The solving Module

Solving module just provides functional wrappers to the LinearSolver class for one-off computations. For repeated use, please use the LinearSolver class

stat_fem.solving.predict_covariance(A, b, G, data, params, coords, unc, ensemble_comm=<Mock name='mock.COMM_SELF' id='140051716830736'>, **kwargs)

Solve for the predicted covariance of the FEM posterior values at new locations

Standalone function to solve for the predicted posterior covariance of the FEM solution at new sensor locations. Solution is returned as a 2D numpy array holding the covariance matrix. If the solves are parallelized, the non-root process arrays will be an empty. This is just a wrapper to the predict_covariance method of the stat-fem LinearSolver class.

Locations where predictions are desired are specified by coords which is a 2D numpy array where the first index indicates the various prediction points and the second index indicates the spatial dimension. unc is the assumed uncertainty of hypothetical measurements taken at these locations and can either be scalar-like or array-like with the same length as coords. All uncertainties must be non-negative.

In addition to the parameters specified here, additional keyword arguments to control the PETSc solver options can be passed here and will be used when creating the LinearSolver class. Please see the documentation for the LinearSolver class for more details.

Parameters:
  • A (Firedrake Matrix) – Assembled Firedrake Matrix for the FEM
  • b (Firedrake Vector) – Assembled RHS vector, must be a Firedrake Vector
  • G (ForcingCovariance) – Forcing covariance matrix
  • data (ObsData) – ObsData object holding sensor data
  • params (ndarray) – Model discrepancy parameters to use in computing the solution. Must by a 1D numpy array of length 3. Note that these parameters are specified on a logarithmic scale to enforce positivity.
  • coords (ndarray) – Spatial locations where predictions will be made. Must be a 2D numpy array where the first index indicates the different points to be predicted and the second index is the spatial coordinates.
  • unc (float or ndarray) – Uncertainty (as a standard deviation) of the hypothetical measurements, as a non-negative scalar or an array with the same length as coords of non-negative floats.
  • ensemble_comm (MPI Communicator) – MPI communicator to use for parallelizing the covariance solves. Optional, default is to do the solves serially.
  • **kwargs –

    Additional keyword arguments for the PETSc solver.
Returns:

Covariance matrix of the posterior predictions as a 2D numpy array.
Return type:

ndarray
stat_fem.solving.predict_mean(A, b, G, data, params, coords, ensemble_comm=<Mock name='mock.COMM_SELF' id='140051716830736'>, scale_mean=True, **kwargs)

Solve for the predicted mean FEM posterior values at new locations

Standalone function to solve for the predicted posterior mean FEM solution at new sensor locations. Solution is returned as a 1D numpy array holding the predicted values. If the solves are parallelized, the non-root process arrays will contain empty arrays. This is just a wrapper to the predict_mean method of the stat-fem LinearSolver class.

Locations where predictions are desired are specified by coords which is a 2D numpy array where the first index indicates the various prediction points and the second index indicates the spatial dimension.

In addition to the parameters specified here, additional keyword arguments to control the PETSc solver options can be passed here and will be used when creating the LinearSolver class. Please see the documentation for the LinearSolver class for more details.

Parameters:
  • A (Firedrake Matrix) – Assembled Firedrake Matrix for the FEM
  • b (Firedrake Vector) – Assembled RHS vector, must be a Firedrake Vector
  • G (ForcingCovariance) – Forcing covariance matrix
  • data (ObsData) – ObsData object holding sensor data
  • params (ndarray) – Model discrepancy parameters to use in computing the solution. Must by a 1D numpy array of length 3. Note that these parameters are specified on a logarithmic scale to enforce positivity.
  • coords (ndarray) – Spatial locations where predictions will be made. Must be a 2D numpy array where the first index indicates the different points to be predicted and the second index is the spatial coordinates.
  • ensemble_comm (MPI Communicator) – MPI communicator to use for parallelizing the covariance solves. Optional, default is to do the solves serially.
  • scale_mean (bool) – If true, scale the output by the model discrepancy scaling factor. Optional, default is True
  • **kwargs –

    Additional keyword arguments for the PETSc solver.
Returns:

Mean of the posterior prediction as a 1D numpy array.
Return type:

ndarray
stat_fem.solving.solve_posterior(A, x, b, G, data, params, ensemble_comm=<Mock name='mock.COMM_SELF' id='140051716830736'>, scale_mean=False, **kwargs)

Solve for the FEM posterior conditioned on the data

Standalone function to solve for the posterior FEM solution on the mesh. Solution is stored in the provided Firedrake function x to be consistent with the Firedrake solving interface. This is just a wrapper to the solve_posterior method of the stat-fem LinearSolver class.

Note that the solution is only stored in the root of the ensemble communicator if the forcing covariance solves are parallelized. The Firedrake function on other processes will not be modified.

In addition to the parameters specified here, additional keyword arguments to control the PETSc solver options can be passed here and will be used when creating the LinearSolver class. Please see the documentation for the LinearSolver class for more details.

Parameters:
  • A (Firedrake Matrix) – Assembled Firedrake Matrix for the FEM
  • x (Firedrake Function) – Firedrake Function to hold the solution. Will be modified in place.
  • b (Firedrake Vector) – Assembled RHS vector, must be a Firedrake Vector
  • G (ForcingCovariance) – Forcing covariance matrix
  • params (ndarray) – Model discrepancy parameters to use in computing the solution. Must by a 1D numpy array of length 3. Note that these parameters are specified on a logarithmic scale to enforce positivity.
  • ensemble_comm (MPI Communicator) – MPI communicator to use for parallelizing the covariance solves. Optional, default is to do the solves serially.
  • scale_mean (bool) – If true, scale the output by the model discrepancy scaling factor. Optional, default is False
  • **kwargs –

    Additional keyword arguments for the PETSc solver.
Returns:

None
stat_fem.solving.solve_posterior_covariance(A, b, G, data, params, ensemble_comm=<Mock name='mock.COMM_SELF' id='140051716830736'>, scale_mean=False, **kwargs)

Solve for the FEM posterior conditioned on the data and the covariance

Standalone function to solve for the posterior FEM solution interpolated to the sensor locations and the posterior covariance. Solution is returned as a pair of numpy arrays holding the solution in a 1D array and the covariance matrix in a 2D array. If the solves are parallelized, the non-root process arrays will contain empty arrays. This is just a wrapper to the solve_posterior_covariance method of the stat-fem LinearSolver class.

In addition to the parameters specified here, additional keyword arguments to control the PETSc solver options can be passed here and will be used when creating the LinearSolver class. Please see the documentation for the LinearSolver class for more details.

Parameters:
  • A (Firedrake Matrix) – Assembled Firedrake Matrix for the FEM
  • b (Firedrake Vector) – Assembled RHS vector, must be a Firedrake Vector
  • G (ForcingCovariance) – Forcing covariance matrix
  • data (ObsData) – ObsData object holding sensor data
  • params (ndarray) – Model discrepancy parameters to use in computing the solution. Must by a 1D numpy array of length 3. Note that these parameters are specified on a logarithmic scale to enforce positivity.
  • ensemble_comm (MPI Communicator) – MPI communicator to use for parallelizing the covariance solves. Optional, default is to do the solves serially.
  • scale_mean (bool) – If true, scale the output by the model discrepancy scaling factor. Optional, default is False
  • **kwargs –

    Additional keyword arguments for the PETSc solver.
Returns:

Mean and covariance matrix of the posterior as numpy arrays. Mean is a 1D array and covariance is a 2D array.
Return type:

tuple of 2 ndarrays
stat_fem.solving.solve_posterior_generating(A, b, G, data, params, ensemble_comm=<Mock name='mock.COMM_SELF' id='140051716830736'>, **kwargs)

Solve for the FEM posterior of the generating process and covariance

Standalone function to solve for the posterior generating process of the FEM solution interpolated to the sensor locations and the covariance. Solution is returned as a pair of numpy arrays holding the solution in a 1D array and the covariance matrix in a 2D array. If the solves are parallelized, the non-root process arrays will contain empty arrays. This is just a wrapper to the solve_posterior_generating method of the stat-fem LinearSolver class.

In addition to the parameters specified here, additional keyword arguments to control the PETSc solver options can be passed here and will be used when creating the LinearSolver class. Please see the documentation for the LinearSolver class for more details.

Parameters:
  • A (Firedrake Matrix) – Assembled Firedrake Matrix for the FEM
  • b (Firedrake Vector) – Assembled RHS vector, must be a Firedrake Vector
  • G (ForcingCovariance) – Forcing covariance matrix
  • data (ObsData) – ObsData object holding sensor data locations
  • params (ndarray) – Model discrepancy parameters to use in computing the solution. Must by a 1D numpy array of length 3. Note that these parameters are specified on a logarithmic scale to enforce positivity.
  • ensemble_comm (MPI Communicator) – MPI communicator to use for parallelizing the covariance solves. Optional, default is to do the solves serially.
  • **kwargs –

    Additional keyword arguments for the PETSc solver.
Returns:

Mean and covariance matrix of the posterior of the generating process as numpy arrays. Mean is a 1D array and covariance is a 2D array.
Return type:

tuple of 2 ndarrays
stat_fem.solving.solve_prior(A, b, G, data, ensemble_comm=<Mock name='mock.COMM_SELF' id='140051716830736'>, **kwargs)

Solve for the FEM prior and covariance

Standalone function to solve for the prior FEM solution interpolated to the sensor locations and the covariance. Solution is returned as a pair of numpy arrays holding the solution in a 1D array and the covariance matrix in a 2D array. If the solves are parallelized, the non-root process arrays will contain empty arrays. This is just a wrapper to the solve_prior method of the stat-fem LinearSolver class.

Note that while the prior solution does not use the data, an ObsData object is still required to determine the sensor positions to which the solution should be interpolated. Any data and uncertainty information in the data object is ignored.

In addition to the parameters specified here, additional keyword arguments to control the PETSc solver options can be passed here and will be used when creating the LinearSolver class. Please see the documentation for the LinearSolver class for more details.

Parameters:
  • A (Firedrake Matrix) – Assembled Firedrake Matrix for the FEM
  • b (Firedrake Vector) – Assembled RHS vector, must be a Firedrake Vector
  • G (ForcingCovariance) – Forcing covariance matrix
  • data (ObsData) – ObsData object holding sensor data locations
  • ensemble_comm (MPI Communicator) – MPI communicator to use for parallelizing the covariance solves. Optional, default is to do the solves serially.
  • **kwargs –

    Additional keyword arguments for the PETSc solver.
Returns:

Mean and covariance matrix of the prior as numpy arrays. Mean is a 1D array and covariance is a 2D array.
Return type:

tuple of 2 ndarrays
stat_fem.solving.solve_prior_generating(A, b, G, data, params, ensemble_comm=<Mock name='mock.COMM_SELF' id='140051716830736'>, **kwargs)

Solve for the FEM prior of the generating process and covariance

Standalone function to solve for the prior generating process of the FEM solution interpolated to the sensor locations and the covariance. Solution is returned as a pair of numpy arrays holding the solution in a 1D array and the covariance matrix in a 2D array. If the solves are parallelized, the non-root process arrays will contain empty arrays. This is just a wrapper to the solve_prior_generating method of the stat-fem LinearSolver class.

Note that while the prior solution does not use the data, an ObsData object is still required to determine the sensor positions to which the solution should be interpolated. Any data and uncertainty information in the data object is ignored.

In addition to the parameters specified here, additional keyword arguments to control the PETSc solver options can be passed here and will be used when creating the LinearSolver class. Please see the documentation for the LinearSolver class for more details.

Parameters:
  • A (Firedrake Matrix) – Assembled Firedrake Matrix for the FEM
  • b (Firedrake Vector) – Assembled RHS vector, must be a Firedrake Vector
  • G (ForcingCovariance) – Forcing covariance matrix
  • data (ObsData) – ObsData object holding sensor data locations
  • params (ndarray) – Model discrepancy parameters to use in computing the solution. Must by a 1D numpy array of length 3. Note that these parameters are specified on a logarithmic scale to enforce positivity.
  • ensemble_comm (MPI Communicator) – MPI communicator to use for parallelizing the covariance solves. Optional, default is to do the solves serially.
  • **kwargs –

    Additional keyword arguments for the PETSc solver.
Returns:

Mean and covariance matrix of the prior of the generating process as numpy arrays. Mean is a 1D array and covariance is a 2D array.
Return type:

tuple of 2 ndarrays