Problem

A Problem holds the inputs of a linear inverse problem: the prior and its covariance, the observations and theirs, the forward operator and an optional offset. It checks their labels once, when it is built, by an exact join: the forward operator, a covariance or the offset must have exactly the labels of the prior or of the observations, in any order, or it raises naming the blocks and labels on one side only. A covariance model is built on the prior’s or the observations’ space. solve() solves a problem. Problem.restrict() holds part of the state at its prior (the cells no observation sees well), and Problem.aggregate() aggregates the observations (see Aggregation and Offsets).

See also

See Terminology for definitions of prior, posterior, forward operator, and other inverse problem terminology.

In [1]: import numpy as np

In [2]: import pandas as pd

In [3]: import fips

In [4]: cells = pd.Index(["a", "b"], name="cell")

In [5]: obs_id = pd.Index(["o1", "o2", "o3"], name="obs_id")

In [6]: prior = fips.Vector.from_series(pd.Series(0.0, index=cells), block="flux")

In [7]: obs = fips.Vector.from_series(pd.Series([1.0, 2.0, 3.0], index=obs_id), block="co2")

In [8]: H = fips.Operator(np.array([[1.0, 0.0], [0.0, 1.0], [1.0, 1.0]]), obs.space, prior.space)

In [9]: problem = fips.Problem(
   ...:     prior=prior,
   ...:     prior_cov=fips.Independent().with_std(1.0),
   ...:     obs=obs,
   ...:     obs_cov=fips.Independent().with_std(0.3),
   ...:     forward=H,
   ...: )
   ...: 

In [10]: problem
Out[10]: Problem(state: Space(flux: 2 x ['cell']), obs: Space(co2: 3 x ['obs_id']))

In [11]: problem.predict(problem.prior)
Out[11]: 
Vector(name=None, Space(co2: 3 x ['obs_id']))
block  obs_id
co2    o1        0.0
       o2        0.0
       o3        0.0
dtype: float64

Constructor

Problem(prior, prior_cov, obs, obs_cov, forward)

A linear inverse problem: what is known before the observations, and the observations.

Inputs

Problem.prior	The prior state, \(x_0\); its space labels the state.
Problem.prior_cov	The prior error covariance, \(S_0\), on the prior's space.
Problem.obs	The observations, \(z\); their space labels the observations.
Problem.obs_cov	The model-data mismatch, \(S_z\), on the observations' space.
Problem.forward	rows on the observations, columns on the prior.
Problem.offset	The offset, \(c\), on the observations' space, or None.

Without a solve

What needs no solve and no \(m \times m\) matrix is on the problem: what the problem predicts, a truth and its synthetic observations (how to use them), and the problems made from it (the observations aggregated, part of the state held at its prior). What depends on the solve is on the Posterior.

Problem.predict(x)	Return the observations the state x models, \(H x + c\).
Problem.simulate([rng, truth])	Return a truth and the observations it would give, \(H x + c + e\).
Problem.aggregate(W)	Return the problem of aggregated observations.
Problem.restrict(keep)	Return the problem on part of the state, the rest held at its prior.

Serialization

A problem pickles with the standard library (pickle.dump(problem, file)).