = 2. * np.pi / (xf - x0)
else:
c = 1./ (xf - x0)
## GET CHEBYSHEV VALUES: *********************************************
tfc = utfc(N,nC,m,basis = basis, x0=-1., xf=1.)
x = tfc.x
H = tfc.H
dH = tfc.dH
H0 = H(tfc.z[0])
Hf = H(tfc.z[-1])
H0p = dH(tfc.z[0])
## DEFINE THE ASSUMED SOLUTION: *************************************
phi1 = lambda a: 1./(1. + 4.*gamma - gamma**2) * ( (1. + gamma) - 2.*gamma*a )
phi2 = lambda a: 1./(1. + 4.*gamma - gamma**2) * ( (1. - gamma)**2 + (1. - gamma**2)*a)
phi3 = lambda a: 1./(1. + 4.*gamma - gamma**2) * ( -gamma*(gamma-3.) + 2.*gamma*a )
y = lambda x, xi: np.dot(H(x),xi) + phi1(x)*(y0 - np.dot(H0,xi)) \
+ phi2(x)*(y0p - np.dot(H0p,xi)) \
+ phi3(x)*(yf - np.dot(Hf,xi))
yp = egrad(y,0)
ypp = egrad(yp,0)
## DEFINE LOSS AND JACOB ********************************************
L = lambda xi: ypp(x,xi) + (np.cos(3.*x**2) -3.*x + 1.)*yp(x,xi) \
+ (6.*np.sin(4.*x**2) - np.exp(np.cos(3.*x)))*y(x,xi) \
- 2. * (1.-np.sin(3.*x))*(3.*x-np.pi)/(4.-x)
## SOLVE THE SYSTEM *************************************************
xi = onp.zeros(H(x).shape[1])
xi,_,_ = NLLS(xi,L,timer=True)
return y(x,xi), L(xi), x
def test_ODE():
# This script will solve the non-linear differential equation
# of the form: y''+f(t)*y*y' = f2(t)
# Constants used in the differential equation:
f = lambda t: np.ones(t.shape)
f2 = lambda t: np.exp(-2. * t) * np.sin(t) * (np.cos(t) - np.sin(
t)) - 2. * np.exp(-t) * np.cos(t)
ti = 0.
tf = np.pi
yi = 0.
yf = 0.
# Real analytical solution:
real = lambda t: np.exp(-t) * np.sin(t)
# Create the ToC Class:
N = 100
m = 30
nC = 2
tfc = TFC(N, nC, m, x0=ti, xf=tf, basis='LeP')
t = tfc.x
# Get the Chebyshev polynomials
H = tfc.H
dH = tfc.dH
Zero = np.zeros_like(t)
End = tf * np.ones_like(t)
H0 = H(Zero)
Hd0 = dH(Zero)
Hf = H(End)
# Create the constraint expression and its derivatives
beta0 = lambda t: (t - tf) / (ti - tf)
beta1 = lambda t: (ti - t) / (ti - tf)
y = lambda t, xi: np.dot(H(t), xi) + beta0(t) * (yi - np.dot(
H0, xi)) + beta1(t) * (yf - np.dot(Hf, xi))
yd = egrad(y)
ydd = egrad(yd)
# Create the residual and jacobians
r = lambda xi, t: ydd(t, xi) + f(t) * y(t, xi) * yd(t, xi) - f2(t)
xi = np.zeros(H(t).shape[1])
xi, it = NLLS(xi, r, t, constant_arg_nums=[1])
assert (np.max(np.abs(r(xi, t))) < 1e-10)
def test_PDE():
## TFC Parameters
maxIter = 10
tol = 1e-13
# Constants and switches:
n = 20
m = 20
x0 = np.array([0., 0.])
xf = np.array([1., 1.])
# Real analytical solution:
real = lambda x, y: y**2 * np.sin(np.pi * x)
# Create the TFC Class:
N = np.array([n, n])
nC = np.array([2, 2])
tfc = TFC(N, nC, m, x0=x0, xf=xf, dim=2, basis='CP')
x = tfc.x
Zero = np.zeros_like(x[0])
One = np.ones_like(x[0])
# Get the basis functions
H = tfc.H
Hy = tfc.Hy
z1 = lambda xi, *x: np.dot(H(*x), xi) - (1. - x[0]) * np.dot(
H(*(Zero, x[1])), xi) - x[0] * np.dot(H(*(One, x[1])), xi)
z = lambda xi, *x: z1(xi, *x) - z1(xi, x[0], Zero) + x[1] * (2. * np.sin(
np.pi * x[0]) - egrad(z1, 2)(xi, x[0], One))
# Create the residual
zxx = egrad(egrad(z, 1), 1)
zyy = egrad(egrad(z, 2), 2)
zy = egrad(z, 2)
r = lambda xi, *x: zxx(xi, *x) + zyy(xi, *x) + z(xi, *x) * zy(
xi, *x) - np.sin(np.pi * x[0]) * (2. - np.pi**2 * x[1]**2 + 2. * x[1]**
3 * np.sin(np.pi * x[0]))
xi = np.zeros(H(*x).shape[1])
xi, it = NLLS(xi, r, *x, constant_arg_nums=[1, 2])
zr = real(x[0], x[1])
ze = z(xi, *x)
err = zr - ze
maxErr = np.max(np.abs(err))
assert (maxErr < 1e-10)
- f(xs2)
L = jit(lambda xi: np.hstack((L1(xi), L2(xi))))
## SOLVE THE SYSTEM *************************************************
xi1 = onp.zeros(Hs1(xs1).shape[1])
xi2 = onp.zeros(Hs2(xs2).shape[1])
m = (yf - y0) / (xf - x0)
y1 = onp.ones(1) * (m * xs1[-1] + y0)
y1d = onp.ones(1) * m
xi0 = TFCDict({'xi1': xi1, 'xi2': xi2, 'y1': y1, 'y1d': y1d})
xi = TFCDict({'xi1': xi1, 'xi2': xi2, 'y1': y1, 'y1d': y1d})
xi, it, time = NLLS(xi, L, timer=True)
## COMPUTE ERROR AND RESIDUAL ***************************************
x = np.hstack((xs1, xs2))
yinit = np.hstack((ys1(xs1, xi0), ys2(xs2, xi0)))
y = np.hstack((ys1(xs1, xi), ys2(xs2, xi)))
yp = np.hstack((yps1(xs1, xi), yps2(xs2, xi)))
err = onp.abs(y - ytrue(x))
res = onp.abs(L(xi))
print()
print('Max Error: ' + str(np.max(err)))
print('Max Loss: ' + str(np.max(res)))
print('Computation time [ms]: ' + str(time * 1000))
Lu = lambda z, xi: -xi['b']**2 * up(z, xi['xi_u']) - beta * x(z, xi[
'xi_x']) + alfa * u(z, xi['xi_u'])
H = lambda z, xi: 0.5 * x(z, xi['xi_x'])**2 - 0.5 * u(z, xi[
'xi_u'])**2 - alfa / beta * x(z, xi['xi_x']) * u(z, xi['xi_u'])
Lf = lambda z, xi: H(z, xi)[-1]
L = lambda xi: np.hstack((Lx(z, xi), Lu(z, xi), H(z, xi)))
xi_x = onp.zeros(Hx(z).shape[1])
xi_u = onp.zeros(Hu(z).shape[1])
b = onp.ones(1) * np.sqrt(2.)
xi = TFCDict({'xi_x': xi_x, 'xi_u': xi_u, 'b': b})
## SOLVE THE SYSTEM *************************************************
xi, it, time = NLLS(xi, L, timer=True, tol=tol, maxIter=iterMax)
t = (z - z[0]) / xi['b']**2
X = x(z, xi['xi_x'])
U = u(z, xi['xi_u'])
Ham = onp.zeros(len(t))
int = onp.zeros(len(t))
for i in range(0, len(t)):
int[i] = 0.5 * (X[i]**2 + U[i]**2)
Ham[i] = int[i] + -U[i] / beta * (alfa * X[i] + beta * U[i])
cost = simps(int, t)
tf = 2. / xi['b']**2
# Create constrained expression:
g = lambda th, xi: np.dot(H(th), xi)
r = lambda th,xi: g(th,xi)+\
(th-thf)/(th0-thf)*(r0-g(th0*np.ones_like(th),xi))+\
(th-th0)/(thf-th0)*(rf-g(thf*np.ones_like(th),xi))
# Create loss function:
dr = egrad(r)
d2r = egrad(dr)
L = lambda xi: -r(th,xi)**2*(dr(th,xi)*np.tan(th)+2.*d2r(th,xi))+\
-np.tan(th)*dr(th,xi)**3+3.*r(th,xi)*dr(th,xi)**2+r(th,xi)**3
# Solve the problem:
xi = np.zeros(H(th).shape[1])
xi, _, time = NLLS(xi, L, timer=True)
# Print out statistics:
print("Solution time: {0} seconds".format(time))
# Plot the solution and residual
p = MakePlot([r"$y$"], [r"$x$"])
p.ax[0].plot(r(th, xi) * np.sin(th), r(th, xi) * np.cos(th), "k")
p.ax[0].axis("equal")
p.ax[0].grid(True)
p.ax[0].invert_yaxis()
p.PartScreen(8, 7)
p.show()
p2 = MakePlot([r"$\theta$"], [r"$L$"])
p2.ax[0].plot(th,
ft = egrad(f,3)
# Create the residual and jacobian
L1 = lambda xiu,xiv,*x: fx(xiu,*x)+fy(xiv,*x)
L2 = lambda xiu,xiv,*x: rho*(ft(xiu,*x)+f(xiu,*x)*fx(xiu,*x)+f(xiv,*x)*fy(xiu,*x))+P-mu*(f2x(xiu,*x)+f2y(xiu,*x))
L3 = lambda xiu,xiv,*x: rho*(ft(xiv,*x)+f(xiu,*x)*fx(xiv,*x)+f(xiv,*x)*fy(xiv,*x))-mu*(f2x(xiv,*x)+f2y(xiv,*x))
L = lambda xi: np.hstack([L1(xi['xiu'],xi['xiv'],*x),L2(xi['xiu'],xi['xiv'],*x),L3(xi['xiu'],xi['xiv'],*x)])
# Calculate the xi values
M = H(*x).shape[1]
xiu = np.zeros(M)
xiv = np.zeros(M)
xi = TFCDict({'xiu':xiu,'xiv':xiv})
if xTfc:
xi,it,time = NLLS(xi,L,maxIter=maxIter,method='lstsq',timer=True)
else:
xi,it,time = NLLS(xi,L,maxIter=maxIter,method='pinv',timer=True)
xiu = xi['xiu']; xiv = xi['xiv']
# Calcualte u and plot for different times
n = 100
X = onp.matlib.repmat(onp.reshape(onp.linspace(0,xf[0],num=n),(n,1)),n,1).flatten()
Y = onp.reshape(onp.matlib.repmat(onp.reshape(onp.linspace(-Hb/2.,Hb/2.,num=n),(n,1)),1,n),(n**2,1)).flatten()
xTest = onp.zeros((3,n**2*3))
xTest[0,:] = onp.hstack([X,]*3)
xTest[1,:] = onp.hstack([Y,]*3)
xTest[2,:] = onp.hstack([onp.ones(n**2)*0.01,onp.ones(n**2)*0.1,onp.ones(n**2)*tf])
p = []; U = [];
vals = [0.01,0.1,tf]
def runLaneEmden(N, m, basis, k, xf):
## user defined parameters: ************************************************************************
# N - number of discretization points
# m - number of basis function terms
# basis - basis function type
# k - specific problem type, k >=0 (analytical solution known for k = 0, 1, and 5)
## problem initial conditions: *****************************************************************
xspan = [0., xf] # problem domain range [x0, xf], where x₀ > 0
y0 = 1. # y(x0) = 1
y0p = 0. # y'(x0) = 0
nC = 2 # number of constraints
## construct univariate tfc class: *************************************************************
tfc = utfc(N, nC, int(m), basis=basis, x0=xspan[0], xf=xspan[1])
x = tfc.x
H = tfc.H
dH = tfc.dH
H0 = H(x[0:1])
H0p = dH(x[0:1])
## define tfc constrained expression and derivatives: ******************************************
# switching function
phi1 = lambda x: np.ones_like(x)
phi2 = lambda x: x
# tfc constrained expression
y = lambda x, xi: np.dot(H(x), xi) + phi1(x) * (y0 - np.dot(
H0, xi)) + phi2(x) * (y0p - np.dot(H0p, xi))
yp = egrad(y)
ypp = egrad(yp)
## define the loss function: *******************************************************************
L = lambda xi: x * ypp(x, xi) + 2. * yp(x, xi) + x * y(x, xi)**k
## solve the problem via nonlinear least-squares ***********************************************
xi = np.zeros(H(x).shape[1])
# if k==0 or k==1, the problem is linear
if k == 0 or k == 1:
xi, time = LS(xi, L, timer=True)
iter = 1
else:
xi, iter, time = NLLS(xi, L, timer=True)
## compute the error (if k = 0, 1, or 5): ******************************************************
if k == 0:
ytrue = 1. - 1. / 6. * x**2
elif k == 1:
ytrue = onp.ones_like(x)
ytrue[1:] = np.sin(x[1:]) / x[1:]
elif k == 5:
ytrue = (1. + x**2 / 3)**(-1 / 2)
else:
ytrue = np.empty_like(x)
err = np.abs(y(x, xi) - ytrue)
## compute the residual of the loss vector: ****************************************************
res = np.abs(L(xi))
return x, y(x, xi), err, res
def laneEmden_tfc(N, m, type, xspan, basis, iterMax, tol):
## Unpack Paramters: *********************************************************
x0 = xspan[0]
xf = xspan[1]
## Initial Conditions: *******************************************************
y0 = 1.
y0p = 0.
nC = 2 # number of constraints
## Determine call tfc class needs to be 1 for ELMs
if basis == 'CP' or basis == 'LeP':
c = 2. / (xf - x0)
elif basis == 'FS':
c = 2. * np.pi / (xf - x0)
else:
c = 1. / (xf - x0)
## Compute true solution
if type == 0:
maxIter = 1
def ytrue(x):
val = onp.zeros_like(x)
val[0] = 1.
val[1:] = 1. - 1. / 6. * x[1:]**2
return val
elif type == 1:
maxIter = 1
def ytrue(x):
val = onp.zeros_like(x)
val[0] = 1.
val[1:] = np.sin(x[1:]) / x[1:]
return val
elif type == 5:
maxIter = iterMax
def ytrue(x):
val = onp.zeros_like(x)
val[0] = 1.
val[1:] = (1. + x[1:]**2 / 3)**(-1 / 2)
return val
else:
def ytrue(x):
return np.nan * np.ones_like(x)
err = np.ones_like(m) * np.nan
res = np.ones_like(m) * np.nan
## GET CHEBYSHEV VALUES: *********************************************
tfc = utfc(N, nC, int(m), basis=basis, x0=x0, xf=xf)
x = tfc.x
H = tfc.H
dH = tfc.dH
H0 = H(x[0])
H0p = dH(x[0])
## DEFINE THE ASSUMED SOLUTION: *************************************
phi1 = lambda x: np.ones_like(x)
phi2 = lambda x: x
y = lambda x,xi: np.dot(H(x),xi) \
+ phi1(x)*(y0 - np.dot(H0,xi)) \
+ phi2(x)*(y0p - np.dot(H0p,xi))
yp = egrad(y)
ypp = egrad(yp)
## DEFINE LOSS AND JACOB ********************************************
L = jit(lambda xi: x * ypp(x, xi) + 2. * yp(x, xi) + x * y(x, xi)**type)
## SOLVE THE SYSTEM *************************************************
# Solve the problem
xi = np.zeros(H(x).shape[1])
xi, _, time = NLLS(xi, L, timer=True, maxIter=maxIter)
## COMPUTE ERROR AND RESIDUAL ***************************************
err = np.linalg.norm(y(x, xi) - ytrue(x))
res = np.linalg.norm(L(xi))
return err, res, time
def CalculateSolutionSplit(Pe):
if Pe > 1e3:
xpBoundL = 0. + 1.e-3
xpBoundU = 1. - 1e-3
else:
xpBoundL = 0. + 1.e-1
xpBoundU = 1. - 1e-1
# Create the ToC Class:
N = 200
m = 190
nC = 3
tfc = utfc(N, nC, m, basis='LeP', x0=-1., xf=1.)
# Get the Chebyshev polynomials
H = tfc.H
dH = tfc.dH
H0 = H(np.array([-1.]))
Hf = H(np.array([1.]))
Hd0 = dH(np.array([-1.]))
Hdf = dH(np.array([1.]))
# Create the constraint expression and its derivatives
z = tfc.z
xp = lambda xi: xi['xpHat'] + (xpBoundU - xi['xpHat']) * step(xi[
'xpHat'] - xpBoundU) + (xpBoundL - xi['xpHat']) * step(xpBoundL - xi[
'xpHat'])
c1 = lambda xi: 2. / (xp(xi))
c2 = lambda xi: 2. / (1. - xp(xi))
x1 = lambda z, xi: (z + 1.) / c1(xi)
x2 = lambda z, xi: (z + 1.) / c2(xi) + xp(xi)
y1 = lambda z,xi: np.dot(H(z),xi['xi1'])+(1.-2.*z+z**2)/4.*(1.-np.dot(H0,xi['xi1']))\
+(3.+2.*z-z**2)/4.*(xi['y']-np.dot(Hf,xi['xi1']))\
+(-1.+z**2)/2.*(xi['yd']/c1(xi)-np.dot(Hdf,xi['xi1']))
ydz1 = egrad(y1, 0)
yddz1 = egrad(ydz1, 0)
yd1 = lambda z, xi: ydz1(z, xi) * c1(xi)
ydd1 = lambda z, xi: yddz1(z, xi) * c1(xi)**2
y2 = lambda z,xi: np.dot(H(z),xi['xi2'])+(3.-2.*z-z**2)/4.*(xi['y']-np.dot(H0,xi['xi2']))\
+(1.-z**2)/2.*(xi['yd']/c2(xi)-np.dot(Hd0,xi['xi2']))\
+(1.+2.*z+z**2)/4.*(0.-np.dot(Hf,xi['xi2']))
ydz2 = egrad(y2, 0)
yddz2 = egrad(ydz2, 0)
yd2 = lambda z, xi: ydz2(z, xi) * c2(xi)
ydd2 = lambda z, xi: yddz2(z, xi) * c2(xi)**2
# Solve the problem
xi = TFCDict({
'xi1': onp.zeros(H(z).shape[1]),
'xi2': onp.zeros(H(z).shape[1]),
'xpHat': onp.array([0.99]),
'y': onp.array([0.]),
'yd': onp.array([0.])
})
L1 = lambda xi: ydd1(z, xi) - Pe * yd1(z, xi)
L2 = lambda xi: ydd2(z, xi) - Pe * yd2(z, xi)
L = lambda xi: np.hstack([L1(xi), L2(xi)])
xi, it = NLLS(xi, L)
# Create the test set:
N = 1000
z = np.linspace(-1., 1., N)
# Calculate the error and return the results
X = np.hstack([x1(z, xi), x2(z, xi)])
Y = np.hstack([y1(z, xi), y2(z, xi)])
err = np.abs(Y - soln(X, Pe))
return np.max(err), np.mean(err)
(xFinal-x[0])/(xFinal-xInit)*(c/alpha-c/alpha*np.tanh(c*(xInit-c*x[1])/(2.*nu))-np.dot(H(xInit*np.ones_like(x[0]),x[1]),xi))+\
(x[0]-xInit)/(xFinal-xInit)*(c/alpha-c/alpha*np.tanh(c*(xFinal-c*x[1])/(2.*nu))-np.dot(H(xFinal*np.ones_like(x[0]),x[1]),xi))
u = lambda xi,*x: u1(xi,*x)+\
c/alpha-c/alpha*np.tanh(c*x[0]/(2.*nu))-u1(xi,x[0],np.zeros_like(x[1]))
# Create the residual
ux = egrad(u, 1)
d2x = egrad(ux, 1)
ut = egrad(u, 2)
r = lambda xi: ut(xi, *x) + alpha * u(xi, *x) * ux(xi, *x) - nu * d2x(xi, *x)
# Solve the problem
xi = np.zeros(H(*x).shape[1])
xi, it, time = NLLS(xi,
r,
method='lstsq',
timer=True,
timerType="perf_counter")
# Calculate error at the test points:
dark = np.meshgrid(np.linspace(xInit, xFinal, nTest),
np.linspace(0., 1., nTest))
xTest = tuple([j.flatten() for j in dark])
err = np.abs(u(xi, *xTest) - real(*xTest))
print("Training time: " + str(time))
print("Max error: " + str(np.max(err)))
print("Mean error: " + str(np.mean(err)))
# Plot analytical solution
if usePlotly:
