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rowanc1
265 lines
8.9 KiB
Python
265 lines
8.9 KiB
Python
import Utils, Model, Parameters, 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, model)
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"""
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__metaclass__ = Utils.SimPEGMetaClass
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modelPair = Model.BaseModel #: Some regularizations only work on specific models
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mesh = None #: A SimPEG.Mesh instance.
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model = None #: A SimPEG.Model instance.
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counter = None
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def __init__(self, mesh, model, **kwargs):
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Utils.setKwargs(self, **kwargs)
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assert isinstance(model, self.modelPair), "Incorrect model for this regularization"
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self.model = model
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self.mesh = mesh
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mref = Parameters.ParameterProperty('mref', default=None, doc='Reference model.')
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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 objFunc(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 data(self): return self.parent.data
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@property
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def mesh(self): return self.model.mesh
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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.model.nP)
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@Utils.timeIt
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def modelObj(self, m):
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r = self.W * self.model.transform(m - self.mref)
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return 0.5*r.dot(r)
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@Utils.timeIt
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def modelObjDeriv(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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return self.W.T * ( self.W * self.model.transform(m - self.mref) )
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@Utils.timeIt
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def modelObj2Deriv(self):
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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 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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return self.W.T * self.W
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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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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, model, **kwargs):
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BaseRegularization.__init__(self, mesh, model, **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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