Merge pull request #625 from CUQI-DTU/access_pde_sol

Enable general observation function for PDE, and accessing original solution (or other observable quantities)

Author: amal-ghamdi

cuqi/pde/__init__.py

@@ -4,3 +4,7 @@
     SteadyStateLinearPDE,
     TimeDependentLinearPDE
 )
+
+from ._observation_map import (
+    FD_spatial_gradient
+)

cuqi/pde/_observation_map.py

@@ -0,0 +1,36 @@
+import scipy
+import numpy as np
+"""
+This module contains observation map examples for PDE problems. The map can
+be passed to the `PDE` object initializer via the `observation_map` argument.
+
+For example on how to use set observation maps in time dependent PDEs, see
+`demos/howtos/TimeDependentLinearPDE.py`.
+"""
+
+# 1. Steady State Observation Maps
+# --------------------------------
+
+# 2. Time-Dependent Observation Maps
+# -----------------------------------
+def FD_spatial_gradient(sol, grid, times):
+    """Time dependent observation map that computes the finite difference (FD) spatial gradient of a solution given at grid points (grid) and times (times). This map is supported for 1D spatial domains only.
+    
+    Parameters
+    ----------
+    sol : np.ndarray
+        The solution array of shape (number of grid points, number of time steps).
+
+    grid : np.ndarray
+        The spatial grid points of shape (number of grid points,).
+
+    times : np.ndarray
+        The discretized time steps of shape (number of time steps,)."""
+
+    if len(grid.shape) != 1:
+        raise ValueError("FD_spatial_gradient only supports 1D spatial domains.")
+    observed_quantity = np.zeros((len(grid)-1, len(times)))
+    for i in range(observed_quantity.shape[0]):
+        observed_quantity[i, :] = ((sol[i, :] - sol[i+1, :])/
+                                       (grid[i] - grid[i+1]))
+    return observed_quantity

cuqi/pde/_pde.py

@@ -15,14 +15,15 @@ class PDE(ABC):
     PDE_form : callable function
         Callable function which returns a tuple of the needed PDE components (expected components are explained in the subclasses) 
 
-    observation_map: a function handle
-        A function that takes the PDE solution as input and the returns the observed solution. e.g. `observation_map=lambda u: u**2` or `observation_map=lambda u: u[0]`
-
     grid_sol: np.ndarray
         The grid on which solution is defined
 
     grid_obs: np.ndarray
-        The grid on which the observed solution should be interpolated (currently only supported for 1D problems).  
+        The grid on which the observed solution should be interpolated (currently only supported for 1D problems).
+
+    observation_map: a function handle
+        A function that takes the PDE solution, interpolated on `grid_obs`, as input and returns the observed solution. e.g., `observation_map=lambda u, grid_obs: u**2`.
+
     """
 
     def __init__(self, PDE_form, grid_sol=None, grid_obs=None, observation_map=None):
@@ -187,14 +188,21 @@ def _solve_linear_system(self, A, b, linalg_solve, kwargs):
             info = None
 
         return solution, info
+    
+    def interpolate_on_observed_domain(self, solution):
+        """Interpolate solution on observed space domain."""
+        raise NotImplementedError("interpolate_on_observed_domain method is not implemented for LinearPDE base class.")
 
 class SteadyStateLinearPDE(LinearPDE):
     """Linear steady state PDE.
     
     Parameters
     -----------   
     PDE_form : callable function
-        Callable function with signature `PDE_form(parameter1, parameter2, ...)` where `parameter1`, `parameter2`, etc. are the Bayesian unknown parameters (the user can choose any names for these parameters, e.g. `a`, `b`, etc.). The function returns a tuple with the discretized differential operator A and right-hand-side b. The types of A and b are determined by what the method :meth:`linalg_solve` accepts as first and second parameters, respectively. 
+        Callable function with signature `PDE_form(parameter1, parameter2, ...)` where `parameter1`, `parameter2`, etc. are the Bayesian unknown parameters (the user can choose any names for these parameters, e.g. `a`, `b`, etc.). The function returns a tuple with the discretized differential operator A and right-hand-side b. The types of A and b are determined by what the method :meth:`linalg_solve` accepts as first and second parameters, respectively.
+
+    observation_map: a function handle
+        A function that takes the PDE solution, interpolated on `grid_obs`, as input and returns the observed solution. e.g. `observation_map=lambda u, grid_obs: u**2`.
 
     kwargs: 
         See :class:`~cuqi.pde.LinearPDE` for the remaining keyword arguments. 
@@ -204,8 +212,8 @@ class SteadyStateLinearPDE(LinearPDE):
     See demo demos/demo24_fwd_poisson.py for an illustration on how to use SteadyStateLinearPDE with varying solver choices. And demos demos/demo25_fwd_poisson_2D.py and demos/demo26_fwd_poisson_mixedBC.py for examples with mixed (Dirichlet and Neumann) boundary conditions problems. demos/demo25_fwd_poisson_2D.py also illustrates how to observe on a specific boundary, for example.
     """
 
-    def __init__(self, PDE_form, **kwargs):
-        super().__init__(PDE_form, **kwargs)
+    def __init__(self, PDE_form, observation_map=None, **kwargs):
+        super().__init__(PDE_form, observation_map=observation_map, **kwargs)
 
     def assemble(self, *args, **kwargs):
         """Assembles differential operator and rhs according to PDE_form"""
@@ -221,17 +229,25 @@ def solve(self):
 
         return self._solve_linear_system(self.diff_op, self.rhs, self._linalg_solve, self._linalg_solve_kwargs)
 
-
-    def observe(self, solution):
-            
+    def interpolate_on_observed_domain(self, solution):
+        """Interpolate solution on observed space grid."""
         if self.grids_equal:
             solution_obs = solution
         else:
             solution_obs = interp1d(self.grid_sol, solution, kind='quadratic')(self.grid_obs)
+        return solution_obs
+
+    def observe(self, solution):
+        """Apply observation operator to the solution. This includes
+        interpolation to observation points (if different from the
+        solution grid) then applying the observation map (if provided)."""
+ 
+        # Interpolate solution on observed domain
+        solution_obs = self.interpolate_on_observed_domain(solution)
 
         if self.observation_map is not None:
-            solution_obs = self.observation_map(solution_obs)
-                
+            solution_obs = self.observation_map(solution_obs, self.grid_obs)
+
         return solution_obs
 
 class TimeDependentLinearPDE(LinearPDE):
@@ -251,16 +267,20 @@ class TimeDependentLinearPDE(LinearPDE):
     method: str
         Time stepping method. Currently two options are available `forward_euler` and  `backward_euler`.
 
+    observation_map: a function handle
+        A function that takes the PDE solution, interpolated on `grid_obs` and `time_obs`, as input and returns the observed solution. e.g. `observation_map=lambda u, grid_obs, time_obs: u**2`.
+
     kwargs: 
         See :class:`~cuqi.pde.LinearPDE` for the remaining keyword arguments 
  
     Example
     -----------  
-    See demos/demo34_TimeDependentLinearPDE.py for 1D heat and 1D wave equations.
+    See demos/howtos/TimeDependentLinearPDE.py for 1D heat and 1D wave equations examples. It demonstrates setting up `TimeDependentLinearPDE` objects, including the choice of time stepping methods, observation domain, and observation map.
     """
 
-    def __init__(self, PDE_form, time_steps, time_obs='final', method='forward_euler', **kwargs):
-        super().__init__(PDE_form, **kwargs)
+    def __init__(self, PDE_form, time_steps, time_obs='final',
+                 method='forward_euler', observation_map=None, **kwargs):
+        super().__init__(PDE_form, observation_map=observation_map, **kwargs)
 
         self.time_steps = time_steps
         self.method = method
@@ -339,8 +359,8 @@ def solve(self):
 
         return u, info
 
-    def observe(self, solution):
-
+    def interpolate_on_observed_domain(self, solution):
+        """Interpolate solution on observed time and space points."""
         # If observation grid is the same as solution grid and observation time
         # is the final time step then no need to interpolate
         if self.grids_equal and np.all(self.time_steps[-1:] == self._time_obs):
@@ -361,15 +381,26 @@ def observe(self, solution):
             # Interpolate solution in space and time to the observation
             # time and space
             solution_obs = scipy.interpolate.RectBivariateSpline(
-                self.grid_sol, self.time_steps, solution)(self.grid_obs,
-                                                          self._time_obs)
+                self.grid_sol, self.time_steps, solution
+            )(self.grid_obs, self._time_obs)
 
+        return solution_obs
+
+    def observe(self, solution):
+        """Apply observation operator to the solution. This includes
+        interpolation to observation points (if different from the
+        solution grid) then applying the observation map (if provided)."""
+
+        # Interpolate solution on observed domain
+        solution_obs = self.interpolate_on_observed_domain(solution)
+        
         # Apply observation map
         if self.observation_map is not None:
-            solution_obs = self.observation_map(solution_obs)
+            solution_obs = self.observation_map(solution_obs, self.grid_obs, 
+                                                self._time_obs)
 
         # squeeze if only one time observation
         if len(self._time_obs) == 1:
             solution_obs = solution_obs.squeeze()
 
-        return solution_obs
+        return solution_obs