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https://github.com/vale981/stocproc
synced 2025-03-04 17:21:42 -05:00
unmerged changes from some late night work
This commit is contained in:
Richard Hartmann
parent
cfbf5fffc8
commit
b7c3908f06
2 changed files with 129 additions and 70 deletions
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@ -1,8 +1,10 @@
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from scipy.optimize import brentq
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from scipy.optimize import brentq, bisect
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from numpy.fft import rfft as np_rfft
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import numpy as np
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from math import fsum
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import logging
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log = logging.getLogger(__name__)
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def find_integral_boundary(integrand, tol, ref_val, max_val, x0):
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"""
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@ -23,23 +25,45 @@ def find_integral_boundary(integrand, tol, ref_val, max_val, x0):
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if integrand(ref_val) <= tol:
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raise ValueError("the integrand at ref_val needs to be greater that tol")
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# find the left boundary called a
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if integrand(x0+ref_val) > tol:
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log.debug("ref_value for search: {} tol: {}".format(ref_val, tol))
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I = integrand(x0+ref_val)
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if I > tol:
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log.debug("x={:.3e} I(x+ref_val) = {:.3e} > tol -> veer x away from ref_value".format(x0, I))
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x = 2*x0
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while integrand(x+ref_val) > tol:
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if abs(x-ref_val) > max_val:
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I = integrand(x + ref_val)
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while I > tol:
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log.debug("x={:.3e} I(x+ref_val) = {:.3e}".format(x, I))
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if abs(x) > max_val:
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raise RuntimeError("|x-ref_val| > max_val was reached")
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x *= 2
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I = integrand(x + ref_val)
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log.debug("x={:.3e} I(x+ref_val) = {:.3e} < tol".format(x, I))
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a = brentq(lambda x: integrand(x)-tol, x+ref_val, x0+ref_val)
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elif integrand(x0+ref_val) < tol:
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log.debug("found I(a={:.3e}) = {:.3e} = tol".format(a, integrand(a)))
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elif I < tol:
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log.debug("x={:.3e} I(x+ref_val) = {:.3e} < tol -> approach x towards ref_value".format(x0, I))
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x = x0/2
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while integrand(x+ref_val) < tol:
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if (1/abs(x-ref_val)) > max_val:
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I = integrand(x + ref_val)
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while I < tol:
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log.debug("x={:.3e} I(x+ref_val) = {:.3e}".format(x, I))
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if (1/abs(x)) > max_val:
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raise RuntimeError("1/|x-ref_val| > max_val was reached")
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x /= 2
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a = brentq(lambda x: integrand(x)-tol, x+ref_val, x0+ref_val)
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I = integrand(x+ref_val)
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log.debug("x={:.3e} I(x+ref_val) = {:.3e} > tol".format(x, I))
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log.debug("search for root in interval [{:.3e},{:.3e}]".format(x0+ref_val, x+ref_val))
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a = brentq(lambda x_: integrand(x_)-tol, x+ref_val, x0+ref_val)
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log.debug("found I(a={:.3e}) = {:.3e} = tol".format(a, integrand(a)))
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else:
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a = x0
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log.debug("I(ref_val) = tol -> no search necessary")
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log.debug("return a={:.5g}".format(a))
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return a
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def find_integral_boundary_auto(integrand, tol, ref_val=0, max_val=1e6,
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@ -51,44 +75,49 @@ def find_integral_boundary_auto(integrand, tol, ref_val=0, max_val=1e6,
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max_val_left = max_val if max_val_left is None else max_val_left
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max_val_right = max_val if max_val_right is None else max_val_right
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log.debug("trigger left search")
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a = find_integral_boundary(integrand, tol, ref_val=ref_val_left, max_val=max_val_left, x0=-1)
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log.debug("trigger right search")
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b = find_integral_boundary(integrand, tol, ref_val=ref_val_right, max_val=max_val_right, x0=+1)
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return a,b
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def fourier_integral(integrand, a, b, N):
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"""
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approximates int_a^b dx integrand(x) by the riemann sum with N terms
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and the most simplest uniform midpoint weights
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"""
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log.debug("integrate over [{:.3e},{:.3e}] using {} points".format(a,b,N))
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delta_x = (b-a)/N
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delta_k = 2*np.pi/(b-a)
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yl = integrand(np.linspace(a+delta_x/2, b+delta_x/2, N, endpoint=False))
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fft_vals = np_rfft(yl)
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tau = np.arange(len(fft_vals))*delta_k
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log.debug("yields d_x={:.3e}, d_k={:.3e} kmax={:.3e}".format(delta_x, delta_k, tau[-1]))
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return tau, delta_x*np.exp(-1j*tau*(a+delta_x/2))*fft_vals
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def get_N_for_accurate_fourier_integral(integrand, a, b, t_0, t_max, tol, ft_ref, N_max = 2**15):
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def get_N_for_accurate_fourier_integral(integrand, a, b, t_max, ft_ref, tol=1e-3, N_max = 2**20):
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"""
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chooses N such that the approximated Fourier integral
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meets the exact solution within a given tolerance of the
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relative deviation for a given interval of interest
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"""
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log.debug("FFT accuracy for k in [0, {:.3e}] (tol={:.3e})".format(t_max, tol))
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i = 10
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while True:
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N = 2**i
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tau, ft_tau = fourier_integral(integrand, a, b, N)
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idx = np.where(tau <= t_max)
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ft_ref_tau = ft_ref(tau[idx])
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rd = np.abs(ft_tau[idx] - ft_ref_tau) / np.abs(ft_ref_tau)
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rd = np.max(np.abs(ft_tau[idx] - ft_ref_tau) / np.abs(ft_ref_tau))
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log.debug("N:{} found rel dif of {:.3e}".format(N, rd))
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if rd < tol:
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log.debug("reached rd ({:.3e}) < tol ({:.3e}), return N={}".format(rd, tol, N))
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return N
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if N > N_max:
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raise RuntimeError("maximum number of points for Fourier Transform reached")
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i += 2
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i += 1
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@ -12,6 +12,11 @@ sys.path.insert(0, str(p.parent.parent))
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import stocproc as sp
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import logging
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logging.basicConfig(level=logging.DEBUG)
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def test_find_integral_boundary():
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def f(x):
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return np.exp(-(x)**2)
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@ -78,6 +83,25 @@ def fourier_integral_trapz(integrand, a, b, N):
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return tau, delta_x*np.exp(-1j*tau*a)*fft_vals
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def fourier_integral_simps(integrand, a, b, N):
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"""
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approximates int_a^b dx integrand(x) by the riemann sum with N terms
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"""
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assert (N % 2) == 1
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yl = integrand(np.linspace(a, b, N, endpoint=True))
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yl[1::2] *= 4
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yl[2:-2:2] *= 2
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delta_x = (b - a) / (N-1)
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delta_k = 2 * np.pi / (N*delta_x)
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fft_vals = np.fft.rfft(yl)
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tau = np.arange(len(fft_vals)) * delta_k
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return tau, delta_x/3 * np.exp(-1j * tau * a) * fft_vals
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def fourier_integral_simple_test(integrand, a, b, N):
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delta_x = (b-a)/N
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delta_k = 2*np.pi/(b-a)
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@ -119,6 +143,23 @@ def fourier_integral_trapz_simple_test(integrand, a, b, N):
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return k, res
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def osd(w, s, wc):
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if not isinstance(w, np.ndarray):
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if w < 0:
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return 0
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else:
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return w ** s * np.exp(-w / wc)
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else:
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res = np.zeros(shape=w.shape)
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w_flat = w.flatten()
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idx_pos = np.where(w_flat > 0)
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fv_res = res.flat
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fv_res[idx_pos] = w_flat ** s * np.exp(-w_flat / wc)
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return res
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def test_fourier_integral():
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intg = lambda x: x**2
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a = -1.23
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@ -149,47 +190,28 @@ def test_fourier_integral():
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assert rd < 4e-6
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N = 512
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tau, ft_n = sp.method_fft.fourier_integral(intg, a, b, N)
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ft_ref_n = ft_ref(tau)
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N = 1024
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for fac in [1,2,4,8]:
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k, ft_n = sp.method_fft.fourier_integral(intg, a, b, fac*N)
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ft_ref_n = ft_ref(k)
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# assert np.max(np.abs(ft_n-ft_ref_n)) < 1e-11
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plt.plot(k, np.abs(ft_n-ft_ref_n)/np.abs(ft_ref_n), label='simple f:{}'.format(fac))
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k, ft_simple = fourier_integral_simple_test(intg, a, b, N)
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assert np.max(np.abs(ft_simple-ft_n)) < 1e-11
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k, ft_n = fourier_integral_trapz(intg, a, b, fac*N)
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ft_ref_n = ft_ref(k)
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# assert np.max(np.abs(ft_n-ft_ref_n)) < 1e-11
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plt.plot(k, np.abs(ft_n-ft_ref_n)/np.abs(ft_ref_n), label='trapz f:{}'.format(fac))
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# plt.plot(np.abs(ft_simple-ft_n))
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# plt.yscale('log')
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# plt.show()
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k, ft_n = fourier_integral_simps(intg, a, b, fac*N-1)
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ft_ref_n = ft_ref(k)
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# assert np.max(np.abs(ft_n-ft_ref_n)) < 1e-11
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plt.plot(k, np.abs(ft_n-ft_ref_n)/np.abs(ft_ref_n), label='simps f:{}'.format(fac))
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N = 512
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tau, ft_n = fourier_integral_trapz(intg, a, b, N)
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ft_ref_n = ft_ref(tau)
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k, ft_simple = fourier_integral_trapz_simple_test(intg, a, b, N)
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assert np.max(np.abs(ft_simple-ft_n)) < 1e-11
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# plt.plot(np.abs(ft_simple-ft_n))
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# plt.yscale('log')
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# plt.show()
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# sys.exit()
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def osd(w, s, wc):
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if not isinstance(w, np.ndarray):
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if w < 0:
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return 0
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else:
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return w**s*np.exp(-w/wc)
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else:
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res = np.zeros(shape=w.shape)
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w_flat = w.flatten()
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idx_pos = np.where(w_flat > 0)
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fv_res = res.flat
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fv_res[idx_pos] = w_flat**s*np.exp(-w_flat/wc)
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return res
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plt.grid()
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plt.yscale('log')
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plt.legend()
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plt.show()
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sys.exit()
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s = 0.1
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wc = 4
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@ -225,11 +247,19 @@ def test_fourier_integral():
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# plt.yscale('log')
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# plt.show()
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def test_get_N_for_accurate_fourier_integral():
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s = 0.1
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wc = 4
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intg = lambda x: osd(x, s, wc)
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bcf_ref = lambda t: gamma_func(s + 1) * wc ** (s + 1) * (1 + 1j * wc * t) ** (-(s + 1))
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a, b = sp.method_fft.find_integral_boundary_auto(integrand=intg, tol=1e-12, ref_val=1)
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N = sp.method_fft.get_N_for_accurate_fourier_integral(intg, a, b, t_max = 40, ft_ref=bcf_ref, tol = 1e-3, N_max = 2**20)
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if __name__ == "__main__":
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# test_find_integral_boundary()
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# test_find_integral_boundary()
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test_fourier_integral()
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# test_get_N_for_accurate_fourier_integral()
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