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Deflated Conjugate Gradient

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Title: Deflated Conjugate Gradient


1
  • Deflated Conjugate Gradient
  • Method
  • for modelling Groundwater Flow
  • in a Layered Grid

Lennart Ros Deltares TU Delft Utrecht July
3 2008 10.30 www.deltares.com
Thesis Committee Prof. Dr. Ir. C. Vuik (TU
Delft) Dr M. Genseberger (Deltares) Ir. J.
Verkaik (Deltares) Dr H.M. Schuttelaars (TU Delft)
2
Outline
3
Outline
  • Introduction
  • Deltares
  • Subsurface, Geohydrology, Clay Faults
  • MODFLOW
  • IBRAHYM problem
  • Equation, Discretization Method
  • A Simple Example (Matlab)
  • Deflation Techniques
  • Results for IBRAHYM
  • Conclusions Recommendations
  • Questions

4
Introduction
5
Introduction
Deltares
January 1st 2008
6
Introduction
Subsurface
  • Subsurface is schematized in layers .
  • Successive sand and clay
  • (aquifers and aquitards)
  • Assumption
  • Horizontal flow in aquifer
  • Vertical flow in aquitard

7
Introduction
Geohydrology
  • The driving force for groundwater flowis the
    difference in height and pressure.
  • To represent this difference we introduce the
    concept of hydraulicheads, h L.

8
Introduction
Clay
  • Very high resistance.
  • Main property
  • Extreme low permeability (vertical)

Medium Faults
  • Vertical barriers inside aquifers.
  • Main property
  • Extreme low permeability (horizontal).

9
Introduction
Clay
  • Very high resistance.
  • Main property
  • Extreme low permeability (vertical)

Consequence Large contrast in medium parameters
Medium Faults
  • Vertical barriers inside aquifers.
  • Main property
  • Extreme low permeability (horizontal).

10
Introduction
MODFLOW
  • MODFLOW is a software package which calculates
    hydraulic heads.
  • Developed by the U.S. Geological Survey.
  • Open-source code everyone can use and improve
    this program
  • Rectangular grid and uses cell-centered
    variables.
  • Quasi-3D model.

11
Introduction
IBRAHYM
  • groundwater model developed for several
    waterboards in Limburg.
  • uses at most 19 layers to model groundwater flow
    area.
  • uses grid cells of 25 times 25 meter to get
    detailed information.
  • a lot of clay and faults
  • these cause model to suffer from bad convergence
    behavior of solver.

12
Equation, Discretization Method
13
Equation, Discretization Method
Governing Equation
Where
hydraulic conductivities along x,y, and z coordinate axes LT-1,
h head L,
W volumetric flux per unit volume representing sources and sinks of water T-1,
Ss specific storage of porous material L-1,
t Time T
14
Equation, Discretization Method
Finite Volume Discretization
15
Equation, Discretization Method
Discretized Equation Using Finite Volume Method
Where
16
Equation, Discretization Method
Solution Method in MODFLOW
stop criteria inner loop or maximal
number of inner iteration is reached
  • MODFLOW uses outer and inner iteration loops
  • We look at inner iteration loop
  • solves a linear system of equations
  • preconditioner
  • iterative method
  • MODFLOW uses a
  • Modified Incomplete Cholesky Conjugate Gradient
    Method
  • with SOR

17
A Simple Example
18
A Simple Example
Simple Testcase
  • 2 Dimensional Problem
  • 15 rows, 15 colums
  • A contrast in the parameter
  • on 1/3th of the domain

19
A Simple Example
Observations for a simple testcase in Matlab
Assume A has eigenvalues Preconditioning
MODFLOW Modified Incomplete Cholesky
Preconditoning generally works
20
A Simple Example
Observations for a simple testcase in Matlab
21
A Simple Example
Observations for a simple testcase in Matlab
Smallest eigenvalue 0.00010283296716 Next
eigenvalue 0.04870854847951
22
A Simple Example
Observations for a simple testcase in Matlab
  • Due to the small eigenvalue we have a slow
    converging model.
  • Want to get rid of this eigenvalue(s)

GOAL IMPROVE CONVERGENCE BEHAVIOUR OF THE
IBRAHYM MODEL
IDEA USE DEFLATION BASED PRECONDITIONER
23
Deflation Techniques
24
Deflation Techniques
Basic Idea of Deflation
General linear system of equations Now
define And instead we solve the deflated
system
25
Deflation Techniques
Deflation using Eigenvectors
Assume that A has eigenvalues and we choose
the corresponding eigenvectors such
that If we now define Then
PROBLEM eigenvalues and eigenvectors are
generally unknown
26
Deflation Techniques
Alternative Deflation Techniques
  • Random Subdomain Deflation
  • Deflation based on Physics
  • Use layers as boundary of domain (1 domain
    is 1 layer)

Original domain Subdomains
27
Deflation Techniques
Results for the simple problem
  • Deflation using subdomain deflation
  • 1 vector represents left part of domain
  • 1 vector represents right part of domain
  • The eigenvector corresponding to the smallest
    eigenvalue is in the span of these two vectors.
  • Eigenvalues of and are
    almost the same, but the smallest is cancelled
    now.

28
Deflation Techniques
Results for the simple problem
  • Less iterates are needed
  • Residuals go faster to zero when
  • using deflation

GOAL IMPROVE CONVERGENCE BEHAVIOUR OF THE
IBRAHYM MODEL or REDUCE NUMBER OF ITERATIONS
29
Results for IBRAHYM
30
Results for IBRAHYM
Process
  • First Subdomain deflation while storing
    matrix Z and AZ
  • Problem Memory limiting for large areas.
  • Optimization concerning memory
  • Claypackages ? Layer based deflation
    (nice structure of Z)
  • Re-using vectors.

31
Results for IBRAHYM
Process
  • Problem No gain of wall-clock times.
  • Optimization concerning wall-clock times
  • Storing AZ using one vector.
  • Pointer-vector instead of IF-loops.

32
Results for IBRAHYM
Results (small area)
  • Claypackages ? Layer based deflation
  • For 3 small areas (7x7 km)
  • 1.489.600 cells

Area original code deflation code
1 (yellow) 62 53
2 (blue) 164 121
3 (orange) 287 168
33
Results for IBRAHYM
34
Results for IBRAHYM
Results (small area)
REMEMBER MODFLOW uses outer and inner iteration
loops
Concerning maximal number of inner iterations Concerning maximal number of inner iterations Concerning maximal number of inner iterations
Max. inner iteration Original code Deflation code
20 566 271
30 383 199
40 335 179
50 287 168
75 230 153
total number of iterations when varying the maximal number of inner iterations per inner loop total number of iterations when varying the maximal number of inner iterations per inner loop total number of iterations when varying the maximal number of inner iterations per inner loop
stop criteria inner loop
less iterations needed
also less fluctuation in solution when varying
maximal number of inner iterations
35
Results for IBRAHYM
Results (small area)
Concerning faults Concerning faults Concerning faults
Faults Original code Deflation code
With 287 171
Without 254 168
Total number of iterations when we take and do not take faults into account Total number of iterations when we take and do not take faults into account Total number of iterations when we take and do not take faults into account
Layer based deflation also solves for the
faults.
36
Results for IBRAHYM
Results (bigger area)
bigger area is 18 x 18 km 9.849.600 cells
Original code Deflation code
Iterations 1082 757
Wall-clock time 1664.40 sec 1542.20 sec
Error -0.31188E-02 0.49375E-03
  • less iterations needed
  • wall-clock time gained
  • error smaller

Error inflow outflow
also less fluctuation in solution when varying
maximal number of inner iterations
37
Conclusions Recommendations
38
Conclusions Recommendations
Conclusions
  • In general deflation works for modelling
    groundwater flow.
  • Less iterations are required.
  • Deflation preconditioner makes solution more
    robust.
  • Wall clock times can be gained, but depends
    strongly in code used.

Recommendations
  • Deflation in horizontal direction.
  • Code should be further optimised for Fortran
  • Minimizing memory,
  • IF-statements,
  • Using smart mappings.
  • Cluster multiple layers in one subdomain.

39
QUESTIONS?
40
Storing AZ
41
Storing AZ
42
Conclusions Recommendations
  • Conclusions
  • In general deflation works for modelling
    groundwater flow.
  • Less iterations are required.
  • Deflation preconditioner makes solution more
    robust.
  • Wall clock times can be gained, but depends
    strongly in code used.
  • Recommendations
  • Deflation in horizontal direction.
  • Code should be further optimised for Fortran
  • Minimizing memory,
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