mirror of
https://github.com/wassname/simpeg.git
synced 2026-07-07 16:01:17 +08:00
Rowan Cockett
223 lines
6.8 KiB
Python
223 lines
6.8 KiB
Python
import numpy as np
|
|
import matplotlib.pyplot as plt
|
|
from SimPEG.utils import mkvc, sdiag
|
|
norm = np.linalg.norm
|
|
|
|
|
|
class Minimize(object):
|
|
"""docstring for Minimize"""
|
|
|
|
name = "GeneralOptimizationAlgorithm"
|
|
|
|
maxIter = 20
|
|
maxIterLS = 10
|
|
LSreduction = 1e-4
|
|
LSshorten = 0.5
|
|
tolF = 1e-4
|
|
tolX = 1e-4
|
|
tolG = 1e-4
|
|
eps = 1e-16
|
|
|
|
def __init__(self, problem, **kwargs):
|
|
self.problem = problem
|
|
|
|
# Set the variables, throw an error if they don't exist.
|
|
for attr in kwargs:
|
|
if hasattr(self, attr):
|
|
setattr(self, attr, kwargs[attr])
|
|
else:
|
|
raise Exception('%s attr is not recognized' % attr)
|
|
|
|
def minimize(self, x0):
|
|
|
|
self.startup(x0)
|
|
self.printInit()
|
|
|
|
while True:
|
|
self.f, self.g, self.H = self.evalFunction(self.xc)
|
|
self.printIter()
|
|
if self.stoppingCriteria(): break
|
|
p = self.findSearchDirection()
|
|
xt, passLS = self.linesearch(p)
|
|
if not passLS:
|
|
xt = self.linesearchBreak(p)
|
|
self.doEndIteration(xt)
|
|
|
|
self.printDone()
|
|
|
|
return self.xc
|
|
|
|
def startup(self, x0):
|
|
self._iter = 0
|
|
self._iterLS = 0
|
|
self._STOP = np.zeros((5,1),dtype=bool)
|
|
|
|
self.x0 = x0
|
|
self.xc = x0
|
|
self.xOld = x0
|
|
|
|
def printInit(self):
|
|
print "%s %s %s" % ('='*22, self.name, '='*22)
|
|
print "iter\tJc\t\tnorm(dJ)\tLS"
|
|
print "%s" % '-'*57
|
|
|
|
def printIter(self):
|
|
print "%3d\t%1.2e\t%1.2e\t%d" % (self._iter, self.f, norm(self.g), self._iterLS)
|
|
|
|
def printDone(self):
|
|
print "%s STOP! %s" % ('-'*25,'-'*25)
|
|
print "%d : |fc-fOld| = %1.4e <= tolF*(1+|fStop|) = %1.4e" % (self._STOP[0], abs(self.f-self.fOld), self.tolF*(1+abs(self.fStop)))
|
|
print "%d : |xc-xOld| = %1.4e <= tolX*(1+|x0|) = %1.4e" % (self._STOP[1], norm(self.xc-self.xOld), self.tolX*(1+norm(self.x0)))
|
|
print "%d : |g| = %1.4e <= tolG*(1+|fStop|) = %1.4e" % (self._STOP[2], norm(self.g), self.tolG*(1+abs(self.fStop)))
|
|
print "%d : |g| = %1.4e <= 1e3*eps = %1.4e" % (self._STOP[3], norm(self.g), 1e3*self.eps)
|
|
print "%d : iter = %3d\t <= maxIter\t = %3d" % (self._STOP[4], self._iter, self.maxIter)
|
|
print "%s DONE! %s\n" % ('='*25,'='*25)
|
|
|
|
def evalFunction(self, x, doDerivative=True):
|
|
f, g, H = self.problem(x)
|
|
return f, g, H
|
|
|
|
def findSearchDirection(self):
|
|
return -self.g
|
|
|
|
def stoppingCriteria(self):
|
|
if self._iter == 0:
|
|
self.fStop = self.f # Save this for stopping criteria
|
|
|
|
# check stopping rules
|
|
self._STOP[0] = self._iter > 0 and (abs(self.f-self.fOld) <= self.tolF*(1+abs(self.fStop)))
|
|
self._STOP[1] = self._iter > 0 and (norm(self.xc-self.xOld) <= self.tolX*(1+norm(self.x0)))
|
|
self._STOP[2] = norm(self.g) <= self.tolG*(1+abs(self.fStop))
|
|
self._STOP[3] = norm(self.g) <= 1e3*self.eps
|
|
self._STOP[4] = self._iter >= self.maxIter
|
|
return all(self._STOP[0:3]) | any(self._STOP[3:])
|
|
|
|
def linesearch(self, p):
|
|
# Armijo linesearch
|
|
descent = np.inner(self.g, p)
|
|
t = 1
|
|
iterLS = 0
|
|
while iterLS < self.maxIterLS:
|
|
xt = self.xc + t*p
|
|
ft, temp, temp = self.evalFunction(xt, doDerivative=False)
|
|
if ft < self.f + t*self.LSreduction*descent:
|
|
break
|
|
iterLS += 1
|
|
t = self.LSshorten*t
|
|
|
|
self._iterLS = iterLS
|
|
return xt, iterLS < self.maxIterLS
|
|
|
|
def linesearchBreak(self, p):
|
|
raise Exception('The linesearch got broken. Boo.')
|
|
|
|
def doEndIteration(self, xt):
|
|
# store old values
|
|
self.fOld = self.f
|
|
self.xOld, self.xc = self.xc, xt
|
|
self._iter += 1
|
|
|
|
|
|
class GaussNewton(Minimize):
|
|
name = 'GaussNewton'
|
|
def findSearchDirection(self):
|
|
return np.linalg.solve(self.H,-self.g)
|
|
|
|
|
|
class SteepestDescent(Minimize):
|
|
name = 'SteepestDescent'
|
|
def findSearchDirection(self):
|
|
return -self.g
|
|
|
|
|
|
|
|
def Rosenbrock(x):
|
|
"""Rosenbrock function for testing GaussNewton scheme"""
|
|
|
|
f = 100*(x[1]-x[0]**2)**2+(1-x[0])**2
|
|
g = np.array([2*(200*x[0]**3-200*x[0]*x[1]+x[0]-1), 200*(x[1]-x[0]**2)])
|
|
H = np.array([[-400*x[1]+1200*x[0]**2+2, -400*x[0]], [-400*x[0], 200]])
|
|
return f, g, H
|
|
|
|
|
|
def checkDerivative(fctn, x0, num=7, plotIt=True, dx=None):
|
|
"""
|
|
Basic derivative check
|
|
|
|
Compares error decay of 0th and 1st order Taylor approximation at point
|
|
x0 for a randomized search direction.
|
|
|
|
Input:
|
|
------
|
|
fctn - function handle
|
|
x0 - point at which to check derivative
|
|
"""
|
|
|
|
print "%s checkDerivative %s" % ('='*20, '='*20)
|
|
print "iter\th\t\t|J0-Jt|\t\t|J0+h*dJ'*dx-Jt|\tOrder\n%s" % ('-'*57)
|
|
|
|
Jc = fctn(x0)
|
|
|
|
x0 = mkvc(x0)
|
|
|
|
if dx is None:
|
|
dx = np.random.randn(len(x0))
|
|
|
|
t = np.logspace(-1, -num, num)
|
|
E0 = np.ones(t.shape)
|
|
E1 = np.ones(t.shape)
|
|
|
|
l2norm = lambda x: np.sqrt(np.inner(x, x)) # because np.norm breaks if they are scalars?
|
|
for i in range(num):
|
|
Jt = fctn(x0+t[i]*dx)
|
|
E0[i] = l2norm(Jt[0]-Jc[0]) # 0th order Taylor
|
|
E1[i] = l2norm(Jt[0]-Jc[0]-t[i]*Jc[1].dot(dx)) # 1st order Taylor
|
|
order0 = np.log10(E0[:-1]/E0[1:])
|
|
order1 = np.log10(E1[:-1]/E1[1:])
|
|
print "%d\t%1.2e\t%1.3e\t\t%1.3e\t\t%1.3f" % (i, t[i], E0[i], E1[i], np.nan if i == 0 else order1[i-1])
|
|
|
|
tolerance = 0.9
|
|
expectedOrder = 2
|
|
eps = 1e-10
|
|
order0 = order0[E0[1:] > eps]
|
|
order1 = order1[E1[1:] > eps]
|
|
belowTol = order1.size == 0 and order0.size > 0
|
|
correctOrder = order1.size > 0 and np.mean(order1) > tolerance * expectedOrder
|
|
|
|
passTest = belowTol or correctOrder
|
|
|
|
if passTest:
|
|
print "%s PASS! %s\n" % ('='*25, '='*25)
|
|
else:
|
|
print "%s\n%s FAIL! %s\n%s" % ('*'*57, '<'*25, '>'*25, '*'*57)
|
|
|
|
if plotIt:
|
|
plt.figure()
|
|
plt.clf()
|
|
plt.loglog(t, E0, 'b')
|
|
plt.loglog(t, E1, 'g--')
|
|
plt.title('checkDerivative')
|
|
plt.xlabel('h')
|
|
plt.ylabel('error of Taylor approximation')
|
|
plt.legend(['0th order', '1st order'], loc='upper left')
|
|
plt.show()
|
|
|
|
return passTest
|
|
|
|
if __name__ == '__main__':
|
|
x0 = np.array([2.6, 3.7])
|
|
checkDerivative(Rosenbrock, x0, plotIt=False)
|
|
xOpt = GaussNewton(Rosenbrock, maxIter=20).minimize(x0)
|
|
print "xOpt=[%f, %f]" % (xOpt[0], xOpt[1])
|
|
xOpt = SteepestDescent(Rosenbrock, maxIter=20, maxIterLS=15).minimize(x0)
|
|
print "xOpt=[%f, %f]" % (xOpt[0], xOpt[1])
|
|
|
|
def simplePass(x):
|
|
return np.sin(x), sdiag(np.cos(x))
|
|
|
|
def simpleFail(x):
|
|
return np.sin(x), -sdiag(np.cos(x))
|
|
|
|
checkDerivative(simplePass, np.random.randn(5), plotIt=False)
|
|
checkDerivative(simpleFail, np.random.randn(5), plotIt=False)
|