Analytical inversions analytic/std

Analytical inversions `analytic/std`#

Description#

Direct (analytical) Bayesian inversion via explicit H-matrix construction.

Mathematical framework#

This mode performs a Best Linear Unbiased Estimator (BLUE) inversion under the Gaussian error assumption. It first assembles the observation operator matrix $\mathbf{H} \in \mathbb{R}^{m \times n}$ explicitly, column by column, then solves for the posterior state analytically.

Step 1 — Building H#

The Jacobian matrix is constructed by running one forward simulation per control-vector dimension $i$:

\[\mathbf{H}_{:,\,i} = \mathcal{H}(\mathbf{e}_i)\]

where $\mathbf{e}_i$ is the $i$-th canonical basis vector (all zeros except element $i$ set to 1). This exploits linearity of the operator:

\[\mathbf{H}\,\mathbf{x} = \sum_i x_i\,\mathcal{H}(\mathbf{e}_i)\]

Each simulation is submitted as an independent pyCIF forward run stored under $workdir/base_functions/.

Step 2 — Analytical inversion (BLUE)#

Given the prior $\mathbf{x}_b$ with background error covariance $\mathbf{B} \in \mathbb{R}^{n \times n}$, observations $\mathbf{y}$ with observation error covariance $\mathbf{R} \in \mathbb{R}^{m \times m}$, the posterior (analysis) state is:

\[\mathbf{x}_a = \mathbf{x}_b + \mathbf{K}\bigl(\mathbf{y} - \mathbf{H}\,\mathbf{x}_b\bigr)\]

where the Kalman gain is

\[\mathbf{K} = \mathbf{B}\mathbf{H}^\top \bigl(\mathbf{R} + \mathbf{H}\mathbf{B}\mathbf{H}^\top\bigr)^{-1}\]

and the posterior error covariance is

\[\mathbf{P}_a = \mathbf{B} - \mathbf{K}\mathbf{H}\mathbf{B}\]

Complexity and scalability#

The dominant cost is the $n$ forward simulations required to build $\mathbf{H}$. The matrix inversion $(\mathbf{R} + \mathbf{H}\mathbf{B}\mathbf{H}^\top)^{-1}$ is $\mathcal{O}(m^3)$ in the observation dimension — feasible for moderate $m$ but prohibitive for large observing systems. For large problems use the variational (4dvar) or response-functions modes instead.

Warning

One forward simulation per control-vector dimension $n$ is required. Check $n$ and the cost of a single forward run before launching. Use the dryrun option to estimate the total wall-clock time without committing to the full computation.

YAML arguments#

The following arguments are used to configure the plugin. pyCIF will return an exception at the initialization if mandatory arguments are not specified, or if any argument does not fit accepted values or type:

dump_nc_base_control : bool, optional, default False: Save each Dirac control vector (base function input) as NetCDF for post-hoc inspection of what was actually run.

dryrun : bool, optional, default False: Submit only the first base function to estimate the per-run cost, then stop without completing the full H matrix.

sequential : bool, optional, default False: Wait for each job to finish before submitting the next. Useful when concurrent submissions are restricted (e.g. GPU queues).

resp_func_only : bool, optional, default False: Does not run the inversion, only the response functions to build the H matrix

Requirements#

The current plugin requires the present plugins to run properly:

Requirement name	Requirement type	Explicit definition	Any valid	Default name	Default version
obsvect	ObsVect	False	True	standard	std
controlvect	ControlVect	True	True	standard	std
obsoperator	ObsOperator	True	True	standard	std
platform	Platform	True	True	None	None

YAML template#

Please find below a template for a YAML configuration:

mode:
  plugin:
    name: analytic
    version: std
    type: mode

  # Optional arguments
  dump_nc_base_control: XXXXX  # bool
  dryrun: XXXXX  # bool
  sequential: XXXXX  # bool
  resp_func_only: XXXXX  # bool