mirror of
https://github.com/wassname/simpeg.git
synced 2026-06-27 21:53:41 +08:00
Lindsey Heagy
317 lines
11 KiB
Python
317 lines
11 KiB
Python
import Utils, Maps, Mesh, numpy as np, scipy.sparse as sp
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class BaseRegularization(object):
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"""
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**Base Regularization Class**
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This is used to regularize the model space::
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reg = Regularization(mesh)
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"""
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__metaclass__ = Utils.SimPEGMetaClass
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counter = None
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mapPair = Maps.IdentityMap #: A SimPEG.Map Class
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mapping = None #: A SimPEG.Map instance.
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mesh = None #: A SimPEG.Mesh instance.
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mref = None #: Reference model.
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def __init__(self, mesh, mapping=None, **kwargs):
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Utils.setKwargs(self, **kwargs)
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self.mesh = mesh
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assert isinstance(mesh, Mesh.BaseMesh), "mesh must be a SimPEG.Mesh object."
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self.mapping = mapping or Maps.IdentityMap(mesh)
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self.mapping._assertMatchesPair(self.mapPair)
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@property
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def parent(self):
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"""This is the parent of the regularization."""
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return getattr(self,'_parent',None)
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@parent.setter
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def parent(self, p):
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if getattr(self,'_parent',None) is not None:
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print 'Regularization has switched to a new parent!'
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self._parent = p
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@property
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def inv(self): return self.parent.inv
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@property
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def invProb(self): return self.parent
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@property
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def reg(self): return self
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@property
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def opt(self): return self.parent.opt
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@property
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def prob(self): return self.parent.prob
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@property
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def survey(self): return self.parent.survey
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@property
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def W(self):
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"""Full regularization weighting matrix W."""
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return sp.identity(self.mapping.nP)
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@Utils.timeIt
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def eval(self, m):
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r = self.W * ( self.mapping * (m - self.mref) )
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return 0.5*r.dot(r)
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@Utils.timeIt
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def evalDeriv(self, m):
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"""
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The regularization is:
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.. math::
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R(m) = \\frac{1}{2}\mathbf{(m-m_\\text{ref})^\\top W^\\top W(m-m_\\text{ref})}
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So the derivative is straight forward:
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.. math::
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R(m) = \mathbf{W^\\top W (m-m_\\text{ref})}
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"""
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mD = self.mapping.deriv(m - self.mref)
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r = self.W * ( self.mapping * (m - self.mref) )
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return mD.T * ( self.W.T * r )
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@Utils.timeIt
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def eval2Deriv(self, m, v=None):
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"""
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:param numpy.array m: geophysical model
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:param numpy.array v: vector to multiply
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:rtype: scipy.sparse.csr_matrix or numpy.ndarray
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:return: WtW or WtW*v
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The regularization is:
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.. math::
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R(m) = \\frac{1}{2}\mathbf{(m-m_\\text{ref})^\\top W^\\top W(m-m_\\text{ref})}
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So the second derivative is straight forward:
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.. math::
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R(m) = \mathbf{W^\\top W}
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"""
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mD = self.mapping.deriv(m - self.mref)
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if v is None:
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return mD.T * self.W.T * self.W * mD
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return mD.T * ( self.W.T * ( self.W * ( mD * v) ) )
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class Tikhonov(BaseRegularization):
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"""**Tikhonov Regularization**
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Here we will define regularization of a model, m, in general however, this should be thought of as (m-m_ref) but otherwise it is exactly the same:
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.. math::
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R(m) = \int_\Omega \\frac{\\alpha_x}{2}\left(\\frac{\partial m}{\partial x}\\right)^2 + \\frac{\\alpha_y}{2}\left(\\frac{\partial m}{\partial y}\\right)^2 \partial v
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Our discrete gradient operator works on cell centers and gives the derivative on the cell faces, which is not where we want to be evaluating this integral. We need to average the values back to the cell-centers before we integrate. To avoid null spaces, we square first and then average. In 2D with ij notation it looks like this:
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.. math::
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R(m) \\approx \sum_{ij} \left[\\frac{\\alpha_x}{2}\left[\left(\\frac{m_{i+1,j} - m_{i,j}}{h}\\right)^2 + \left(\\frac{m_{i,j} - m_{i-1,j}}{h}\\right)^2\\right]
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+ \\frac{\\alpha_y}{2}\left[\left(\\frac{m_{i,j+1} - m_{i,j}}{h}\\right)^2 + \left(\\frac{m_{i,j} - m_{i,j-1}}{h}\\right)^2\\right]
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\\right]h^2
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If we let D_1 be the derivative matrix in the x direction
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.. math::
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\mathbf{D}_1 = \mathbf{I}_2\otimes\mathbf{d}_1
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.. math::
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\mathbf{D}_2 = \mathbf{d}_2\otimes\mathbf{I}_1
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Where d_1 is the one dimensional derivative:
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.. math::
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\mathbf{d}_1 = \\frac{1}{h} \left[ \\begin{array}{cccc}
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-1 & 1 & & \\\\
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& \ddots & \ddots&\\\\
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& & -1 & 1\end{array} \\right]
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.. math::
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R(m) \\approx \mathbf{v}^\\top \left[\\frac{\\alpha_x}{2}\mathbf{A}_1 (\mathbf{D}_1 m) \odot (\mathbf{D}_1 m) + \\frac{\\alpha_y}{2}\mathbf{A}_2 (\mathbf{D}_2 m) \odot (\mathbf{D}_2 m) \\right]
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Recall that this is really a just point wise multiplication, or a diagonal matrix times a vector. When we multiply by something in a diagonal we can interchange and it gives the same results (i.e. it is point wise)
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.. math::
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\mathbf{a\odot b} = \\text{diag}(\mathbf{a})\mathbf{b} = \\text{diag}(\mathbf{b})\mathbf{a} = \mathbf{b\odot a}
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and the transpose also is true (but the sizes have to make sense...):
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.. math::
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\mathbf{a}^\\top\\text{diag}(\mathbf{b}) = \mathbf{b}^\\top\\text{diag}(\mathbf{a})
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So R(m) can simplify to:
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.. math::
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R(m) \\approx \mathbf{m}^\\top \left[\\frac{\\alpha_x}{2}\mathbf{D}_1^\\top \\text{diag}(\mathbf{A}_1^\\top\mathbf{v}) \mathbf{D}_1 + \\frac{\\alpha_y}{2}\mathbf{D}_2^\\top \\text{diag}(\mathbf{A}_2^\\top \mathbf{v}) \mathbf{D}_2 \\right] \mathbf{m}
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We will define W_x as:
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.. math::
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\mathbf{W}_x = \sqrt{\\alpha_x}\\text{diag}\left(\sqrt{\mathbf{A}_1^\\top\mathbf{v}}\\right) \mathbf{D}_1
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And then W as a tall matrix of all of the different regularization terms:
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.. math::
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\mathbf{W} = \left[ \\begin{array}{c}
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\mathbf{W}_s\\\\
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\mathbf{W}_x\\\\
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\mathbf{W}_y\end{array} \\right]
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Then we can write
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.. math::
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R(m) \\approx \\frac{1}{2}\mathbf{m^\\top W^\\top W m}
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"""
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smoothModel = True #: SMOOTH and SMOOTH_MOD_DIF options
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alpha_s = Utils.dependentProperty('_alpha_s', 1e-6, ['_W', '_Ws'], "Smallness weight")
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alpha_x = Utils.dependentProperty('_alpha_x', 1.0, ['_W', '_Wx'], "Weight for the first derivative in the x direction")
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alpha_y = Utils.dependentProperty('_alpha_y', 1.0, ['_W', '_Wy'], "Weight for the first derivative in the y direction")
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alpha_z = Utils.dependentProperty('_alpha_z', 1.0, ['_W', '_Wz'], "Weight for the first derivative in the z direction")
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alpha_xx = Utils.dependentProperty('_alpha_xx', 0.0, ['_W', '_Wxx'], "Weight for the second derivative in the x direction")
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alpha_yy = Utils.dependentProperty('_alpha_yy', 0.0, ['_W', '_Wyy'], "Weight for the second derivative in the y direction")
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alpha_zz = Utils.dependentProperty('_alpha_zz', 0.0, ['_W', '_Wzz'], "Weight for the second derivative in the z direction")
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def __init__(self, mesh, mapping=None, **kwargs):
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BaseRegularization.__init__(self, mesh, mapping=mapping, **kwargs)
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@property
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def Ws(self):
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"""Regularization matrix Ws"""
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if getattr(self,'_Ws', None) is None:
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self._Ws = Utils.sdiag((self.mesh.vol*self.alpha_s)**0.5)
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return self._Ws
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@property
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def Wx(self):
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"""Regularization matrix Wx"""
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if getattr(self, '_Wx', None) is None:
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Ave_x_vol = self.mesh.aveF2CC[:,:self.mesh.nFx].T*self.mesh.vol
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self._Wx = Utils.sdiag((Ave_x_vol*self.alpha_x)**0.5)*self.mesh.cellGradx
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return self._Wx
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@property
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def Wy(self):
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"""Regularization matrix Wy"""
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if getattr(self, '_Wy', None) is None:
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Ave_y_vol = self.mesh.aveF2CC[:,self.mesh.nFx:np.sum(self.mesh.vnF[:2])].T*self.mesh.vol
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self._Wy = Utils.sdiag((Ave_y_vol*self.alpha_y)**0.5)*self.mesh.cellGrady
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return self._Wy
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@property
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def Wz(self):
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"""Regularization matrix Wz"""
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if getattr(self, '_Wz', None) is None:
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Ave_z_vol = self.mesh.aveF2CC[:,np.sum(self.mesh.vnF[:2]):].T*self.mesh.vol
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self._Wz = Utils.sdiag((Ave_z_vol*self.alpha_z)**0.5)*self.mesh.cellGradz
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return self._Wz
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@property
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def Wxx(self):
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"""Regularization matrix Wxx"""
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if getattr(self, '_Wxx', None) is None:
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self._Wxx = Utils.sdiag((self.mesh.vol*self.alpha_xx)**0.5)*self.mesh.faceDivx*self.mesh.cellGradx
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return self._Wxx
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@property
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def Wyy(self):
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"""Regularization matrix Wyy"""
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if getattr(self, '_Wyy', None) is None:
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self._Wyy = Utils.sdiag((self.mesh.vol*self.alpha_yy)**0.5)*self.mesh.faceDivy*self.mesh.cellGrady
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return self._Wyy
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@property
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def Wzz(self):
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"""Regularization matrix Wzz"""
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if getattr(self, '_Wzz', None) is None:
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self._Wzz = Utils.sdiag((self.mesh.vol*self.alpha_zz)**0.5)*self.mesh.faceDivz*self.mesh.cellGradz
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return self._Wzz
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@property
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def W(self):
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"""Full regularization matrix W"""
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if getattr(self, '_W', None) is None:
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wlist = (self.Ws, self.Wx, self.Wxx)
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if self.mesh.dim > 1:
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wlist += (self.Wy, self.Wyy)
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if self.mesh.dim > 2:
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wlist += (self.Wz, self.Wzz)
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self._W = sp.vstack(wlist)
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return self._W
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@Utils.timeIt
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def eval(self, m):
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if self.smoothModel == True:
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r1 = self.W * ( self.mapping * (m - self.mref) )
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r2 = self.Ws * ( self.mapping * (m - self.mref) )
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return 0.5*(r1.dot(r1)+r2.dot(r2))
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elif self.smoothModel == False:
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r = self.W * ( self.mapping * (m - self.mref) )
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return 0.5*r.dot(r)
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@Utils.timeIt
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def evalDeriv(self, m):
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"""
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The regularization is:
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.. math::
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R(m) = \\frac{1}{2}\mathbf{(m-m_\\text{ref})^\\top W^\\top W(m-m_\\text{ref})}
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So the derivative is straight forward:
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.. math::
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R(m) = \mathbf{W^\\top W (m-m_\\text{ref})}
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"""
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if self.smoothModel == True:
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mD1 = self.mapping.deriv(m)
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mD2 = self.mapping.deriv(self.mref)
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r1 = self.W * ( self.mapping * (m) )
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r2 = self.Ws * ( self.mapping * (self.mref) )
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out1 = mD1.T * ( self.W.T * r1 )
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out2 = mD2.T * ( self.Ws.T * r2 )
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out = out1-out2
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elif self.smoothModel == False:
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mD = self.mapping.deriv(m - self.mref)
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r = self.W * ( self.mapping * (m - self.mref) )
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out = mD.T * ( self.W.T * r )
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return out
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