CovarianceVectorMultiplySolver.md

Covariance Vector Multiply Solver {#Covariance_Vector_product}

General Information

• Solver Fortran File: CovarianceVectorMultiplySolver.F90

• Solver Name: CovarianceVectorMultiplySolver

• Required Input Variable(s):

  • A nodal input variable $x$

• Required Output Variable(s):

  • The product of a covariance matrix with the input variable

• Required Input Keywords:

  • Solver Section:

    • Input Variable = String : Name of the input variable

    • standard deviation = Real : the standard deviation $\sigma$

    • Covariance type = String : Available choices to construct the covariance matrix

      • "diffusion operator"

      • "full matrix"

      • "diagonal"

    • covariance type specific keywords: see CovarianceUtils

• Optional Input Keywords:

  • Solver Section:

    • Normalize = Logical: wether to normalize the output (default: False)

Remark

This documentation contains equations and is part of a generic documentation that can be converted to pdf using pandoc:

> pandoc -d MakeDoc_CovarianceUtils.yml

General Description

Compute the product $$y=C.x$$ with:

• $x$ and input variable

• $C$ a covariance matrix

Applications:

• covariance visualization and code validation: the spatial correlation function at a given node $z_i$ corresponds to the $i$-th column of the covariance matrix $C$. It can be visualized by plotting the result of applying $C$ to a vector that has a value of one at $z_i$ and a value of zero at all other points (Guillet et al., 2019).

• Filtering: When Normalize = Logical True, the output is normalized by the results of applying $C$ to a vector full of ones. If the kernel is a Gaussian correlation function, this would be equivalent to applying a Gaussian filter and this will thus smooth the input variable. The Matérn covariance, obtained with the diffusion operator method, converges to the Gaussian correlation function when the smoothness parameters $\nu$ tends to infinity.

Implementation

See the generic documentation for CovarianceUtils for details on the possible choices to construct the covariance matrix $C$.

Known Bugs and Limitations

• Limited to serial if using the "full matrix" covariance method.

SIF Contents

Solver 1
 Equation = "Filter"
 Variable = "y"
 Procedure = "CovarianceVectorMultiplySolver" "CovarianceVectorMultiplySolver"

 input variable = string "x"

 Normalize = Logical True !#(default: False)

!# Covariance types
 !############################################################################
 !# keywords for the "diffusion operator" method
 !# see CovarianceUtils.md for other choices
 !############################################################################
  Covariance type = String "diffusion operator"

  Matern exponent m = Integer $m
  correlation range = Real $range
  standard deviation = Real $std

!# The diffusion operator method requires to solve symmetric positive definite
!# linear systems
  Linear System Solver = Direct
  Linear System Direct Method = umfpack

  Linear System Refactorize = Logical False
  Linear System Symmetric = Logical True
  Linear System Positive Definite = Logical True

end

Examples

• ElmerIce unitary tests:

  • [ELMER_TRUNK]/elmerice/Tests/CovarianceVector

  • [ELMER_TRUNK]/elmerice/Tests/CovarianceVector2

• Validation test cases:

  • https://gricad-gitlab.univ-grenoble-alpes.fr/gilletcf/CovarianceUtils/-/tree/master/CovarianceTestCase

• Filtering test case:

  • https://gricad-gitlab.univ-grenoble-alpes.fr/gilletcf/CovarianceUtils/-/tree/master/FilterTestCase

References

• Guillet O., Weaver A.T., Vasseur X., Michel Y., Gratton S., Gurol S. Modelling spatially correlated observation errors in variational data assimilation using a diffusion operator on an unstructured mesh. Q. J. R. Meteorol. Soc., 2019. https://doi.org/10.1002/qj.3537