• Example 1: Rosenbrock function solved using Levenberg-Marquardt (LM) optimization technique (1 initial guess for unknown model parameters)
• Example 2: Rosenbrock function solved using Levenberg-Marquardt (LM) optimization technique (10 initial guesses for unknown model parameters)
• Example 3: Rosenbrock function solved using Squads optimization technique (10 initial guesses for unknown model parameters)

### Example 1: MADS solving 5-dimensional Rosenbrock function (5 parameters, 8 observations) using random initial guesses for model parameters and the Levenberg-Marquardt (LM) optimization technique

\$> mads t01 test=33 dim=5 igrnd real=1
MADS: Model Analyses & Decision Support (v1.1) 2011
---------------------------------------------------
Velimir Vesselinov (monty) vvv@lanl.gov

Test Function #33 Dimensionality 5
Problem type: calibration

Calibration technique: sequential calibration using a set of random initial values (realizations = 1)
Optimization method: opt=lm | Levenberg-Marquardt optimization
Number of Levenberg-Marquardt iterations = will be computed internally

Global termination criteria:
1: Maximum number of evaluations = 5000
2: Objective function cutoff value: NOT implemented (ADD keyword cutoff=[value] to implement)
3: Observations within defined calibration ranges: NOT implemented (ADD keyword success to implement)
Objective function: sum of squared residuals

Sin transformation of the model parameters is applied!

Debug level (general): debug=0

WARNING: Input file is not needed because this is a test problem!

Model: internal | test optimization problem #33: Rosenbrock (with observations = (d-1)*2) dimensionality 5

Execution date & time stamp: 20111107-190742

SEQUENTIAL RUNS using random initial guesses for model parameters (realizations = 1):
No flagged parameters; all optimizable parameters are randomized
New Seed: 1096757678
Random sampling (variables 5; realizations 1) using
Pure random sampling method ... done.

Random set #1:
Parameter #1 13.0576138179329
Parameter #2 -40.6845060604347
Parameter #3 15.5066505344963
Parameter #4 20.2755913894203
Parameter #5 -28.1354556701211
Levenberg-Marquardt Optimization ...
Model parameters:
Parameter #1 -0.751434
Parameter #2 0.575961
Parameter #3 0.338018
Parameter #4 0.123299
Parameter #5 0.0127839
Objective function: 4.47966 Success: 0
At least one of the estimated model parameters has an absolute error from the true parameters (>0.1)!
Number of function evaluations = 96

Total number of evaluations = 96
Total number of jacobians = 10
Minimum objective function: 4.47966
Model parameters:
Parameter #1 -0.751434
Parameter #2 0.575961
Parameter #3 0.338018
Parameter #4 0.123299
Parameter #5 0.0127839
Objective function: 4.47966 Success: 0
At least one of the estimated model parameters has an absolute error from the true parameters (>0.1)!
Number of function evaluations = 96
Results are saved in t01.igrnd.results and t01.igrnd-opt=lm_eval=5000_real=1
Simulation time = 1 seconds
Functional evaluations = 96
Jacobian evaluations = 10
Levenberg-Marquardt optimizations = 1
Functional evaluations per second = 96
Seed = 1096757678
Execution started on Mon Nov 7 19:07:42 2011
Execution completed on Mon Nov 7 19:07:43 2011
Execution date & time stamp: 20111107-190742

### Example 2: MADS solving 5-dimensional Rosenbrock function (5 parameters, 8 observations) using 10 different random initial guesses for model parameters and the Levenberg-Marquardt (LM) optimization technique

>\$ mads t01 test=33 dim=5 igrnd real=10 truth=0.1
MADS: Model Analyses & Decision Support (v1.1) 2011
---------------------------------------------------
Velimir Vesselinov (monty) vvv@lanl.gov

Test Function #33 Dimensionality 5
Problem type: calibration

Calibration technique: sequential calibration using a set of random initial values (realizations = 10)
Optimization method: opt=lm | Levenberg-Marquardt optimization
Number of Levenberg-Marquardt iterations = will be computed internally

Global termination criteria:
1: Maximum number of evaluations = 5000
2: Objective function cutoff value: NOT implemented (ADD keyword cutoff=[value] to implement)
3: Observations within defined calibration ranges: implemented (keyword success)
Objective function: sum of squared residuals

Sin transformation of the model parameters is applied!

Debug level (general): debug=0

WARNING: Input file is not needed because this is a test problem!

Model: internal | test optimization problem #33: Rosenbrock (with observations = (d-1)*2) dimensionality 5

Execution date & time stamp: 20111107-192407

SEQUENTIAL RUNS using random initial guesses for model parameters (realizations = 10):
No flagged parameters; all optimizable parameters are randomized
New Seed: 1145722293
Random sampling (variables 5; realizations 10) using
Improved Distances LHS method ( real < 500 ) ... done.

Random set #1: Levenberg-Marquardt Optimization ... Evaluations: 45 Jacobians: 5 Objective function: 3177.81 Success: 0
Random set #2: Levenberg-Marquardt Optimization ... Evaluations: 37 Jacobians: 4 Objective function: 229706 Success: 0
Random set #3: Levenberg-Marquardt Optimization ... Evaluations: 29 Jacobians: 4 Objective function: 0.546085 Success: 1
Random set #4: Levenberg-Marquardt Optimization ... Evaluations: 54 Jacobians: 7 Objective function: 3.93084 Success: 0
Random set #5: Levenberg-Marquardt Optimization ... Evaluations: 127 Jacobians: 11 Objective function: 1.61683 Success: 0
Random set #6: Levenberg-Marquardt Optimization ... Evaluations: 37 Jacobians: 5 Objective function: 3.93085 Success: 0
Random set #7: Levenberg-Marquardt Optimization ... Evaluations: 101 Jacobians: 11 Objective function: 2.49285 Success: 0
Random set #8: Levenberg-Marquardt Optimization ... Evaluations: 80 Jacobians: 9 Objective function: 0.366393 Success: 0
Random set #9: Levenberg-Marquardt Optimization ... Evaluations: 54 Jacobians: 5 Objective function: 521.221 Success: 0
Random set #10: Levenberg-Marquardt Optimization ... Evaluations: 80 Jacobians: 10 Objective function: 0.00382584 Success: 1
Total number of evaluations = 644
Total number of jacobians = 71
Minimum objective function: 0.00382584
Model parameters:
Parameter #1 0.994713
Parameter #2 0.989551
Parameter #3 0.979227
Parameter #4 0.956186
Parameter #5 0.911826
Objective function: 0.00382584 Success: 1
All the estimated model parameters have an absolute error from the true parameters (<0.1)!
Number of function evaluations = 644
Number of the sequential calibration runs producing predictions within calibration ranges = 2 (out of 10; success ratio 0.2)
Statistics of successful number of evaluations : 29 - 29 29 80 - 80 : 2
Statistics of total number of evaluations : 29 - 37 54 80 - 127 : 10
Results are saved in t01.igrnd.results and t01.igrnd-opt=lm_eval=5000_real=10
Simulation time = 0 seconds
Functional evaluations = 644
Jacobian evaluations = 71
Levenberg-Marquardt optimizations = 10
Seed = 1145722293
Execution started on Mon Nov 7 19:24:07 2011
Execution completed on Mon Nov 7 19:24:07 2011
Execution date & time stamp: 20111107-192407

### Example 3: MADS solving 5-dimensional Rosenbrock function (5 parameters, 8 observations) using 10 different random initial guesses for model parameters and the Squads optimization technique

MADS: Model Analyses & Decision Support (v1.1) 2011
---------------------------------------------------
Velimir Vesselinov (monty) vvv@lanl.gov

Test Function #33 Dimensionality 5
Problem type: calibration

Calibration technique: sequential calibration using a set of random initial values (realizations = 10)
Number of Levenberg-Marquardt iterations = will be computed internally
Number of particles = will be computed internally

Global termination criteria:
1: Maximum number of evaluations = 5000
2: Objective function cutoff value: NOT implemented (ADD keyword cutoff=[value] to implement)
3: Observations within defined calibration ranges: implemented (keyword success)
Objective function: sum of squared residuals

Sin transformation of the model parameters is applied!

Debug level (general): debug=0

WARNING: Input file is not needed because this is a test problem!

Model: internal | test optimization problem #33: Rosenbrock (with observations = (d-1)*2) dimensionality 5

Execution date & time stamp: 20111107-192534

SEQUENTIAL RUNS using random initial guesses for model parameters (realizations = 10):
No flagged parameters; all optimizable parameters are randomized
New Seed: 1150047087
Random sampling (variables 5; realizations 10) using
Improved Distances LHS method ( real < 500 ) ... done.

Random set #1: SQUADS: Coupled Particle-Swarm and Levenberg-Marquardt Optimization ... Evaluations: 303 Jacobians: 30 Objective function: 0.00995423 Success: 1
Random set #2: SQUADS: Coupled Particle-Swarm and Levenberg-Marquardt Optimization ... Evaluations: 154 Jacobians: 12 Objective function: 0.00259125 Success: 1
Random set #3: SQUADS: Coupled Particle-Swarm and Levenberg-Marquardt Optimization ... Evaluations: 494 Jacobians: 48 Objective function: 0.000377564 Success: 1
Random set #4: SQUADS: Coupled Particle-Swarm and Levenberg-Marquardt Optimization ... Evaluations: 126 Jacobians: 11 Objective function: 0.0451372 Success: 1
Random set #5: SQUADS: Coupled Particle-Swarm and Levenberg-Marquardt Optimization ... Evaluations: 968 Jacobians: 98 Objective function: 0.00244585 Success: 1
Random set #6: SQUADS: Coupled Particle-Swarm and Levenberg-Marquardt Optimization ... Evaluations: 264 Jacobians: 27 Objective function: 0.00387452 Success: 1
Random set #7: SQUADS: Coupled Particle-Swarm and Levenberg-Marquardt Optimization ... Evaluations: 166 Jacobians: 13 Objective function: 0.0219389 Success: 1
Random set #8: SQUADS: Coupled Particle-Swarm and Levenberg-Marquardt Optimization ... Evaluations: 440 Jacobians: 41 Objective function: 0.0977437 Success: 1
Random set #9: SQUADS: Coupled Particle-Swarm and Levenberg-Marquardt Optimization ... Evaluations: 352 Jacobians: 36 Objective function: 0.00476497 Success: 1
Random set #10: SQUADS: Coupled Particle-Swarm and Levenberg-Marquardt Optimization ... Evaluations: 353 Jacobians: 32 Objective function: 0.00200607 Success: 1
Total number of evaluations = 3620
Total number of jacobians = 348
Minimum objective function: 0.000377564
Model parameters:
Parameter #1 1.00117
Parameter #2 1.00075
Parameter #3 1.00054
Parameter #4 1.00065
Parameter #5 1.00101
Objective function: 0.000377564 Success: 1
All the estimated model parameters have an absolute error from the true parameters (<0.1)!
Number of function evaluations = 3620
Number of the sequential calibration runs producing predictions within calibration ranges = 10 (out of 10; success ratio 1)
Statistics of successful number of evaluations : 126 - 166 303 440 - 968 : 10
Results are saved in t01.igrnd.results and t01.igrnd-opt=squads_eval=5000_real=10
Simulation time = 0 seconds
Functional evaluations = 3620
Jacobian evaluations = 348
Levenberg-Marquardt optimizations = 43
Seed = 1150047087
Execution started on Mon Nov 7 19:25:34 2011
Execution completed on Mon Nov 7 19:25:34 2011
Execution date & time stamp: 20111107-192534