AdjointThickness_GradientSolver.md

AdjointThickness : Gradient Solver {#thickness_gradient_solver}

Module name: AdjointThickness_GradientSolver.F90 Module subroutines: AdjointThickness_GradientSolver Module authors: Fabien Gillet-Chaulet (IGE-Grenoble) Document authors: Fabien Gillet-Chaulet Document edited: 10/12/2020

Required input variables:

• Variable solution of the direct problem (Thickness solver)

• The adjoint variable computed from the adjoint linear system

output variables: [Optional]

• Nodal derivatives with respect to:

  • convection velocity $u$: [DJDuv]

  • the top surface accumulation $Ms$: [DJDsmbTop]

  • the bottom surface accumulation $Mb$: [DJDsmbBot]

Introduction

This is the adjoint code of the Thickness solver.

Computes the derivatives of a cost function that involves the thickness $H$, solution of the direct solver, with respect to the input parameters (the convection velocity and the mass balance forcing).

The sequence in the .sif is usually as follow:

1. Compute the setady-state thickness using the thickness solver

2. Compute a cost function that measures the difference between the model and some observations, usually the observations are discrete along flow lines and the Adjoint_CostDiscSolver will be used.

3. Impose some constraint on the spacial variations of $H$. This can be done by penalising first spatial derivatives of $H$ with the Adjoint_CostRegSolver (This has to be done before the next step as it involves the variable $H$.)

4. Compute the solution of the adjoint linear system

5. Compute the gradient of your cost function with respect to your input parameters

Note that the step 3 is usually crucial as it will impose spatial correlation in the retrieved ice thickness field and thus drives the results far from the observations.

Keywords

Solver Section:

Solver *id*
  Equation = "DJDp"

  procedure = "ElmerIceSolvers" "AdjointThickness_GradientSolver"

 !# as for the direct solver the convection velocity 
 !#  can be provided here or in the body forces
  Flow Solution Name = String "...."

 !# name of the variable solution of the direct problem
  Thickness Solution Name = String ...

 !# name of the variable solution of the adjoint problem
  Adjoint Solution Name = String ...

 !# Compute the derivatives

 !# with respect to the velocity; results will be stored in the variable DJDUV
  ComputeDJDUV = Logical ... [default: False]
 !# Should we initialise the gradient to 0 here?
  Reset DJDUV = Logical ... [default: True]

 !# with respect to the top surface accumulation; 
 !#  results will be stored in the variable ComputeDJDsmbTop
  ComputeDJDsmbTop = Logical ... [default: False]
  Reset DJDsmbTop = Logical ... [default: True]
 
 !# with respect to the top surface accumulation;
 !#  results will be stored in the variable ComputeDJDsmbBot
  ComputeDJDsmbBot = Logical ... [default: False]
  Reset DJDsmbBot = Logical ... [default: True]

end

Body Forces:

!# Ms and Mb
Body Force *id*
  Top Surface Accumulation = Real ...
  Bottom Surface Accumulation = Real ...

  !# If <Flow Solution Name> is not provided in the solver section
  !#  the convection velocity can be read here:
  ! x-componenent
  Convection Velocity 1 = Real ...
  ! y-componenent if Convection Dimension = 2
  Convection Velocity 1 = Real ...
End

Tests and Examples

• See examples for the Mass Conservation methods