Documentation updates for forward problem.

docs/api_Problem.rst

@@ -0,0 +1,71 @@
+.. _api_Problem:
+
+
+Problem
+*******
+
+The problem is a partial differential equation of the form:
+
+.. math::
+
+    c(m, u) = 0
+
+Here, \\(m\\) is the model and u is the field (or fields).
+Given the model, \\(m\\), we can calculate the fields \\(u(m)\\),
+however, the data we collect is a subset of the fields,
+and can be defined by a linear projection, \\(P\\).
+
+.. math::
+
+    d_\text{pred} = P u(m)
+
+For the inverse problem, we are interested in how changing the model transforms the data,
+as such we can take write the Taylor expansion:
+
+.. math::
+
+    Pu(m + hv) = Pu(m) + hP\frac{\partial u(m)}{\partial m} v + \mathcal{O}(h^2 \left\| v \right\| )
+
+We can linearize and define the sensitivity matrix as:
+
+.. math::
+
+    J = P\frac{\partial u}{\partial m}
+
+The sensitivity matrix, and it's transpose will be used in the inverse problem
+to (locally) find how model parameters change the data, and optimize!
+
+
+Working with the general PDE, \\(c(m, u) = 0\\), where m is the model and u is the field,
+the sensitivity is defined as:
+
+.. math::
+
+    J = P\frac{\partial u}{\partial m}
+
+We can take the derivative of the PDE:
+
+.. math::
+
+    \nabla_m c(m, u) \partial m + \nabla_u c(m, u) \partial u = 0
+
+If the forward problem is invertible, then we can rearrange for \\(\\frac{\\partial u}{\\partial m}\\):
+
+.. math::
+
+    J = - P \left( \nabla_u c(m, u) \right)^{-1} \nabla_m c(m, u)
+
+This can often be computed given a vector (i.e. \\(J(v)\\)) rather than stored, as \\(J\\) is a large dense matrix.
+
+
+.. math::
+
+    u(m)
+
+The API
+=======
+
+.. automodule:: SimPEG.Problem
+    :members:
+    :undoc-members:
+