Creating the inverse problem framework. Feedback welcome!

SimPEG/forward/Problem.py

@@ -0,0 +1,115 @@
+import numpy as np
+from SimPEG.utils import mkvc, sdiag
+norm = np.linalg.norm
+
+
+class Problem(object):
+    """Problem is the base class for all geophysical forward problems in SimPEG"""
+    def __init__(self, mesh):
+        self.mesh = mesh
+        pass
+
+    def residual(self, m):
+        pass
+
+    def modelTransform(self, m):
+        """
+            :param numpy.array m: model
+            :rtype: numpy.array
+            :return: transformed model
+
+            The modelTransform changes the model into the physical property.
+
+            A common example of this is to invert for electrical conductivity
+            in log space. In this case, your model will be log(sigma) and to
+            get back to sigma, you can take the exponential:
+
+            .. math::
+
+                m = \log{\sigma}
+
+                \exp{m} = \exp{\log{\sigma}} = \sigma
+        """
+        return np.exp(mkvc(m))
+
+    def modelTransformDeriv(self, m):
+        """
+            :param numpy.array m: model
+            :rtype: scipy.csr_matrix
+            :return: derivative of transformed model
+
+            The modelTransform changes the model into the physical property.
+            The modelTransformDeriv provides the derivative of the modelTransform.
+
+            If the model transform is:
+
+            .. math::
+
+                m = \log{\sigma}
+
+                \exp{m} = \exp{\log{\sigma}} = \sigma
+
+            Then the derivative is:
+
+            .. math::
+
+                \\frac{\partial \exp{m}}{\partial m} = \\text{sdiag}(\exp{m})
+        """
+        return sdiag(np.exp(mkvc(m)))
+
+    def _test_modelTransformDeriv(self):
+        m = np.random.rand(5)
+        return checkDerivative(lambda m : [self.modelTransform(m), self.modelTransformDeriv(m)], m)
+
+    def misfit(self, field):
+        """
+            :param numpy.array field: geophysical field of interest
+            :rtype: float
+            :return: data misfit
+
+            The data misfit using an l_2 norm is:
+
+            .. math::
+
+                \mu_\\text{data} = {1\over 2}\left| \mathbf{W} (\mathbf{Pu} - d_\\text{obs}) \\right|_2^2
+
+            Where P is a projection matrix that brings the field on the full domain to the data measurement locations;
+            u is the field of interest; d_obs is the observed data; and W is the weighting matrix.
+        """
+        R = self.W*(self.P*field - self.dobs)
+        return 0.5*mkvc(R).inner(mkvc(R))
+
+    def misfitDeriv(self, field):
+        """
+            TODO: Change this documentation.
+
+            :param numpy.array field: geophysical field of interest
+            :rtype: float
+            :return: data misfit derivative
+
+            The data misfit using an l_2 norm is:
+
+            .. math::
+
+                \mu_\\text{data} = {1\over 2}\left| \mathbf{W} (\mathbf{Pu} - d_\\text{obs}) \\right|_2^2
+
+            Where P is a projection matrix that brings the field on the full domain to the data measurement locations;
+            u is the field of interest; d_obs is the observed data; and W is the weighting matrix.
+        """
+
+        R = self.W*(self.P*field - self.dobs)
+        # 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])
+        return mkvc(R)
+
+    def J(self, u):
+        pass
+
+    def Jt(self, v):
+        pass
+
+if __name__ == '__main__':
+    from SimPEG.inverse import checkDerivative
+
+    p = Problem(None)
+    m = np.random.rand(5)
+    checkDerivative(lambda m : [p.modelTransform(m), p.modelTransformDeriv(m)], m)