Adjoint_CostRegSolver.md

Regularisation solver {#adjoint_CostReg}

Module name: Adjoint_CostRegSolver Module subroutines: Adjoint_CostRegSolver Module authors: Fabien Gillet-Chaulet (IGE-Grenoble) Document authors: Fabien Gillet-Chaulet Document edited: 24.04.2020

Introduction

This solver computes a cost function classically used for regularisation of the inverse problems and it's derivative with respect to the input nodal variable V.

Two regularisations are possible:

• First option penalises 1st spatial derivatives [default]:

$$J_{reg} = \int_{\Omega} 0.5 (|dV/dx|)^2 d\Omega$$

The dimension of the spatial derivatives is CoordinateSystemDimension() or CoordinateSystemDimension()-1 depending if the solver is executed on the bulk on a boundary.

• Second option penalises difference from a prior:

$$J_{reg} = \int_{\Omega} 0.5 ((V-V^{prior})/s^2)^2 d\Omega$$

where $V^{prior}$ is the prior estimate and $s^2$ the variance

Keywords

Below are the related keywords in the .sif file:

Solver *solver id* 
  
    Equation = String "CostReg"  
    procedure = "ElmerIceSolvers" "Adjoint_CostRegSolver"
    !## No need to declare a variable, this is done internally to ensure
    !    that Solver structures exist
 
     !# True if cost function and gradient must be initialised 
     !   to 0 in this solver
     Reset Cost Value = Logical [Default: True]

     !# Name of an ascii file that will contain the cost value as
     !## Time, J_{reg}, RMS=sqrt(2J_{reg}/Area)
     Cost Filename = File ""
     
     !# Name of the variable that contain the cost value
     !#  must exist and can be a global variable
     Cost Variable Name = String ""
     
     !# a multiplicatif factor can be used to scale the cost function
     Lambda=Real [default 1.0]
     
     !# The name of the model variable that will contain the derivative
     !  of J w.r.t. the input variable
     Gradient Variable Name = String ""

     !# The name of the model variable V
     !## if not found will look for the keyword "CostReg Nodal Variable" 
     !    in the body force.
     Optimized Variable Name = String ""

     !# Switch to regularisation from *prior*
     A priori Regularisation = Logical [default:False]
     
      
  End

Values for the nodal variable V can be given in the body force if Optimized Variable Name was not provided in the solver parameters. This can be useful if we want to apply some change of variable compared to the variable that is optimised, However it will be to the user responsibility to provide the exact derivative of his change of variable as, by default, the solver computes the derivative with respect to the nodal value. In this case the user can directly provide the derivative with the keyword CostReg Nodal Variable der = Real ... see below and [examples]

If using a priori regularisation, nodal values for the prior and standard deviation are given in the body force too.

Body Force i
 # value for the nodal *V*
  CostReg Nodal Variable = Real ...
 # If the definition above implies a change of variable
 #  the derivative of the function above can be performed using 
 #  the following keyword
  CostReg Nodal Variable derivative = Real ...

 # value for the nodal *V^{prior}*
  CostReg Nodal Prior = Real ...
 # value for the nodal std *s*
  CostReg Nodal std = Real ...
End

It is possible to use a passive condition in the body force, if we want to skip the evaluation of the cost function within passive elements.

Body Force i
 # the name of the  solver variable is NameOfEquation_var
 # keywords relative with passive elements can be used
 CostReg_var Passive = ...
End

Limitations and possible improvements

Below is a list of features that are not currently possible in this solver but that could be implemented:

• If running on a boundary, we use only the first spatial derivatives, so this solver is intended to be used on a bottom or top surface.

• If Regularisation with respect to the prior is used, for the moment we assume no spatial error in the statistics and use only a standard deviation; The cost function could easily be improved if a full background error covariance matrix is known

Tests and Examples

Validation examples available under:

• ELMER_SRC/elmerice/examples/Adjoint_CostRegSolver

  Tested:

  • Regularisation penalising first spatial derivatives on a 2D mesh and on the bottom boundary of a 3D mesh

  • Regularisation from prior

  • Optimisation

  • Prescribing nodal value from the body forces using a change of variable

• See examples for the inverse methods