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PINENE.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME PINENE
* Problem :
* *********
* Determine the reaction coefficients in the thermal isometrization
* of alpha-Pinene. The linear kinetic model is
* y_1' = - (theta_1 + theta_2) y_1
* y_2' = theta_1 y_1
* y_3' = theta_2 y_1 - ( theta_3 + theta_4 ) y_3 + theta_5 y_5
* y_4' = theta_3 y_3
* y_5' = theta_4 y_3 - theta_5 y_5
* where the reaction coefficients theta_i are positive,
* with given initial conditions. The problem is to minimize
* sum{i=1,8} || y(tau_i,theta) - z_i||^2
* where the z_i are concentration measurements for y at times tau_i (i=1,8)
* This is problem 5 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 5
* The number of subintervals
*IE NH 25 $-PARAMETER
*IE NH 50 $-PARAMETER
*IE NH 100 $-PARAMETER
IE NH 200 $-PARAMETER
* The number of ODE parameters
IE NP 5 $-PARAMETER
* The number of measurements
IE NM 8 $-PARAMETER
* 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 1230.0
RE TAU2 3060.0
RE TAU3 4920.0
RE TAU4 7800.0
RE TAU5 10680.0
RE TAU6 15030.0
RE TAU7 22620.0
RE TAU8 36420.0
* 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 88.35
RE Z2,1 76.4
RE Z3,1 65.1
RE Z4,1 50.4
RE Z5,1 37.5
RE Z6,1 25.9
RE Z7,1 14.0
RE Z8,1 4.5
RE Z1,2 7.3
RE Z2,2 15.6
RE Z3,2 23.1
RE Z4,2 32.9
RE Z5,2 42.7
RE Z6,2 49.1
RE Z7,2 57.4
RE Z8,2 63.1
RE Z1,3 2.3
RE Z2,3 4.5
RE Z3,3 5.3
RE Z4,3 6.0
RE Z5,3 6.0
RE Z6,3 5.9
RE Z7,3 5.1
RE Z8,3 3.8
RE Z1,4 0.4
RE Z2,4 0.7
RE Z3,4 1.1
RE Z4,4 1.5
RE Z5,4 1.9
RE Z6,4 2.2
RE Z7,4 2.6
RE Z8,4 2.9
RE Z1,5 1.75
RE Z2,5 2.8
RE Z3,5 5.8
RE Z4,5 9.3
RE Z5,5 12.0
RE Z6,5 17.0
RE Z7,5 21.0
RE Z8,5 25.7
* ODE initial conditions
RE BC1 100.0
RE BC2 0.0
RE BC3 0.0
RE BC4 0.0
RE BC5 0.0
* Other useful values
IE 1 1
IE 2 2
IE 3 3
IE 4 4
IE 5 5
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
DO I 1 NP
X THETA(I) $ ODE parameters
ND
* 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
* U[i,j,s] is u evaluated at the collocation points
* DU[i,j,s] is u' evaluated at the collocation points
X U(I,J,1)
X DU(I,J,1)
X DU(I,J,2)
X U(I,J,3)
X DU(I,J,3)
X DU(I,J,4)
X U(I,J,5)
X DU(I,J,5)
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,3) U(I,J,3) -1.0 V(I,3) 1.0
XE U(I,J,5) U(I,J,5) -1.0 V(I,5) 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,3) W(I,K,3) COEF
ZE U(I,J,5) W(I,K,5) 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
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
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 PINENE OBJ(J,S) Z(J,S)
ND
BOUNDS
XR PINENE 'DEFAULT'
DO I 1 NP
XL PINENE THETA(I) 0.0
ND
DO S 1 NE
ZX PINENE V(1,S) BC(S)
ND
START POINT
DO I 1 NP
X PINENE THETA(I) 0.0
ND
DO I 1 NH
DO J 1 NE
X PINENE V(I,J)
OD J
ND
IE I1 1
A= RITAU ITAU(1)
IR I2 RITAU
DO I I1 I2
IE S 1
Z PINENE V(I,S) BC(S)
DO J 1 NC
X PINENE W(I,J,S) 0.0
Z PINENE U(I,J,S) BC(S)
X PINENE DU(I,J,S) 0.0
OD J
IE S 2
Z PINENE V(I,S) BC(S)
DO J 1 NC
X PINENE W(I,J,S) 0.0
X PINENE DU(I,J,S) 0.0
OD J
IE S 3
Z PINENE V(I,S) BC(S)
DO J 1 NC
X PINENE W(I,J,S) 0.0
Z PINENE U(I,J,S) BC(S)
X PINENE DU(I,J,S) 0.0
OD J
IE S 4
Z PINENE V(I,S) BC(S)
DO J 1 NC
X PINENE W(I,J,S) 0.0
X PINENE DU(I,J,S) 0.0
OD J
IE S 5
Z PINENE V(I,S) BC(S)
DO J 1 NC
X PINENE W(I,J,S) 0.0
Z PINENE U(I,J,S) BC(S)
X PINENE DU(I,J,S) 0.0
OD J
ND
DO K 2 NM
IA I1 I2 1
A= RITAU ITAU(K)
IR I2 RITAU
DO I I1 I2
IE S 1
Z PINENE V(I,S) Z(K,S)
DO J 1 NC
X PINENE W(I,J,S) 0.0
Z PINENE U(I,J,S) Z(K,S)
X PINENE DU(I,J,S) 0.0
OD J
IE S 2
Z PINENE V(I,S) Z(K,S)
DO J 1 NC
X PINENE W(I,J,S) 0.0
X PINENE DU(I,J,S) 0.0
OD J
IE S 3
Z PINENE V(I,S) Z(K,S)
DO J 1 NC
X PINENE W(I,J,S) 0.0
Z PINENE U(I,J,S) Z(K,S)
X PINENE DU(I,J,S) 0.0
OD J
IE S 4
Z PINENE V(I,S) Z(K,S)
DO J 1 NC
X PINENE W(I,J,S) 0.0
X PINENE DU(I,J,S) 0.0
OD J
IE S 5
Z PINENE V(I,S) Z(K,S)
DO J 1 NC
X PINENE W(I,J,S) 0.0
Z PINENE U(I,J,S) Z(K,S)
X PINENE DU(I,J,S) 0.0
OD J
ND
IA I1 I2 1
I= I2 NH
DO I I1 I2
IE S 1
Z PINENE V(I,S) Z(NM,S)
DO J 1 NC
X PINENE W(I,J,S) 0.0
Z PINENE U(I,J,S) Z(NM,S)
X PINENE DU(I,J,S) 0.0
OD J
IE S 2
Z PINENE V(I,S) Z(NM,S)
DO J 1 NC
X PINENE W(I,J,S) 0.0
X PINENE DU(I,J,S) 0.0
OD J
IE S 3
Z PINENE V(I,S) Z(NM,S)
DO J 1 NC
X PINENE W(I,J,S) 0.0
Z PINENE U(I,J,S) Z(NM,S)
X PINENE DU(I,J,S) 0.0
OD J
IE S 4
Z PINENE V(I,S) Z(NM,S)
DO J 1 NC
X PINENE W(I,J,S) 0.0
X PINENE DU(I,J,S) 0.0
OD J
IE S 5
Z PINENE V(I,S) Z(NM,S)
DO J 1 NC
X PINENE W(I,J,S) 0.0
Z PINENE U(I,J,S) Z(NM,S)
X PINENE DU(I,J,S) 0.0
OD J
ND
ELEMENT TYPE
EV PROD THETA U
EV PROD2 THETAA THETAB
EV PROD2 U
IV PROD2 THETA U
ELEMENT USES
DO I 1 NH
DO J 1 NC
XT C1(I,J) PROD2
ZV C1(I,J) THETAA THETA(1)
ZV C1(I,J) THETAB THETA(2)
ZV C1(I,J) U U(I,J,1)
XT C2(I,J) PROD
ZV C2(I,J) THETA THETA(1)
ZV C2(I,J) U U(I,J,1)
XT C3(I,J) PROD
ZV C3(I,J) THETA THETA(2)
ZV C3(I,J) U U(I,J,1)
XT C4(I,J) PROD2
ZV C4(I,J) THETAA THETA(3)
ZV C4(I,J) THETAB THETA(4)
ZV C4(I,J) U U(I,J,3)
XT C5(I,J) PROD
ZV C5(I,J) THETA THETA(5)
ZV C5(I,J) U U(I,J,5)
XT C6(I,J) PROD
ZV C6(I,J) THETA THETA(3)
ZV C6(I,J) U U(I,J,3)
XT C7(I,J) PROD
ZV C7(I,J) THETA THETA(4)
ZV C7(I,J) U U(I,J,3)
ND
GROUP TYPE
GV L2 X
GROUP USES
* collocation for equation 1 {i in 1..nh,j in 1..nc}:
* du[i,j,1] + (theta[1]+theta[2])*u[i,j,1] = 0
* collocation for equation 2 {i in 1..nh,j in 1..nc}:
* du[i,j,2] - theta[1]*u[i,j,1] = 0
* collocation for equation 3 {i in 1..nh,j in 1..nc}:
* du[i,j,3] - theta[2]*u[i,j,1] + (theta[3]+theta[4])*u[i,j,3]
* - theta[5]*u[i,j,5] = 0
* collocation for equation 4 {i in 1..nh,j in 1..nc}:
* du[i,j,4] - theta[3]*u[i,j,3] = 0
* collocation for equation 5 {i in 1..nh,j in 1..nc}:
* du[i,j,5] - theta[4]*u[i,j,3] + theta[5]*u[i,j,5] = 0
DO I 1 NH
DO J 1 NC
XE CO(I,J,1) C1(I,J) 1.0
XE CO(I,J,2) C2(I,J) -1.0
XE CO(I,J,3) C3(I,J) -1.0 C4(I,J) 1.0
XE CO(I,J,3) C5(I,J) -1.0
XE CO(I,J,4) C6(I,J) -1.0
XE CO(I,J,5) C7(I,J) -1.0 C5(I,J) 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 1.98715D+01 $ (NH=25)
*LO SOLUTION 1.98721D+01 $ (NH=50)
*LO SOLUTION 1.98721D+01 $ (NH=100
*LO SOLUTION 1.98721D+01 $ (NH=200)
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS PINENE
INDIVIDUALS
T PROD
F THETA * U
G THETA U
G U THETA
H THETA U 1.0
T PROD2
R THETA THETAA 1.0 THETAB 1.0
R U U 1.0
F THETA * U
G THETA U
G U THETA
H THETA U 1.0
ENDATA
*********************
* SET UP THE GROUPS *
* ROUTINE *
*********************
GROUPS PINENE
INDIVIDUALS
T L2
F X * X
G X + X
H 2.0
ENDATA