mirror of
https://github.com/wassname/simpeg.git
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Creating the inverse problem framework. Feedback welcome!
This commit is contained in:
Rowan Cockett
+32
-32
@@ -2,25 +2,25 @@ import numpy as np
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import matplotlib.pyplot as plt
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from pylab import norm
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def GaussNewton(fctn, x0,maxIter=20, maxIterLS=10, LSreduction=1e-4, tolJ=1e-3, tolX=1e-3,
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def GaussNewton(fctn, x0,maxIter=20, maxIterLS=10, LSreduction=1e-4, tolJ=1e-3, tolX=1e-3,
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tolG=1e-3, eps=1e-16, xStop=[]):
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"""
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GaussNewton Optimization
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Input:
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------
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fctn - objective Function (lambda function)
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x0 - starting guess
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Output:
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-------
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xOpt - numerical optimizer
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"""
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# initial output
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print "%s GaussNewton %s" % ('='*22,'='*22)
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print "iter\tJc\t\tnorm(dJ)\tLS"
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print "iter\tJc\t\tnorm(dJ)\tLS"
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print "%s" % '-'*57
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# evaluate stopping criteria
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if xStop==[]:
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xStop=x0
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@@ -31,26 +31,26 @@ def GaussNewton(fctn, x0,maxIter=20, maxIterLS=10, LSreduction=1e-4, tolJ=1e-3,
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xc = x0
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STOP = np.zeros((5,1),dtype=bool)
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iterLS=0; iter=0
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Jold = Jstop
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xOld=xc
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while 1:
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# evaluate objective function
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Jc,dJ,H = fctn(xc)
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print "%3d\t%1.2e\t%1.2e\t%d" % (iter, Jc[0],norm(dJ),iterLS)
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# check stopping rules
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STOP[0] = (iter>0) & (abs(Jc[0]-Jold[0]) <= tolJ*(1+abs(Jstop[0])))
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STOP[1] = (iter>0) & (norm(xc-xOld) <= tolX*(1+norm(x0)))
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STOP[2] = norm(dJ) <= tolG*(1+abs(Jstop[0]))
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STOP[3] = norm(dJ) <= 1e3*eps
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STOP[4] = (iter >= maxIter)
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if all(STOP[0:3]) | any(STOP[3:]):
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if all(STOP[0:3]) | any(STOP[3:]):
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break
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# get search direction
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dx = np.linalg.solve(H,-dJ)
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# Armijo linesearch
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descent = np.dot(dJ.T,dx)
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LS =0; t = 1; iterLS=1
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@@ -62,13 +62,13 @@ def GaussNewton(fctn, x0,maxIter=20, maxIterLS=10, LSreduction=1e-4, tolJ=1e-3,
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break
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iterLS = iterLS+1
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t = .5*t
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# store old values
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Jold = Jc; xOld = xc
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# update
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# update
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xc = xt
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iter = iter +1
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print "%s STOP! %s" % ('-'*25,'-'*25)
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print "%d : |Jc-Jold| = %1.4e <= tolJ*(1+|Jstop|) = %1.4e" % (STOP[0],abs(Jc[0]-Jold[0]),tolJ*(1+abs(Jstop[0])))
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print "%d : |xc-xOld| = %1.4e <= tolX*(1+|x0|) = %1.4e" % (STOP[1],norm(xc-xOld),tolX*(1+norm(x0)))
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@@ -76,50 +76,50 @@ def GaussNewton(fctn, x0,maxIter=20, maxIterLS=10, LSreduction=1e-4, tolJ=1e-3,
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print "%d : |dJ| = %1.4e <= 1e3*eps = %1.4e" % (STOP[3],norm(dJ),1e3*eps)
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print "%d : iter = %3d\t <= maxIter\t = %3d" % (STOP[4],iter,maxIter)
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print "%s DONE! %s\n" % ('='*25,'='*25)
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return xc
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def Rosenbrock(x):
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"""
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Rosenbrock function for testing GaussNewton scheme
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"""
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J = 100*(x[1]-x[0]**2)**2+(1-x[0])**2
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J = 100*(x[1]-x[0]**2)**2+(1-x[0])**2
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dJ = np.array([2*(200*x[0]**3-200*x[0]*x[1]+x[0]-1),200*(x[1]-x[0]**2)])
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H = np.array([[-400*x[1]+1200*x[0]**2+2, -400*x[0]],[ -400*x[0], 200]],dtype=float);
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return J,dJ,H
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def checkDerivative(fctn,x0):
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"""
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Basic derivative check
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Compares error decay of 0th and 1st order Taylor approximation at point
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Compares error decay of 0th and 1st order Taylor approximation at point
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x0 for a randomized search direction.
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Input:
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------
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fctn - function handle
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x0 - point at which to check derivative
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x0 - point at which to check derivative
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"""
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print "%s checkDerivative %s" % ('='*20,'='*20)
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print "iter\th\t\t|J0-Jt|\t\t|J0+h*dJ'*dx-Jt|"
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Jc,dJ,H = fctn(x0)
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dx = np.random.randn(len(x0),1)
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t = np.logspace(-1,-10,10)
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E0 = np.zeros(t.shape)
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E1 = np.zeros(t.shape)
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for i in range(0,10):
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Jt = fctn(x0+t[i]*dx)
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E0[i] = norm(Jt[0]-Jc[0]) # 0th order Taylor
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E1[i] = norm(Jt[0]-Jc[0]-t[i]*np.dot(dJ.T,dx)) # 1st order Taylor
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E0[i] = norm(Jt[0]-Jc[0]) # 0th order Taylor
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E1[i] = norm(Jt[0]-Jc[0]-t[i]*np.dot(dJ.T,dx)) # 1st order Taylor
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print "%d\t%1.2e\t%1.3e\t%1.3e" % (i,t[i],E0[i],E1[i])
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print "%s DONE! %s\n" % ('='*25,'='*25)
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plt.figure()
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plt.clf()
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@@ -131,11 +131,11 @@ def checkDerivative(fctn,x0):
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plt.legend(['0th order', '1st order'],loc='upper left')
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plt.show()
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return
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if __name__ == '__main__':
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x0 = np.array([[2.6],[3.7]])
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fctn = lambda x:Rosenbrock(x)
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checkDerivative(fctn,x0)
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xOpt = GaussNewton(fctn,x0,maxIter=20)
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print "xOpt=[%f,%f]" % (xOpt[0],xOpt[1])
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@@ -1,3 +1,4 @@
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from TensorMesh import TensorMesh
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from LogicallyOrthogonalMesh import LogicallyOrthogonalMesh
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import utils
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import inverse
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@@ -0,0 +1,115 @@
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import numpy as np
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from SimPEG.utils import mkvc, sdiag
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norm = np.linalg.norm
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class Problem(object):
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"""Problem is the base class for all geophysical forward problems in SimPEG"""
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def __init__(self, mesh):
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self.mesh = mesh
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pass
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def residual(self, m):
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pass
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def modelTransform(self, m):
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"""
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:param numpy.array m: model
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:rtype: numpy.array
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:return: transformed model
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The modelTransform changes the model into the physical property.
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A common example of this is to invert for electrical conductivity
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in log space. In this case, your model will be log(sigma) and to
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get back to sigma, you can take the exponential:
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.. math::
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m = \log{\sigma}
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\exp{m} = \exp{\log{\sigma}} = \sigma
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"""
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return np.exp(mkvc(m))
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def modelTransformDeriv(self, m):
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"""
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:param numpy.array m: model
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:rtype: scipy.csr_matrix
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:return: derivative of transformed model
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The modelTransform changes the model into the physical property.
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The modelTransformDeriv provides the derivative of the modelTransform.
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If the model transform is:
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.. math::
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m = \log{\sigma}
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\exp{m} = \exp{\log{\sigma}} = \sigma
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Then the derivative is:
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.. math::
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\\frac{\partial \exp{m}}{\partial m} = \\text{sdiag}(\exp{m})
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"""
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return sdiag(np.exp(mkvc(m)))
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def _test_modelTransformDeriv(self):
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m = np.random.rand(5)
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return checkDerivative(lambda m : [self.modelTransform(m), self.modelTransformDeriv(m)], m)
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def misfit(self, field):
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"""
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:param numpy.array field: geophysical field of interest
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:rtype: float
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:return: data misfit
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The data misfit using an l_2 norm is:
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.. math::
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\mu_\\text{data} = {1\over 2}\left| \mathbf{W} (\mathbf{Pu} - d_\\text{obs}) \\right|_2^2
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Where P is a projection matrix that brings the field on the full domain to the data measurement locations;
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u is the field of interest; d_obs is the observed data; and W is the weighting matrix.
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"""
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R = self.W*(self.P*field - self.dobs)
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return 0.5*mkvc(R).inner(mkvc(R))
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def misfitDeriv(self, field):
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"""
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TODO: Change this documentation.
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:param numpy.array field: geophysical field of interest
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:rtype: float
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:return: data misfit derivative
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The data misfit using an l_2 norm is:
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.. math::
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\mu_\\text{data} = {1\over 2}\left| \mathbf{W} (\mathbf{Pu} - d_\\text{obs}) \\right|_2^2
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Where P is a projection matrix that brings the field on the full domain to the data measurement locations;
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u is the field of interest; d_obs is the observed data; and W is the weighting matrix.
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"""
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R = self.W*(self.P*field - self.dobs)
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# TODO: make in terms of the field and call Jt, e.g. if looping over RHSs using i: self.Jt(field[:,i],self.W[:,i]*R[:,i])
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return mkvc(R)
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def J(self, u):
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pass
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def Jt(self, v):
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pass
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if __name__ == '__main__':
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from SimPEG.inverse import checkDerivative
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p = Problem(None)
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m = np.random.rand(5)
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checkDerivative(lambda m : [p.modelTransform(m), p.modelTransformDeriv(m)], m)
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@@ -0,0 +1 @@
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from Problem import *
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@@ -0,0 +1,222 @@
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import numpy as np
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import matplotlib.pyplot as plt
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from SimPEG.utils import mkvc, sdiag
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norm = np.linalg.norm
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class Minimize(object):
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"""docstring for Minimize"""
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name = "GeneralOptimizationAlgorithm"
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maxIter = 20
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maxIterLS = 10
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LSreduction = 1e-4
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LSshorten = 0.5
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tolF = 1e-4
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tolX = 1e-4
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tolG = 1e-4
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eps = 1e-16
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def __init__(self, problem, **kwargs):
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self.problem = problem
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# Set the variables, throw an error if they don't exist.
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for attr in kwargs:
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if hasattr(self, attr):
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setattr(self, attr, kwargs[attr])
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else:
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raise Exception('%s attr is not recognized' % attr)
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def minimize(self, x0):
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self.startup(x0)
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self.printInit()
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while True:
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self.f, self.g, self.H = self.evalFunction(self.xc)
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self.printIter()
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if self.stoppingCriteria(): break
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p = self.findSearchDirection()
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xt, passLS = self.linesearch(p)
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if not passLS:
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xt = self.linesearchBreak(p)
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self.doEndIteration(xt)
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self.printDone()
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return self.xc
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def startup(self, x0):
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self._iter = 0
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self._iterLS = 0
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self._STOP = np.zeros((5,1),dtype=bool)
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self.x0 = x0
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self.xc = x0
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self.xOld = x0
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def printInit(self):
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print "%s %s %s" % ('='*22, self.name, '='*22)
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print "iter\tJc\t\tnorm(dJ)\tLS"
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print "%s" % '-'*57
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def printIter(self):
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print "%3d\t%1.2e\t%1.2e\t%d" % (self._iter, self.f, norm(self.g), self._iterLS)
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def printDone(self):
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print "%s STOP! %s" % ('-'*25,'-'*25)
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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)))
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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)))
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print "%d : |g| = %1.4e <= tolG*(1+|fStop|) = %1.4e" % (self._STOP[2], norm(self.g), self.tolG*(1+abs(self.fStop)))
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print "%d : |g| = %1.4e <= 1e3*eps = %1.4e" % (self._STOP[3], norm(self.g), 1e3*self.eps)
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print "%d : iter = %3d\t <= maxIter\t = %3d" % (self._STOP[4], self._iter, self.maxIter)
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print "%s DONE! %s\n" % ('='*25,'='*25)
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def evalFunction(self, x, doDerivative=True):
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f, g, H = self.problem(x)
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return f, g, H
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def findSearchDirection(self):
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return -self.g
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def stoppingCriteria(self):
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if self._iter == 0:
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self.fStop = self.f # Save this for stopping criteria
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# check stopping rules
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self._STOP[0] = self._iter > 0 and (abs(self.f-self.fOld) <= self.tolF*(1+abs(self.fStop)))
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self._STOP[1] = self._iter > 0 and (norm(self.xc-self.xOld) <= self.tolX*(1+norm(self.x0)))
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self._STOP[2] = norm(self.g) <= self.tolG*(1+abs(self.fStop))
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self._STOP[3] = norm(self.g) <= 1e3*self.eps
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self._STOP[4] = self._iter >= self.maxIter
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return all(self._STOP[0:3]) | any(self._STOP[3:])
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def linesearch(self, p):
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# Armijo linesearch
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descent = np.inner(self.g, p)
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t = 1
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iterLS = 0
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while iterLS < self.maxIterLS:
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xt = self.xc + t*p
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ft, temp, temp = self.evalFunction(xt, doDerivative=False)
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if ft < self.f + t*self.LSreduction*descent:
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break
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iterLS += 1
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t = self.LSshorten*t
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self._iterLS = iterLS
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return xt, iterLS < self.maxIterLS
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def linesearchBreak(self, p):
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raise Exception('The linesearch got broken. Boo.')
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def doEndIteration(self, xt):
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# store old values
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self.fOld = self.f
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self.xOld, self.xc = self.xc, xt
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self._iter += 1
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class GaussNewton(Minimize):
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name = 'GaussNewton'
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def findSearchDirection(self):
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return np.linalg.solve(self.H,-self.g)
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class SteepestDescent(Minimize):
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name = 'SteepestDescent'
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def findSearchDirection(self):
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return -self.g
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def Rosenbrock(x):
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"""Rosenbrock function for testing GaussNewton scheme"""
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f = 100*(x[1]-x[0]**2)**2+(1-x[0])**2
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g = np.array([2*(200*x[0]**3-200*x[0]*x[1]+x[0]-1), 200*(x[1]-x[0]**2)])
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H = np.array([[-400*x[1]+1200*x[0]**2+2, -400*x[0]], [-400*x[0], 200]])
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return f, g, H
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def checkDerivative(fctn, x0, num=7, plotIt=True, dx=None):
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"""
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Basic derivative check
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Compares error decay of 0th and 1st order Taylor approximation at point
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x0 for a randomized search direction.
|
||||
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Input:
|
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------
|
||||
fctn - function handle
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x0 - point at which to check derivative
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"""
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print "%s checkDerivative %s" % ('='*20, '='*20)
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print "iter\th\t\t|J0-Jt|\t\t|J0+h*dJ'*dx-Jt|\tOrder\n%s" % ('-'*57)
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Jc = fctn(x0)
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x0 = mkvc(x0)
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if dx is None:
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dx = np.random.randn(len(x0))
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t = np.logspace(-1, -num, num)
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E0 = np.ones(t.shape)
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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)
|
||||
@@ -0,0 +1 @@
|
||||
from Optimize import *
|
||||
@@ -0,0 +1,8 @@
|
||||
.. _api_Problem:
|
||||
|
||||
Problem
|
||||
*******
|
||||
|
||||
.. automodule:: SimPEG.forward.Problem
|
||||
:members:
|
||||
:undoc-members:
|
||||
+2
-1
@@ -38,12 +38,13 @@ Inversion
|
||||
|
||||
api_GaussNewton
|
||||
|
||||
Example Problems
|
||||
Forward Problems
|
||||
================
|
||||
|
||||
.. toctree::
|
||||
:maxdepth: 2
|
||||
|
||||
api_Problem
|
||||
|
||||
|
||||
Project Index & Search
|
||||
|
||||
Reference in New Issue
Block a user