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MARINE.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME MARINE
* Problem :
* *********
* Given estimates of the abundance of the population of a marine
* spieces at each stage as a function of time, determine specific
* growth and mortality rates. Th emodel for the population dynamics
* of the n_e -stage population is
* y_j' = g_j-1 y_j-1 - (m_j + g_j) y_j (1 <= j <= n_e)
* where m_i and g_i are unknown mortality and growth rates at stage i
* with g_0 = 0 = g_n_e. This model assumes that the species eventually
* dies or grows into the next stage, with the implicit assumption that
* the species cannot skip a stage. Initial conditions for the differntial
* equation are unknown, since the stage abundance measurements at the
* initial time might also be contaminated with experimental error. The
* problem is to minimize the error between the observed and computed data,
* sum{i=1,nm} || y(tau_i,m,g) - z_i||^2
* where the z_i are stage abundance measurements.
* This is problem 6 in the COPS (Version 2) collection of
* E. Dolan and J. More'
* see "Benchmarking Optimization Software with COPS"
* Argonne National Labs Technical Report ANL/MCS-246 (2000)
* SIF input: Nick Gould, November 2000
* classification OOR2-AN-V-V
* The number of differential equations
IE NE 8
* The number of subintervals
*IE NH 25 $-PARAMETER
*IE NH 50 $-PARAMETER
*IE NH 100 $-PARAMETER
IE NH 200 $-PARAMETER
* The number of measurements
IE NM 21
* The number of collocation points
*IE NC 1 $-PARAMETER
IE NC 2 $-PARAMETER
*IE NC 3 $-PARAMETER
*IE NC 4 $-PARAMETER
* roots of NC-th degree Legendre polynomial
*RE RHO1 0.5 $-PARAMETER NC = 1
RE RHO1 0.7886751346 $-PARAMETER NC = 2
*RE RHO1 0.5 $-PARAMETER NC = 3
*RE RHO1 0.0694318442 $-PARAMETER NC = 4
RE RHO2 0.2113248654 $-PARAMETER NC = 2
*RE RHO2 0.8872983346 $-PARAMETER NC = 3
*RE RHO2 0.3300094782 $-PARAMETER NC = 4
*RE RHO3 0.1127016654 $-PARAMETER NC = 3
*RE RHO3 0.6699905218 $-PARAMETER NC = 4
*RE RHO4 0.9305681558 $-PARAMETER NC = 4
* times at which observations made
RE TAU1 0.0
RE TAU2 0.5
RE TAU3 1.0
RE TAU4 1.5
RE TAU5 2.0
RE TAU6 2.5
RE TAU7 3.0
RE TAU8 3.5
RE TAU9 4.0
RE TAU10 4.5
RE TAU11 5.0
RE TAU12 5.5
RE TAU13 6.0
RE TAU14 6.5
RE TAU15 7.0
RE TAU16 7.5
RE TAU17 8.0
RE TAU18 8.5
RE TAU19 9.0
RE TAU20 9.5
RE TAU21 10.0
* The ODE is defined in [0,TF]
A= TF TAU(NM)
* The uniform interval length
RI RNH NH
R/ H TF RNH
* Concentrations
RE Z1,1 20000.0
RE Z2,1 12445.0
RE Z3,1 7705.0
RE Z4,1 4664.0
RE Z5,1 2977.0
RE Z6,1 1769.0
RE Z7,1 943.0
RE Z8,1 581.0
RE Z9,1 355.0
RE Z10,1 223.0
RE Z11,1 137.0
RE Z12,1 87.0
RE Z13,1 49.0
RE Z14,1 32.0
RE Z15,1 17.0
RE Z16,1 11.0
RE Z17,1 7.0
RE Z18,1 4.0
RE Z19,1 2.0
RE Z20,1 1.0
RE Z21,1 1.0
RE Z1,2 17000.0
RE Z2,2 15411.0
RE Z3,2 13074.0
RE Z4,2 8579.0
RE Z5,2 7053.0
RE Z6,2 5054.0
RE Z7,2 3907.0
RE Z8,2 2624.0
RE Z9,2 1744.0
RE Z10,2 1272.0
RE Z11,2 821.0
RE Z12,2 577.0
RE Z13,2 337.0
RE Z14,2 228.0
RE Z15,2 168.0
RE Z16,2 99.0
RE Z17,2 65.0
RE Z18,2 44.0
RE Z19,2 27.0
RE Z20,2 20.0
RE Z21,2 12.0
RE Z1,3 10000.0
RE Z2,3 13040.0
RE Z3,3 14623.0
RE Z4,3 12434.0
RE Z5,3 11219.0
RE Z6,3 10065.0
RE Z7,3 9473.0
RE Z8,3 7421.0
RE Z9,3 5369.0
RE Z10,3 4713.0
RE Z11,3 3451.0
RE Z12,3 2649.0
RE Z13,3 2058.0
RE Z14,3 1440.0
RE Z15,3 1178.0
RE Z16,3 919.0
RE Z17,3 647.0
RE Z18,3 509.0
RE Z19,3 345.0
RE Z20,3 231.0
RE Z21,3 198.0
RE Z1,4 15000.0
RE Z2,4 13338.0
RE Z3,4 11976.0
RE Z4,4 12603.0
RE Z5,4 11340.0
RE Z6,4 11232.0
RE Z7,4 10334.0
RE Z8,4 10297.0
RE Z9,4 7748.0
RE Z10,4 6869.0
RE Z11,4 6050.0
RE Z12,4 5454.0
RE Z13,4 4115.0
RE Z14,4 3790.0
RE Z15,4 3087.0
RE Z16,4 2596.0
RE Z17,4 1873.0
RE Z18,4 1571.0
RE Z19,4 1227.0
RE Z20,4 934.0
RE Z21,4 707.0
RE Z1,5 12000.0
RE Z2,5 13484.0
RE Z3,5 12453.0
RE Z4,5 11738.0
RE Z5,5 13665.0
RE Z6,5 12112.0
RE Z7,5 11115.0
RE Z8,5 12427.0
RE Z9,5 10057.0
RE Z10,5 9564.0
RE Z11,5 8671.0
RE Z12,5 8430.0
RE Z13,5 7435.0
RE Z14,5 6474.0
RE Z15,5 6524.0
RE Z16,5 5360.0
RE Z17,5 4556.0
RE Z18,5 4009.0
RE Z19,5 3677.0
RE Z20,5 3197.0
RE Z21,5 2562.0
RE Z1,6 9000.0
RE Z2,6 8426.0
RE Z3,6 9272.0
RE Z4,6 9710.0
RE Z5,6 8534.0
RE Z6,6 9600.0
RE Z7,6 8826.0
RE Z8,6 8747.0
RE Z9,6 8698.0
RE Z10,6 8766.0
RE Z11,6 8291.0
RE Z12,6 7411.0
RE Z13,6 7627.0
RE Z14,6 6658.0
RE Z15,6 5880.0
RE Z16,6 5762.0
RE Z17,6 5058.0
RE Z18,6 4527.0
RE Z19,6 4229.0
RE Z20,6 3695.0
RE Z21,6 3163.0
RE Z1,7 7000.0
RE Z2,7 6615.0
RE Z3,7 6891.0
RE Z4,7 6821.0
RE Z5,7 6242.0
RE Z6,7 6647.0
RE Z7,7 6842.0
RE Z8,7 7199.0
RE Z9,7 6542.0
RE Z10,7 6810.0
RE Z11,7 6827.0
RE Z12,7 6423.0
RE Z13,7 6268.0
RE Z14,7 5859.0
RE Z15,7 5562.0
RE Z16,7 4480.0
RE Z17,7 4944.0
RE Z18,7 4233.0
RE Z19,7 3805.0
RE Z20,7 3159.0
RE Z21,7 3232.0
RE Z1,8 3000.0
RE Z2,8 4022.0
RE Z3,8 5020.0
RE Z4,8 5722.0
RE Z5,8 5695.0
RE Z6,8 7034.0
RE Z7,8 7348.0
RE Z8,8 7684.0
RE Z9,8 7410.0
RE Z10,8 6961.0
RE Z11,8 7525.0
RE Z12,8 8388.0
RE Z13,8 7189.0
RE Z14,8 7467.0
RE Z15,8 7144.0
RE Z16,8 7256.0
RE Z17,8 7538.0
RE Z18,8 6649.0
RE Z19,8 6378.0
RE Z20,8 6454.0
RE Z21,8 5566.0
* Other useful values
IE 1 1
IE 2 2
IE 3 3
IE 4 4
IE 5 5
IE 6 6
IE 7 7
IE 8 8
IA NE-1 NE -1
IA NH-1 NH -1
* Factorials
RE FACT0 1.0
DO I 1 NC
RI RI I
IA I-1 I -1
A* FACT(I) FACT(I-1) RI
ND
* itau[i] is the largest integer k with t[k] <= tau[i]
* itau {i in 1..nm} := min(nh,floor(tau[i]/h)+1);
DO I 1 NM
A/ TAU/H TAU(I) H
IR IT/H TAU/H
IA IT/H+1 IT/H 1
I= A IT/H+1
I= B NH
* this is how we find min(a,b) !!
* first set a <- - a and b <- - b
IM A A -1
IM B B -1
* now add |a| + |b| to both
RI A A
R( ABSA ABS A
IR ABSA ABSA
RI B B
R( ABSB ABS B
IR ABSB ABSB
I+ ABSA+B ABSA ABSB
I+ A A ABSA+B
I+ B B ABSA+B
* only one of [a/b] and [b/a] is nonzero (unless a=b)
I/ A/B A B
I/ B/A B A
* form max(a,b) = (a.[a/b] + b.[b/a]) / ([a/b] + [b/a])
I+ SUM A/B B/A
I* A A A/B
I* B B B/A
I+ MAXA,B A B
I/ MAXA,B MAXA,B SUM
* subtract |a| + |b| and negate to find the required minimum
I- MINA,B ABSA+B MAXA,B
AI ITAU(I) MINA,B
ND
VARIABLES
* The growth rates G and mortality rates M
DO I 1 NE-1
X M(I)
X G(I)
OD J
X M(NE)
* The collocation approximation u is defined by the parameters V and W
DO I 1 NH
DO J 1 NE
X V(I,J)
OD J
DO K 1 NC
DO S 1 NE
X W(I,K,S)
OD S
OD K
DO J 1 NC
DO S 1 NE
* U[i,j,s] is u evaluated at the collocation points
X U(I,J,S)
* DU[i,j,s] is u' evaluated at the collocation points
X DU(I,J,S)
ND
GROUPS
* param t {i in 1..nh+1} := (i-1)*h
* obj(j,s) = v[itau[j],s] - z[j,s] + (
* sum {k in 1..nc} w[itau[j],k,s]*(tau[j]-t[itau[j]])^k/(fact[k]*h^(k-1)))
DO J 1 NM
A= RITAU ITAU(J)
IR I RITAU
RA T RITAU -1.0
R* T T H
A- DIFF TAU(J) T
DO S 1 NE
R= RATIO DIFF
XN OBJ(J,S) V(I,S) 1.0
DO K 1 NC
A/ COEF RATIO FACT(K)
ZN OBJ(J,S) W(I,K,S) COEF
R* RATIO RATIO DIFF
R/ RATIO RATIO H
ND
* - u {i in 1..nh, j in 1..nc, s in 1..ne} +
* v[i,s] + h*sum {k in 1..nc} w[i,k,s]*(rho[j]^k/fact[k]) = 0
DO I 1 NH
DO J 1 NC
A= RH RHO(J)
XE U(I,J,1) U(I,J,1) -1.0 V(I,1) 1.0
XE U(I,J,2) U(I,J,2) -1.0 V(I,2) 1.0
XE U(I,J,3) U(I,J,3) -1.0 V(I,3) 1.0
XE U(I,J,4) U(I,J,4) -1.0 V(I,4) 1.0
XE U(I,J,5) U(I,J,5) -1.0 V(I,5) 1.0
XE U(I,J,6) U(I,J,6) -1.0 V(I,6) 1.0
XE U(I,J,7) U(I,J,7) -1.0 V(I,7) 1.0
XE U(I,J,8) U(I,J,8) -1.0 V(I,8) 1.0
R* PROD RH H
DO K 1 NC
A/ COEF PROD FACT(K)
ZE U(I,J,1) W(I,K,1) COEF
ZE U(I,J,2) W(I,K,2) COEF
ZE U(I,J,3) W(I,K,3) COEF
ZE U(I,J,4) W(I,K,4) COEF
ZE U(I,J,5) W(I,K,5) COEF
ZE U(I,J,6) W(I,K,6) COEF
ZE U(I,J,7) W(I,K,7) COEF
ZE U(I,J,8) W(I,K,8) COEF
R* PROD PROD RH
OD K
* - du {i in 1..nh, j in 1..nc, s in 1..ne} +
* sum {k in 1..nc} w[i,k,s]*(rho[j]^(k-1)/fact[k-1]) = 0
XE DU(I,J,1) DU(I,J,1) -1.0
XE DU(I,J,2) DU(I,J,2) -1.0
XE DU(I,J,3) DU(I,J,3) -1.0
XE DU(I,J,4) DU(I,J,4) -1.0
XE DU(I,J,5) DU(I,J,5) -1.0
XE DU(I,J,6) DU(I,J,6) -1.0
XE DU(I,J,7) DU(I,J,7) -1.0
XE DU(I,J,8) DU(I,J,8) -1.0
RE PROD 1.0
DO K 1 NC
IA K-1 K -1
A/ COEF PROD FACT(K-1)
ZE DU(I,J,1) W(I,K,1) COEF
ZE DU(I,J,2) W(I,K,2) COEF
ZE DU(I,J,3) W(I,K,3) COEF
ZE DU(I,J,4) W(I,K,4) COEF
ZE DU(I,J,5) W(I,K,5) COEF
ZE DU(I,J,6) W(I,K,6) COEF
ZE DU(I,J,7) W(I,K,7) COEF
ZE DU(I,J,8) W(I,K,8) COEF
R* PROD PROD RH
ND
* continuity {i in 1..nh-1, s in 1..ne}:
* v[i,s] - v[i+1,s] + (sum {j in 1..nc} w[i,j,s]*h/fact[j]) = 0
DO I 1 NH-1
IA I+1 I 1
DO S 1 NE
XE C(I,S) V(I,S) 1.0 V(I+1,S) -1.0
DO J 1 NC
A/ COEF H FACT(J)
ZE C(I,S) W(I,J,S) COEF
ND
* collocation eqns (linear parts)
DO I 1 NH
DO J 1 NC
DO S 1 NE
XE CO(I,J,S) DU(I,J,S) 1.0
ND
CONSTANTS
DO J 1 NM
DO S 1 NE
Z MARINE OBJ(J,S) Z(J,S)
ND
BOUNDS
XR MARINE 'DEFAULT'
DO I 1 NE-1
XL MARINE M(I) 0.0
XL MARINE G(I) 0.0
OD J
XL MARINE M(NE) 0.0
START POINT
DO I 1 NH
DO J 1 NE
X MARINE V(I,J)
OD J
ND
IE I1 1
A= RITAU ITAU(1)
IR I2 RITAU
DO I I1 I2
DO S 1 NE
Z MARINE V(I,S) Z(1,S)
DO J 1 NC
X MARINE W(I,J,S) 0.0
Z MARINE U(I,J,S) Z(1,S)
X MARINE DU(I,J,S) 0.0
ND
DO K 2 NM
IA I1 I2 1
A= RITAU ITAU(K)
IR I2 RITAU
DO I I1 I2
IE S 1
Z MARINE V(I,S) Z(K,S)
DO J 1 NC
X MARINE W(I,J,S) 0.0
Z MARINE U(I,J,S) Z(K,S)
X MARINE DU(I,J,S) 0.0
OD J
IE S 2
Z MARINE V(I,S) Z(K,S)
DO J 1 NC
X MARINE W(I,J,S) 0.0
Z MARINE U(I,J,S) Z(K,S)
X MARINE DU(I,J,S) 0.0
OD J
IE S 3
Z MARINE V(I,S) Z(K,S)
DO J 1 NC
X MARINE W(I,J,S) 0.0
Z MARINE U(I,J,S) Z(K,S)
X MARINE DU(I,J,S) 0.0
OD J
ND
IA I1 I2 1
I= I2 NH
DO I I1 I2
DO S 1 NE
Z MARINE V(I,S) Z(NM,S)
DO J 1 NC
X MARINE W(I,J,S) 0.0
Z MARINE U(I,J,S) Z(NM,S)
X MARINE DU(I,J,S) 0.0
ND
ELEMENT TYPE
EV PROD THETA U
ELEMENT USES
DO I 1 NH
DO J 1 NC
DO S 1 NE
XT M(I,J,S) PROD
ZV M(I,J,S) THETA M(S)
ZV M(I,J,S) U U(I,J,S)
OD S
DO S 1 NE-1
XT G(I,J,S) PROD
ZV G(I,J,S) THETA G(S)
ZV G(I,J,S) U U(I,J,S)
ND
GROUP TYPE
GV L2 X
GROUP USES
* collocation for equation 1 {i in 1..nh,j in 1..nc}:
* du[i,j,1] + (m[1]+g[1])*u[i,j,1] = 0
* collocation for equations {i in 1..nh,j in 1..nc,s in 2..ne-1}:
* du[i,j,s] - g[s-1]*u[i,j,s-1] + (m[s]+g[s])*u[i,j,s] = 0
* collocation for equation {i in 1..nh,j in 1..nc}:
* du[i,j,ne] - g[ne-1]*u[i,j,ne-1] + m[ne]*u[i,j,ne] = 0
DO I 1 NH
DO J 1 NC
I= S 1
XE CO(I,J,S) G(I,J,S) 1.0 M(I,J,S) 1.0
DO S 2 NE-1
IA S1 S -1
XE CO(I,J,S) G(I,J,S1) -1.0 M(I,J,S) 1.0
XE CO(I,J,S) G(I,J,S) 1.0
OD S
I= S NE
IA S1 S -1
XE CO(I,J,S) G(I,J,S1) -1.0 M(I,J,S) 1.0
ND
* objective function sum {j in 1..nm} obj(j)^2
DO J 1 NM
DO S 1 NE
XT OBJ(J,S) L2
ND
OBJECT BOUND
* Solution
*LO SOLUTION 1.97522D+07 $ (NH=25)
*LO SOLUTION 1.97465D+07 $ (NH=50)
*LO SOLUTION 1.97465D+07 $ (NH=100)
*LO SOLUTION 1.97465D+07 $ (NH=200)
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS MARINE
INDIVIDUALS
T PROD
F THETA * U
G THETA U
G U THETA
H THETA U 1.0
ENDATA
*********************
* SET UP THE GROUPS *
* ROUTINE *
*********************
GROUPS MARINE
INDIVIDUALS
T L2
F X * X
G X + X
H 2.0
ENDATA