FMINSRF2.SIF

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* SET UP THE INITIAL DATA *
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NAME          FMINSRF2

*   Problem :
*   *********

*   The free boundary minimum surface problem.

*   The problem comes from the discretization of the minimum surface
*   problem on the unit square with "free boundary conditions"
*   one must find the minumum surface over the unit square 
*   (which is clearly 1.0).  Furthermore, the distance of the surface
*   from zero at the centre of the unit square is also minimized.

*   The unit square is discretized into (p-1)**2 little squares. The
*   heights of the considered surface above the corners of these little
*   squares are the problem variables,  There are p**2 of them.
*   Given these heights, the area above a little square is
*   approximated by the
*     S(i,j) = sqrt( 1 + 0.5(p-1)**2 ( a(i,j) + b(i,j) ) ) / (p-1)**2
*   where
*     a(i,j) = x(i,j) - x(i+1,j+1)
*   and
*     b(i,j) = x(i+1,j) - x(i,j+1)

*   Source: setting the boundary free in 
*   A Griewank and Ph. Toint,
*   "Partitioned variable metric updates for large structured
*   optimization problems",
*   Numerische Mathematik 39:429-448, 1982.

*   SIF input: Ph. Toint, November 1991.

*   classification OUR2-MY-V-0

*   P is the number of points in one side of the unit square

*IE P                   4              $-PARAMETER n = 16
*IE P                   7              $-PARAMETER n = 49
*IE P                   8              $-PARAMETER n = 64     original value
*IE P                   11             $-PARAMETER n = 121
*IE P                   31             $-PARAMETER n = 961
*IE P                   32             $-PARAMETER n = 1024
 IE P                   75             $-PARAMETER n = 5625
*IE P                   100            $-PARAMETER n = 10000
*IE P                   125            $-PARAMETER n = 15625

*   Define the plane giving the boundary conditions

 RE H00                 1.0
 RE SLOPEJ              4.0
 RE SLOPEI              8.0

*   Define a few helpful parameters

 I+ TWOP      P                        P
 IA P-1       P         -1
 I* PP-1      P                        P-1
 RI RP-1      P-1
 RD INVP-1    RP-1      1.0
 R* RP-1SQ    INVP-1                   INVP-1
 RD SCALE     RP-1SQ    1.0
 R* SQP-1     RP-1                     RP-1
 RM PARAM     SQP-1     0.5
 RI RP        P
 R* P2        RP                       RP
 R* P4        P2                       P2

 IE 1                   1
 IE 2                   2

 R* STON      INVP-1                   SLOPEI
 R* WTOE      INVP-1                   SLOPEJ
 R+ H01       H00                      SLOPEJ
 R+ H10       H00                      SLOPEI
 I/ MID       P                        2

VARIABLES

*   Define one variable per discretized point in the unit square

 DO J         1                        P
 DO I         1                        P
 X  X(I,J)
 ND

GROUPS

*   Define a group per little square

 DO I         1                        P-1
 DO J         1                        P-1
 ZN S(I,J)    'SCALE'                  SCALE
 ND

*   Cental height group

 XN MID       X(MID,MID)1.0

 ZN MID       'SCALE'                  P2

CONSTANTS

 X  FMINSRF2  'DEFAULT' -1.0
    FMINSRF2  MID       0.0

BOUNDS

 FR FMINSRF2  'DEFAULT'

START POINT

*   All variables not on the boundary are set to 0.0

 XV FMINSRF2  'DEFAULT' 0.0

*   Starting values on the lower and upper edges

 DO J         1                        P
 IA J-1       J         -1
 RI RJ-1      J-1
 R* TH        RJ-1                     WTOE
 R+ TL        TH                       H00
 R+ TU        TH                       H10
 Z  FMINSRF2  X(1,J)                   TL
 Z  FMINSRF2  X(P,J)                   TU
 ND

*   Starting values on the left and right edges

 DO I         2                        P-1
 IA I-1       I         -1
 RI RI-1      I-1
 R* TV        RI-1                     STON
 R+ TR        TV                       H00
 R+ TL        TV                       H01
 Z  FMINSRF2  X(I,P)                   TL
 Z  FMINSRF2  X(I,1)                   TR
 ND

ELEMENT TYPE

*   The only element type.

 EV ISQ       V1                       V2
 IV ISQ       U

ELEMENT USES

*   Each little square has two elements using diagonal and
*   antidiagonal corner values

 DO I         1                        P-1
 IA I+1       I         1
 DO J         1                        P-1
 IA J+1       J         1
 XT A(I,J)    ISQ
 ZV A(I,J)    V1                       X(I,J)
 ZV A(I,J)    V2                       X(I+1,J+1)
 XT B(I,J)    ISQ
 ZV B(I,J)    V1                       X(I+1,J)
 ZV B(I,J)    V2                       X(I,J+1)
 ND

GROUP TYPE

*   Groups are of the square root or least-squares type

 GV SQROOT    ALPHA
 GV L2        GVAR

GROUP USES

*   The overage distance group of of least-squares type

 T  MID       L2

*   All other groups are of SQRT type.

 DO I         1                        P-1
 DO J         1                        P-1
 XT S(I,J)    SQROOT
 ZE S(I,J)    A(I,J)                   PARAM
 ZE S(I,J)    B(I,J)                   PARAM
 ND

OBJECT BOUND

 LO FMINSRF2            0.0

*   Solution

*LO SOLTN               1.0

ENDATA

***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES  *
***********************

ELEMENTS      FMINSRF2

INDIVIDUALS

*   Difference squared

 T  ISQ
 R  U         V1        1.0            V2        -1.0
 F                      U * U
 G  U                   U + U
 H  U         U         2.0

ENDATA

*********************
* SET UP THE GROUPS *
* ROUTINE           *
*********************

GROUPS        FMINSRF2

TEMPORARIES

 M  SQRT
 R  SQRAL

INDIVIDUALS

*   square root groups

 T  SQROOT
 A  SQRAL               SQRT(ALPHA)
 F                      SQRAL
 G                      0.5D0 / SQRAL
 H                      -0.25D0 / ( SQRAL * ALPHA )

*   least-squares groups

 T  L2
 F                      GVAR * GVAR
 G                      GVAR + GVAR
 H                      2.0D0

ENDATA