Heat1D#

class cuqi.testproblem.Heat1D(dim=128, endpoint=1, max_time=0.2, field_type=None, field_params=None, map=None, imap=None, SNR=200, exactSolution=None, observation_grid_map=None)#

1D Heat test problem. Discretized Heat equation (time-dependent linear PDE).

Parameters:

  • dim (int) –

    size of the grid for the heat problem

  • endpoint (float) –

    Location of end-point of grid.

  • max_time (float) –

    The last time step.

  • field_type (str or cuqi.geometry.Geometry) –

    Field type of domain. The accepted values are:
    a Geometry object.
    ”KL”: a cuqi.geometry.KLExpansion geometry object will be created and set as a domain geometry.
    ”KL_Full”: a cuqi.geometry.KLExpansion_Full geometry object will be created and set as a domain geometry.
    ”Step”: a cuqi.geometry.StepExpansion geometry object will be created and set as a domain geometry.
    ”CustomKL”: a cuqi.geometry.CustomKL geometry object will be created and set as a domain geometry.
    None: a cuqi.geometry.Continuous1D geometry object will be created and set as a domain geometry.

  • field_params (dict) –

    A dictionary of key word arguments that the underlying geometry accepts. (Passed to the underlying geometry when field type is KL, KL_Full, CustomKL, Step). For example, for Step field type, the dictionary can be {“n_steps”: 3}.

  • map (lambda function) –

    If given, an underlying MappedGeometry geometry object is created which applies the mapping on the field, e.g. for log parameterization: map = lambda x:np.exp(x).

  • imap (lambda function) –

    The inverse of the provided map.

  • SNR (int) –

    Signal-to-noise ratio

  • exactSolution (ndarray or CUQIarray) –

    The exact solution of the problem which is the heat model initial condition in this test problem. If provided as None, an exact solution is generated, if provided as ndarray, it is assumed to be function values and it is converted to a CUQIarray.

observation_grid_maplambda function
Function that takes the grid as input and returns a sub-grid of the nodes where observations are available, e.g. observation_grid_map = lambda x: x[np.where(x>5.0)].
data#

Generated (noisy) data

Type:

ndarray

model#

Heat 1D model

Type:

cuqi.model.PDEModel

prior#

Distribution of the prior

Type:

cuqi.distribution.Distribution

likelihood#

Likelihood function

Type:

cuqi.likelihood.Likelihood

exactSolution#

Exact solution (ground truth)

Type:

ndarray

exactData#

Noise free data

Type:

ndarray

MAP()#

Compute MAP estimate of posterior. NB: Requires prior to be defined.

sample_posterior(Ns)#

Sample Ns samples of the posterior. NB: Requires prior to be defined.

__init__(dim=128, endpoint=1, max_time=0.2, field_type=None, field_params=None, map=None, imap=None, SNR=200, exactSolution=None, observation_grid_map=None)#

Methods

MAP([disp, x0])

Compute the Maximum A Posteriori (MAP) estimate of the posterior.

ML([disp, x0])

Compute the Maximum Likelihood (ML) estimate of the posterior.

UQ([Ns, Nb, percent, exact, experimental])

Run an Uncertainty Quantification (UQ) analysis on the Bayesian problem and provide a summary of the results.

__init__([dim, endpoint, max_time, ...])

get_components()

Method that returns the model, the data and additional information to be used in formulating the Bayesian problem.

sample_posterior(Ns[, Nb, callback, ...])

Sample the posterior.

sample_prior(Ns[, callback])

Sample the prior distribution.

set_data(**kwargs)

Set the data of the problem.

Attributes

data

Extract the observed data from likelihood

likelihood

The likelihood function.

model

Extract the cuqi model from likelihood.

posterior

Create posterior distribution from likelihood and prior.

prior

The prior distribution