Planning Multiple Experiments with Pyomo.DoE - Pyomo.DoE Workshop and Tutorials

Multi-experiment design is useful when several experiments can be run in parallel. Instead of designing one experiment at a time, we optimize multiple experiments together so that they provide complementary information about the model parameters. In Pyomo.DoE, this is done by maximizing a scalar function of the sum of the Fisher information matrices from each experiment, together with any prior information.

Maximize a scalar-valued function $\psi(\cdot)$ of the sum of the Fisher information matrices $\mathbf{M}$ :

\begin{align*} \max_{u_1,.., u_{N_{exp}}} \quad & \psi \left(\mathbf{M}_0+ \sum_{k\in E}\mathbf{M}_k(u_k)\right) \\ \mathrm{s.t.} \quad & C_p^H \frac{dT_{H,k}}{dt} = U_a (T_{amb, k} - T_{H, k}) + U_b (T_{S,k} - T_{H,k}) + \alpha P u_k(t); \; \forall k \in E\\ & C_p^S \frac{dT_{S, k}}{dt} = U_b (T_{H,k} - T_{S,k}); \; \forall k \in E \\ & 0\% \leq u_k(t) \leq 100 \%; \; \forall k \in E\\ & T_{H,k}(t_0) = T_{amb, k}; \; \forall k \in E\\ & T_{S, k}(t_0) = T_{amb, k}; \; \forall k \in E \end{align*}

(1)

Here, $E=\{1, 2, ..., N_{exp}\}$ is the set of experiments in the experimental campaign. For multi-experiment design, Pyomo.DoE automatically formulates, initializes, and solves this optimization problem for the following optimality criteria:

Maximizing $\log_{10}(\mathrm{trace (FIM)})$ , i.e., Pseudo A-optimality
Minimizing $\log_{10}(\mathrm{trace (FIM^{-1})})$ , i.e., A-optimality,
Maximizing $\log_{10}(\mathrm{det (FIM)})$ , i.e., D-optimality.
Maximizing $\log_{10}(\mathrm{min\; eigenvalue (FIM)})$ , i.e., E-optimality.
Minimizing $\log_{10}(\mathrm{condition\; number (FIM)})$ , i.e., ME-optimality.

For documentation follow the Pyomo documentation page.

Import Necessary Functions for the Two-State TC Lab Model¶

import sys

# If running on Google Colab, install Pyomo and Ipopt via IDAES
on_colab = "google.colab" in sys.modules
if on_colab:
    !wget "https://raw.githubusercontent.com/dowlinglab/pyomo-doe/main/notebooks/tclab_pyomo.py"

# import TCLab model, simulation, and data analysis functions
from tclab_pyomo import (
    TC_Lab_data,
    TC_Lab_experiment,
    extract_plot_results,
    results_summary,
)

# set default number of states in the TCLab model
number_tclab_states = 2

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Load Experimental Data (sine test)¶

We will load the sine test experimental data to serve as an initial point.

import pandas as pd

if on_colab:
    file = "https://raw.githubusercontent.com/dowlinglab/pyomo-doe/main/data/tclab_sine_test_5min_period.csv"
else:
    file = "../data/tclab_sine_test_5min_period.csv"
df = pd.read_csv(file)
df.head()

Use Prior Parameter Information¶

We will use the parameter estimates and covariance matrix obtained in the previous notebook. Since the Fisher information matrix (FIM) is the inverse of the covariance matrix, we can use that result as prior information for the new design problem.

import numpy as np

# Load Pyomo.DoE class
from pyomo.contrib.doe import DesignOfExperiments
from pyomo.environ import SolverFactory

# Theta values estimated from the regularized regresssion in the previous notebook
# L2 regularization
theta_values = {
    "Ua": 0.041705,
    "Ub": 0.012009,
    "inv_CpH": 0.167457,
    "inv_CpS": 4.545432,
}

# Cov estimated from the regularized regression in the previous notebook
# L2 regularization
cov = np.array(
    [
        [1.857017e-10, -2.576198e-10, 1.402148e-09, -2.242347e-12],
        [-2.576198e-10, 1.624383e-07, 9.109870e-08, -6.325555e-05],
        [1.402148e-09, 9.109870e-08, 1.031454e-07, -3.890789e-05],
        [-2.242347e-12, -6.325555e-05, -3.890789e-05, 2.499914e-02],
    ]
)

PRIOR_FIM = np.linalg.inv(cov)
results_summary(PRIOR_FIM)

======Results Summary======
Five design criteria log10() value:
Pseudo A-optimality: 9.910298890895081
A-optimality: -1.6020703142934367
D-optimality: 27.781982236650116
E-optimality: 1.602071098791314
Modified E-optimality: 8.254407420205549

FIM:
 [[ 7.17766325e+09  1.00811757e+08 -2.18902252e+08 -8.56071665e+04]
 [ 1.00811757e+08  8.99331580e+08  1.51888631e+08  2.51198161e+06]
 [-2.18902252e+08  1.51888631e+08  5.68992000e+07  4.72881315e+05]
 [-8.56071665e+04  2.51198161e+06  4.72881315e+05  7.13206905e+03]]

eigenvalues:
 [7.18585615e+09 9.25285788e+08 2.27591861e+07 4.00010230e+01]

Eigenvector matrix:
          eigvec_1  eigvec_2  eigvec_3  eigvec_4
Ua         0.9994    0.0098    0.0326   -0.0000
Ub         0.0153   -0.9846   -0.1743   -0.0025
inv_CpH   -0.0304   -0.1747    0.9842   -0.0016
inv_CpS   -0.0000   -0.0028    0.0011    1.0000

from pyomo.contrib.doe.utils import rescale_FIM

theta_ref = np.array(
    [
        theta_values["Ua"],
        theta_values["Ub"],
        theta_values["inv_CpH"],
        theta_values["inv_CpS"],
    ]
)
# Because the parameter values differ substantially in magnitude, we can use parameter
# scaling in DesignOfExperiments to improve numerical conditioning.
SCALE_BY_NOMINAL_PARAM = True
SCALE_CONSTANT_VALUE = 1e-2
if SCALE_BY_NOMINAL_PARAM:
    PRIOR_FIM_SCALED = rescale_FIM(PRIOR_FIM, theta_ref) * SCALE_CONSTANT_VALUE**2
else:
    PRIOR_FIM_SCALED = PRIOR_FIM * SCALE_CONSTANT_VALUE**2

Initialize Two Experiments¶

We now create two experiment objects. The first uses the measured sine-test data, and the second uses a different initial input profile so that the optimizer can optimize both experiments simultaneously.

# Create the first experiment object
skip = 6

# Create the data object considering the new control points every 15 seconds
tc_data_1 = TC_Lab_data(
    name="Sine Wave Test for Heater 1",
    time=df["Time"].values[::skip],
    T1=df["T1"].values[::skip],
    u1=df["Q1"].values[::skip],
    P1=200,
    TS1_data=None,
    T2=df["T2"].values[::skip],
    u2=df["Q2"].values[::skip],
    P2=200,
    TS2_data=None,
    Tamb=df["T1"].values[0],
)

# Create experiment object for design of experiments
doe_experiment_1 = TC_Lab_experiment(
    data=tc_data_1,
    theta_initial=theta_values,
    number_of_states=number_tclab_states,
)

# Create the second experiment object
from dataclasses import replace

# Set a random seed for reproducibility
SEED: int = 11  # Choose any integer seed you like
PDF = "uniform"  # Choose between "normal" or "uniform" distribution for the new design of u1


def get_initial_u1_design(PDF="uniform", seed=11, sample_data=tc_data_1):
    rng = np.random.default_rng(seed=seed)
    if PDF == "normal":
        u1_design = rng.normal(loc=50.0, scale=10.0, size=len(sample_data.time))
        # Normal distribution is unbounded, so we need to clip the values to be within the
        # bounds of the control input, which is [0, 100] in this case.
        u1_design = np.clip(u1_design, 0.0, 100.0)
    elif PDF == "uniform":
        u1_design = rng.uniform(low=0.0, high=100.0, size=len(sample_data.time))

    # To break permutation symmetry, we can enforce that the first value of u1_design
    # is greater than the first value of the original u1 data.
    if u1_design[0] <= sample_data.u1[0] and sample_data.u1[0] < 90.0:
        u1_design[0] = sample_data.u1[0] + 10.0
    else:
        u1_design[0] = 100.0

    # Create a new data object with only u1 replaced
    tc_data_new = replace(sample_data, u1=u1_design)
    return tc_data_new


tc_data_2 = get_initial_u1_design(PDF=PDF, seed=SEED, sample_data=tc_data_1)
# Build the experiment with the new design variable
doe_experiment_2 = TC_Lab_experiment(
    data=tc_data_2,
    theta_initial=theta_values,
    number_of_states=number_tclab_states,
)

Optimize Two Experiments (D-Optimality)¶

We now create a DesignOfExperiments object for the two experiments and choose a D-optimality objective. We also pass prior_FIM, which represents information already available from previous parameter estimation.

# Add a solver object to pass to DesignOfExperiments
solver = SolverFactory("ipopt")
solver.options["max_iter"] = 3000
solver.options["tol"] = 1e-5
solver.options["linear_solver"] = "ma57"
solver.options["nlp_scaling_method"] = "gradient-based"

# Create the `design of experiments` object using our experiment instance from above
TC_Lab_DoE_D = DesignOfExperiments(
    experiment=[doe_experiment_1, doe_experiment_2],
    # We are optimizing two experiments simultaneously!
    step=1e-2,
    scale_constant_value=SCALE_CONSTANT_VALUE,
    scale_nominal_param_value=SCALE_BY_NOMINAL_PARAM,
    objective_option="determinant",
    # Now we specify a type of objective, D-opt = "determinant"
    prior_FIM=PRIOR_FIM_SCALED,
    tee=True,
    solver=solver,
)


# Optimize the experimental design
TC_Lab_DoE_D.optimize_experiments()

Fetching long content....

WARNING:pyomo.contrib.doe.doe:No symmetry breaking variable specified. Automatically using the first experiment input 'U1[0.0]' for ordering constraints. To specify a different variable, add: m.sym_break_cons = pyo.Suffix(direction=pyo.Suffix.LOCAL); m.sym_break_cons[m.your_variable] = None

Ipopt 3.13.2: max_iter=3000
tol=1e-05
linear_solver=ma57
nlp_scaling_method=gradient-based


******************************************************************************
This program contains Ipopt, a library for large-scale nonlinear optimization.
 Ipopt is released as open source code under the Eclipse Public License (EPL).
         For more information visit http://projects.coin-or.org/Ipopt

This version of Ipopt was compiled from source code available at
    https://github.com/IDAES/Ipopt as part of the Institute for the Design of
    Advanced Energy Systems Process Systems Engineering Framework (IDAES PSE
    Framework) Copyright (c) 2018-2019. See https://github.com/IDAES/idaes-pse.

This version of Ipopt was compiled using HSL, a collection of Fortran codes
    for large-scale scientific computation.  All technical papers, sales and
    publicity material resulting from use of the HSL codes within IPOPT must
    contain the following acknowledgement:
        HSL, a collection of Fortran codes for large-scale scientific
        computation. See http://www.hsl.rl.ac.uk.
******************************************************************************

This is Ipopt version 3.13.2, running with linear solver ma57.

Number of nonzeros in equality constraint Jacobian...:    39418
Number of nonzeros in inequality constraint Jacobian.:        0
Number of nonzeros in Lagrangian Hessian.............:     3020

Total number of variables............................:    12984
                     variables with only lower bounds:        0
                variables with lower and upper bounds:     6914
                     variables with only upper bounds:        0
Total number of equality constraints.................:    10884
Total number of inequality constraints...............:        1
        inequality constraints with only lower bounds:        0
   inequality constraints with lower and upper bounds:        0
        inequality constraints with only upper bounds:        1

iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0  0.0000000e+00 6.24e+08 0.00e+00  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  0.0000000e+00 2.13e+02 1.02e-02  -1.0 1.25e+09    -  9.90e-01 1.00e+00h  1
   2  0.0000000e+00 2.24e+03 1.00e-06  -1.0 9.60e+02    -  1.00e+00 1.00e+00f  1
   3  0.0000000e+00 5.88e+04 5.77e-04  -1.0 2.60e+04    -  1.00e+00 8.03e-01f  1
   4  0.0000000e+00 8.00e+02 1.00e-06  -1.0 6.65e+04    -  1.00e+00 1.00e+00h  1
   5  0.0000000e+00 4.81e+01 1.00e-06  -1.0 1.61e+03    -  1.00e+00 1.00e+00h  1
   6  0.0000000e+00 6.39e-01 1.00e-06  -1.0 3.19e+02    -  1.00e+00 1.00e+00h  1
   7  0.0000000e+00 3.62e-07 2.00e-07  -1.7 4.21e-01    -  1.00e+00 1.00e+00h  1
   8  0.0000000e+00 5.96e-08 1.50e-09  -3.8 7.59e-03    -  1.00e+00 1.00e+00h  1
   9  0.0000000e+00 8.15e-10 1.84e-11  -5.7 1.23e-02    -  1.00e+00 1.00e+00h  1

Number of Iterations....: 9

                                   (scaled)                 (unscaled)
Objective...............:   0.0000000000000000e+00    0.0000000000000000e+00
Dual infeasibility......:   1.8449144625272179e-11    1.8449144625272179e-11
Constraint violation....:   8.1490725278854370e-10    8.1490725278854370e-10
Complementarity.........:   1.8450305182103931e-06    1.8450305182103931e-06
Overall NLP error.......:   1.8450305182103931e-06    1.8450305182103931e-06


Number of objective function evaluations             = 10
Number of objective gradient evaluations             = 10
Number of equality constraint evaluations            = 10
Number of inequality constraint evaluations          = 10
Number of equality constraint Jacobian evaluations   = 10
Number of inequality constraint Jacobian evaluations = 10
Number of Lagrangian Hessian evaluations             = 9
Total CPU secs in IPOPT (w/o function evaluations)   =      0.863
Total CPU secs in NLP function evaluations           =      0.021

EXIT: Optimal Solution Found.
Ipopt 3.13.2: max_iter=3000
tol=1e-05
linear_solver=ma57
nlp_scaling_method=gradient-based


******************************************************************************
This program contains Ipopt, a library for large-scale nonlinear optimization.
 Ipopt is released as open source code under the Eclipse Public License (EPL).
         For more information visit http://projects.coin-or.org/Ipopt

This version of Ipopt was compiled from source code available at
    https://github.com/IDAES/Ipopt as part of the Institute for the Design of
    Advanced Energy Systems Process Systems Engineering Framework (IDAES PSE
    Framework) Copyright (c) 2018-2019. See https://github.com/IDAES/idaes-pse.

This version of Ipopt was compiled using HSL, a collection of Fortran codes
    for large-scale scientific computation.  All technical papers, sales and
    publicity material resulting from use of the HSL codes within IPOPT must
    contain the following acknowledgement:
        HSL, a collection of Fortran codes for large-scale scientific
        computation. See http://www.hsl.rl.ac.uk.
******************************************************************************

This is Ipopt version 3.13.2, running with linear solver ma57.

Number of nonzeros in equality constraint Jacobian...:    39774
Number of nonzeros in inequality constraint Jacobian.:        2
Number of nonzeros in Lagrangian Hessian.............:     3040

Total number of variables............................:    13296
                     variables with only lower bounds:        4
                variables with lower and upper bounds:     7216
                     variables with only upper bounds:        0
Total number of equality constraints.................:    10894
Total number of inequality constraints...............:        1
        inequality constraints with only lower bounds:        0
   inequality constraints with lower and upper bounds:        0
        inequality constraints with only upper bounds:        1

iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0 -2.2106523e+01 4.72e-03 6.67e-01  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1 -2.2107585e+01 1.12e-02 4.82e+00  -1.0 4.87e-02   2.0 9.90e-01 1.00e+00f  1
   2 -2.2110773e+01 2.05e-04 5.10e-01  -1.0 1.40e-02   1.5 9.90e-01 1.00e+00f  1
   3 -2.2120374e+01 4.72e-03 8.05e-01  -1.0 7.24e-02   1.0 1.00e+00 1.00e+00f  1
   4 -2.2147363e+01 1.38e-03 1.44e-01  -1.7 3.88e-02   0.6 1.00e+00 1.00e+00f  1
   5 -2.2235399e+01 2.87e-02 1.73e-01  -2.5 1.40e-01   0.1 1.00e+00 1.00e+00f  1
   6 -2.2435675e+01 1.49e-01 2.55e-01  -2.5 4.22e-01  -0.4 1.00e+00 1.00e+00h  1
   7 -2.2639739e+01 8.89e-02 1.18e-01  -2.5 7.25e-01  -0.9 1.00e+00 1.00e+00h  1
   8 -2.2871315e+01 1.96e-01 1.37e-01  -2.5 1.33e+00  -1.3 1.00e+00 1.00e+00h  1
   9 -2.3107216e+01 3.36e-01 8.15e-02  -2.5 2.35e+00  -1.8 1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  10 -2.3157043e+01 2.07e-02 4.22e-02  -2.5 1.04e+00  -1.4 1.00e+00 1.00e+00h  1
  11 -2.3283698e+01 1.64e-01 2.93e-02  -2.5 2.12e+00  -1.9 1.00e+00 1.00e+00h  1
  12 -2.3456076e+01 5.62e-01 3.54e-02  -2.5 4.08e+00  -2.3 1.00e+00 1.00e+00h  1
  13 -2.3657596e+01 2.69e+00 3.56e-02  -2.5 7.37e+00  -2.8 1.00e+00 1.00e+00h  1
  14 -2.3876650e+01 1.19e+01 3.10e-02  -2.5 1.30e+01  -3.3 1.00e+00 1.00e+00h  1
  15 -2.4105041e+01 4.83e+01 2.51e-02  -2.5 2.28e+01  -3.8 1.00e+00 1.00e+00h  1
  16 -2.4338393e+01 1.78e+02 1.97e-02  -2.5 3.96e+01  -4.3 1.00e+00 1.00e+00h  1
  17 -2.4574292e+01 5.75e+02 1.53e-02  -2.5 6.88e+01  -4.7 1.00e+00 1.00e+00h  1
  18 -2.4811477e+01 1.54e+03 1.17e-02  -2.5 1.19e+02  -5.2 1.00e+00 1.00e+00h  1
  19 -2.5049298e+01 2.85e+03 8.91e-03  -2.5 2.07e+02  -5.7 1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  20 -2.5287400e+01 2.27e+03 6.78e-03  -2.5 3.58e+02  -6.2 1.00e+00 1.00e+00h  1
  21 -2.5525585e+01 3.68e+02 5.15e-03  -2.5 6.20e+02  -6.6 1.00e+00 1.00e+00h  1
  22 -2.5763762e+01 1.55e+02 3.91e-03  -2.5 1.07e+03  -7.1 1.00e+00 1.00e+00h  1
  23 -2.6001976e+01 2.67e+02 2.97e-03  -2.5 1.85e+03  -7.6 1.00e+00 1.00e+00h  1
  24 -2.6240626e+01 4.59e+02 2.24e-03  -2.5 3.19e+03  -8.1 1.00e+00 1.00e+00h  1
  25 -2.6481255e+01 7.84e+02 1.69e-03  -2.5 5.48e+03  -8.5 1.00e+00 1.00e+00h  1
  26 -2.6728504e+01 2.13e+03 1.27e-03  -2.5 9.36e+03  -9.0 1.00e+00 1.00e+00h  1
  27 -2.6704984e+01 1.62e+03 4.46e-05  -2.5 7.58e+01  -6.8 1.00e+00 1.00e+00H  1
  28 -2.6709570e+01 2.01e+02 1.22e-05  -2.5 2.27e+02  -7.3 1.00e+00 1.00e+00H  1
  29 -2.6722767e+01 1.61e+03 1.18e-05  -2.5 6.60e+02  -7.7 1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  30 -2.6757833e+01 3.06e+02 2.53e-05  -2.5 1.82e+03  -8.2 1.00e+00 1.00e+00h  1
  31 -2.6836864e+01 5.68e+02 1.21e-04  -2.5 4.47e+03  -8.7 1.00e+00 1.00e+00h  1
  32 -2.6976741e+01 6.99e+02 3.17e-04  -2.5 9.34e+03  -9.2 1.00e+00 1.00e+00h  1
  33 -2.7180558e+01 1.71e+03 4.52e-04  -2.5 1.72e+04  -9.7 1.00e+00 1.00e+00h  1
  34 -2.7180557e+01 1.71e+03 4.52e-04  -2.5 2.83e+03  -8.3 1.70e-01 3.48e-04h 10
  35 -2.7166819e+01 3.83e+02 6.13e-06  -2.5 5.68e+00  -6.1 1.00e+00 1.00e+00h  1
  36 -2.7166886e+01 1.70e+02 4.57e-06  -2.5 1.70e+01  -6.6 1.00e+00 1.00e+00h  1
  37 -2.7167377e+01 2.71e+02 4.57e-06  -2.5 5.11e+01  -7.0 1.00e+00 1.00e+00h  1
  38 -2.7168848e+01 7.31e+01 4.55e-06  -2.5 1.53e+02  -7.5 1.00e+00 1.00e+00h  1
  39 -2.7173211e+01 1.27e+01 4.51e-06  -2.5 4.54e+02  -8.0 1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  40 -2.7185880e+01 5.25e+01 4.39e-06  -2.5 1.33e+03  -8.5 1.00e+00 1.00e+00h  1
  41 -2.7220586e+01 3.46e+02 1.07e-05  -2.5 3.70e+03  -9.0 1.00e+00 1.00e+00h  1
  42 -2.7303631e+01 1.06e+03 5.54e-05  -2.5 9.24e+03  -9.4 1.00e+00 1.00e+00h  1
  43 -2.7461529e+01 1.30e+03 1.59e-04  -2.5 1.95e+04  -9.9 1.00e+00 1.00e+00h  1
  44 -2.7523300e+01 3.07e+03 1.34e-04  -2.5 3.49e+04 -10.4 1.00e+00 2.50e-01h  3
  45 -2.7523423e+01 3.10e+03 1.33e-04  -2.5 2.58e+03  -9.1 7.59e-01 1.17e-02h  7
  46 -2.7523354e+01 3.15e+03 1.32e-04  -2.5 1.20e+03  -7.7 5.60e-01 1.29e-02h  6
  47 -2.7522599e+01 3.53e+03 1.15e-04  -2.5 2.22e+01  -5.5 1.00e+00 1.25e-01h  4
  48 -2.7517307e+01 7.39e+03 1.11e-03  -2.5 1.24e+01  -6.0 1.00e+00 1.00e+00h  1
  49 -2.7517339e+01 3.16e+03 1.14e-04  -2.5 2.42e+01  -6.5 1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  50 -2.7517356e+01 2.97e+03 2.65e-04  -2.5 1.18e+03  -6.9 1.00e+00 1.25e-01h  4
  51 -2.7517757e+01 5.47e+02 3.02e-03  -2.5 2.70e+03  -7.4 1.00e+00 1.00e+00h  1
  52 -2.7518912e+01 5.60e+00 2.02e-04  -2.5 5.05e+02  -7.9 1.00e+00 1.00e+00h  1
  53 -2.7522341e+01 1.42e+01 1.39e-05  -2.5 5.28e+02  -8.4 1.00e+00 1.00e+00h  1
  54 -2.7532381e+01 7.22e+01 3.72e-06  -2.5 1.55e+03  -8.8 1.00e+00 1.00e+00h  1
  55 -2.7560505e+01 3.87e+02 8.91e-06  -2.5 4.40e+03  -9.3 1.00e+00 1.00e+00h  1
  56 -2.7630924e+01 3.04e+02 2.24e-05  -2.5 1.14e+04  -9.8 1.00e+00 1.00e+00h  1
  57 -2.7772300e+01 1.29e+03 7.82e-05  -2.5 2.50e+04 -10.3 1.00e+00 1.00e+00h  1
  58 -2.7772319e+01 1.25e+03 7.58e-05  -2.5 1.24e+03  -8.9 1.00e+00 3.12e-02h  6
  59 -2.7766989e+01 2.59e+02 1.39e-05  -2.5 1.06e+01  -6.7 1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  60 -2.7767089e+01 7.89e+01 4.58e-06  -2.5 2.19e+01  -7.2 1.00e+00 1.00e+00H  1
  61 -2.7767131e+01 7.67e+00 1.43e-06  -2.5 8.24e+00  -6.8 1.00e+00 1.00e+00H  1
  62 -2.7767146e+01 2.15e-01 1.44e-06  -2.5 3.09e+00  -6.3 1.00e+00 1.00e+00H  1
  63 -2.7767193e+01 1.84e+02 1.85e-06  -2.5 9.33e+00  -6.8 1.00e+00 1.00e+00H  1
  64 -2.7767210e+01 5.20e+00 1.45e-06  -2.5 3.52e+00  -6.4 1.00e+00 1.00e+00H  1
  65 -2.7767211e+01 2.72e+01 5.16e-05  -2.5 2.22e+01  -6.9 1.00e+00 1.56e-02h  7
  66 -2.7767231e+01 9.54e+01 2.58e-06  -2.5 3.99e+00  -6.4 1.00e+00 1.00e+00H  1
  67 -2.7767291e+01 6.72e+02 2.26e-05  -2.5 1.24e+01  -6.9 1.00e+00 1.00e+00H  1
  68 -2.7767472e+01 1.01e+03 2.84e-06  -2.5 3.69e+01  -7.4 1.00e+00 1.00e+00h  1
  69 -2.7768014e+01 3.41e+01 1.50e-06  -2.5 1.10e+02  -7.9 1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  70 -2.7769632e+01 3.17e-01 1.49e-06  -2.5 3.30e+02  -8.3 1.00e+00 1.00e+00h  1
  71 -2.7774428e+01 1.68e+00 1.48e-06  -2.5 9.80e+02  -8.8 1.00e+00 1.00e+00h  1
  72 -2.7788329e+01 1.42e+01 1.44e-06  -2.5 2.86e+03  -9.3 1.00e+00 1.00e+00h  1
  73 -2.7826158e+01 1.07e+02 4.93e-06  -2.5 7.94e+03  -9.8 1.00e+00 1.00e+00h  1
  74 -2.7914819e+01 1.16e+03 2.55e-05  -2.5 1.96e+04 -10.3 1.00e+00 1.00e+00h  1
  75 -2.7942267e+01 3.60e+03 7.05e-06  -2.5 7.08e+03  -9.8 1.00e+00 1.00e+00h  1
  76 -2.7946991e+01 4.45e+03 9.96e-05  -2.5 2.18e+04 -10.3 8.49e-01 5.48e-02h  5
  77 -2.7946787e+01 2.48e+02 6.66e-06  -2.5 2.54e+00  -6.3 1.00e+00 1.00e+00h  1
  78 -2.7946811e+01 2.49e+00 1.62e-06  -2.5 5.79e+00  -6.7 1.00e+00 1.00e+00h  1
  79 -2.7946883e+01 6.71e-01 1.05e-06  -2.5 1.74e+01  -7.2 1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  80 -2.7947098e+01 3.30e-01 1.05e-06  -2.5 5.21e+01  -7.7 1.00e+00 1.00e+00h  1
  81 -2.7947743e+01 1.78e+00 1.05e-06  -2.5 1.56e+02  -8.2 1.00e+00 1.00e+00h  1
  82 -2.7949671e+01 1.54e+01 1.04e-06  -2.5 4.66e+02  -8.7 1.00e+00 1.00e+00h  1
  83 -2.7955392e+01 1.35e+02 1.03e-06  -2.5 1.38e+03  -9.1 1.00e+00 1.00e+00h  1
  84 -2.7972002e+01 1.06e+03 2.61e-06  -2.5 4.01e+03  -9.6 1.00e+00 1.00e+00h  1
  85 -2.8017020e+01 4.13e+03 1.07e-05  -2.5 1.09e+04 -10.1 1.00e+00 1.00e+00h  1
  86 -2.8116699e+01 6.49e+03 2.33e-05  -2.5 2.60e+04 -10.6 1.00e+00 1.00e+00h  1
  87 -2.8137923e+01 9.62e+03 8.88e-05  -2.5 6.94e+04 -11.0 1.00e+00 1.25e-01h  4
  88 -2.8137900e+01 9.77e+03 1.89e-04  -2.5 5.51e+03  -8.8 1.00e+00 1.90e-02h  6
  89 -2.8138223e+01 9.21e+03 6.52e-05  -2.5 5.30e+03  -9.3 1.00e+00 1.25e-01h  4
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  90 -2.8149857e+01 2.05e+04 2.62e-04  -2.5 3.91e+03  -9.8 1.00e+00 1.00e+00h  1
  91 -2.8186023e+01 6.07e+03 7.74e-05  -2.5 1.07e+04 -10.2 1.00e+00 1.00e+00h  1
  92 -2.8266784e+01 2.52e+03 1.39e-05  -2.5 2.58e+04 -10.7 1.00e+00 1.00e+00h  1
  93 -2.8397601e+01 9.07e+02 3.23e-05  -2.5 4.85e+04 -11.2 1.00e+00 1.00e+00H  1
  94 -2.8438411e+01 2.20e+02 2.28e-06  -2.5 1.67e+04 -10.8 1.00e+00 1.00e+00h  1
  95 -2.8513416e+01 9.71e+02 1.00e-05  -2.5 3.37e+04 -11.2 1.00e+00 1.00e+00H  1
  96 -2.8536878e+01 4.38e+02 8.38e-07  -2.5 1.12e+04 -10.8 1.00e+00 1.00e+00h  1
  97 -2.8582235e+01 3.64e+03 4.31e-06  -2.5 3.95e+04 -11.3 1.00e+00 1.00e+00H  1
  98 -2.8595309e+01 2.22e+03 7.12e-07  -2.5 1.62e+04 -10.9 1.00e+00 1.00e+00h  1
  99 -2.8597827e+01 3.08e+03 4.34e-05  -2.5 8.30e+04 -11.3 1.00e+00 1.25e-01h  4
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
 100 -2.8607984e+01 3.07e+02 3.43e-06  -2.5 2.40e+04 -10.9 1.00e+00 1.00e+00H  1
 101 -2.8613043e+01 3.51e+03 3.31e-05  -2.5 9.03e+04 -11.4 1.00e+00 2.50e-01h  3
 102 -2.8612820e+01 3.58e+03 1.45e-04  -2.5 1.00e+06 -11.9 1.75e-01 2.66e-03h  7
 103 -2.8615847e+01 3.18e+03 2.77e-06  -2.5 1.08e+04 -10.5 1.00e+00 1.00e+00h  1
 104 -2.8616186e+01 3.42e+03 6.43e-05  -2.5 4.61e+04 -11.0 1.00e+00 6.25e-02h  5
 105 -2.8616682e+01 2.27e+03 1.46e-06  -2.5 1.53e+03  -9.7 1.00e+00 1.00e+00h  1
 106 -2.8616726e+01 2.33e+03 4.17e-05  -2.5 6.37e+03 -10.2 1.00e+00 6.25e-02h  5
 107 -2.8616726e+01 2.33e+03 1.46e-04  -2.5 1.47e+03  -7.9 7.61e-01 3.33e-03h  9
 108 -2.8616726e+01 2.33e+03 5.73e-05  -2.5 6.52e+02  -7.5 6.05e-01 5.77e-03h  7
 109 -2.8616727e+01 2.33e+03 1.32e-04  -2.5 6.62e+02  -7.1 1.00e+00 6.06e-03h  8
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
 110 -2.8616729e+01 2.36e+03 7.26e-05  -2.5 5.74e+02  -7.6 7.61e-01 1.29e-02h  6
 111 -2.8616751e+01 2.35e+03 4.85e-05  -2.5 1.24e+03  -8.0 1.00e+00 3.12e-02h  6
 112 -2.8617373e+01 2.18e+03 1.21e-04  -2.5 2.06e+03  -8.5 1.00e+00 5.00e-01h  2
 113 -2.8620123e+01 2.19e+03 1.20e-04  -2.5 9.88e+05    -  1.19e-01 6.28e-03h  5
 114 -2.8621135e+01 1.76e+02 2.88e-05  -2.5 1.37e+03  -9.0 1.00e+00 1.00e+00h  1
 115 -2.8623208e+01 1.13e+03 1.32e-04  -2.5 9.48e+05    -  1.02e-01 5.08e-03h  5
 116 -2.8623728e+01 1.04e+03 5.78e-06  -2.5 7.33e+02  -9.5 1.00e+00 1.00e+00h  1
 117 -2.8623729e+01 1.04e+03 4.87e-05  -2.5 3.47e+02  -7.2 1.00e+00 1.56e-02h  7
 118 -2.8624488e+01 1.05e+03 9.55e-05  -2.5 1.34e+06    -  2.90e-01 2.82e-03h  7
 119 -2.8624488e+01 1.05e+03 8.16e-05  -2.5 1.02e+03  -7.7 1.00e+00 3.03e-03h  9
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
 120 -2.8624723e+01 7.67e+02 7.75e-05  -2.5 7.85e+02  -8.2 1.00e+00 5.00e-01h  2
 121 -2.8786450e+01 4.57e+02 5.76e-05  -2.5 9.43e+04    -  1.00e+00 1.00e+00H  1
 122 -2.8807639e+01 2.04e+01 6.07e-07  -2.5 1.38e+04    -  1.00e+00 1.00e+00H  1
 123 -2.8807311e+01 2.39e+00 2.88e-08  -2.5 3.77e+02    -  1.00e+00 1.00e+00h  1
 124 -2.8807310e+01 2.39e-05 2.83e-08  -2.5 3.66e-01    -  1.00e+00 1.00e+00h  1
 125 -2.9414832e+01 1.99e+03 5.89e-04  -3.8 4.22e+05    -  7.32e-01 1.00e+00F  1
 126 -2.9973868e+01 2.16e+03 3.49e-05  -3.8 7.60e+05    -  9.97e-01 1.00e+00H  1
 127 -3.0001794e+01 2.56e+03 3.32e-05  -3.8 1.07e+06    -  1.00e+00 6.02e-02h  4
 128 -3.0365243e+01 7.73e+02 8.84e-05  -3.8 9.41e+05    -  1.00e+00 1.00e+00H  1
 129 -3.0373521e+01 1.18e+02 3.72e-06  -3.8 2.90e+04    -  1.00e+00 1.00e+00H  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
 130 -3.0369609e+01 8.37e+01 1.49e-08  -3.8 1.90e+04    -  1.00e+00 1.00e+00h  1
 131 -3.0369685e+01 3.45e-02 1.51e-09  -3.8 4.33e+02    -  1.00e+00 1.00e+00h  1
 132 -3.0369685e+01 1.19e-07 1.50e-09  -3.8 3.21e-02    -  1.00e+00 1.00e+00h  1
 133 -3.0581248e+01 3.30e+02 2.43e-05  -5.7 7.87e+05    -  8.33e-01 1.00e+00F  1
 134 -3.0587018e+01 2.04e+03 1.97e-05  -5.7 6.68e+05    -  7.69e-01 1.87e-01h  3
Reallocating memory for MA57: lfact (412977)
 135 -3.0587707e+01 2.27e+03 1.93e-05  -5.7 2.04e+06    -  7.90e-01 2.23e-02h  6
 136 -3.0587913e+01 2.30e+03 2.28e-05  -5.7 3.15e+06    -  1.00e+00 5.76e-03h  8
 137 -3.0588130e+01 2.32e+03 2.74e-05  -5.7 2.91e+06    -  1.00e+00 6.25e-03h  8
 138 -3.0588364e+01 2.36e+03 2.73e-05  -5.7 2.94e+06    -  1.00e+00 6.79e-03h  8
 139 -3.0588483e+01 2.37e+03 2.85e-05  -5.7 2.82e+06    -  1.00e+00 3.52e-03h  9
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
 140 -3.0588604e+01 2.37e+03 2.86e-05  -5.7 2.85e+06    -  1.00e+00 3.56e-03h  9
 141 -3.0588668e+01 2.37e+03 2.88e-05  -5.7 2.80e+06    -  1.00e+00 1.90e-03h 10
 142 -3.0588732e+01 2.37e+03 2.97e-05  -5.7 2.82e+06    -  1.00e+00 1.92e-03h 10
 143 -3.0588850e+01 2.37e+03 3.01e-05  -5.7 2.79e+06    -  1.00e+00 3.53e-03h  9
 144 -3.0616219e+01 7.27e+05 5.27e-06  -5.7 2.79e+06    -  1.00e+00 8.26e-01w  1
 145 -3.0621843e+01 1.23e+05 6.38e-08  -5.7 2.06e+06    -  1.00e+00 1.00e+00w  1
 146 -3.0621756e+01 4.88e+03 3.01e-09  -5.7 1.34e+05    -  1.00e+00 1.00e+00h  1
 147 -3.0621759e+01 2.76e+00 1.85e-11  -5.7 6.85e+03    -  1.00e+00 1.00e+00h  1
 148 -3.0621759e+01 1.20e-06 1.84e-11  -5.7 1.73e+00    -  1.00e+00 1.00e+00h  1

Number of Iterations....: 148

                                   (scaled)                 (unscaled)
Objective...............:  -3.0621759113959421e+01   -3.0621759113959421e+01
Dual infeasibility......:   1.8449144690832783e-11    1.8449144690832783e-11
Constraint violation....:   3.7917877910781740e-07    1.2014061212539673e-06
Complementarity.........:   1.8449152194934589e-06    1.8449152194934589e-06
Overall NLP error.......:   1.8449152194934589e-06    1.8449152194934589e-06


Number of objective function evaluations             = 207
Number of objective gradient evaluations             = 149
Number of equality constraint evaluations            = 452
Number of inequality constraint evaluations          = 452
Number of equality constraint Jacobian evaluations   = 149
Number of inequality constraint Jacobian evaluations = 149
Number of Lagrangian Hessian evaluations             = 148
Total CPU secs in IPOPT (w/o function evaluations)   =     23.932
Total CPU secs in NLP function evaluations           =      0.516

EXIT: Optimal Solution Found.

multiexp_results = extract_plot_results(None, TC_Lab_DoE_D)

Interesting! The cold experiment (exp 1) operates near room temperature to isolate internal heat transfer ( $U_b$ ) by eliminating ambient losses. It yields information along the coordinate axes of internal parameters ( $U_b, C_{p,S}, C_{p,H}$ ). The hot experiment (exp 2) drives steep thermal gradients to capture the exact behavior of ambient heat dissipation ( $U_a$ ) and heat capacities.

FIM_D = np.asarray(TC_Lab_DoE_D.results["solution"]["param_scenarios"][0]["total_fim"])
if SCALE_BY_NOMINAL_PARAM:
    D = np.diag(theta_ref)
    FIM_unscaled_D = D @ FIM_D @ D / SCALE_CONSTANT_VALUE**2
else:
    FIM_unscaled_D = FIM_D / SCALE_CONSTANT_VALUE**2
results_summary(np.asarray(FIM_unscaled_D))

======Results Summary======
Five design criteria log10() value:
Pseudo A-optimality: 11.914399955393007
A-optimality: -8.672600624735777
D-optimality: 39.78410599788649
E-optimality: 8.942827955880793
Modified E-optimality: 2.9666353990745264

FIM:
 [[ 7.21899524e+09  8.78022397e+07 -5.78467897e+06 -1.04510545e+07]
 [ 8.78022397e+07  9.03427868e+08  8.47502225e+07  2.60112894e+06]
 [-5.78467897e+06  8.47502225e+07  1.15821212e+09  2.54091911e+07]
 [-1.04510545e+07  2.60112894e+06  2.54091911e+07  8.11826745e+11]]

eigenvalues:
 [8.11826745e+11 7.22021904e+09 8.76653469e+08 1.18376178e+09]

Eigenvector matrix:
          eigvec_1  eigvec_2  eigvec_3  eigvec_4
Ua            0.0    0.9999    0.0135   -0.0033
Ub           -0.0    0.0139   -0.9574    0.2884
inv_CpH      -0.0   -0.0008    0.2885    0.9575
inv_CpS      -1.0    0.0000   -0.0000   -0.0000

Compare the Reduction in Predicted Uncertainty¶

# Helper function to print parameter values with their standard deviations
def param_std(
    std: np.ndarray,
    parameters: dict = theta_values,
):
    for i, key in enumerate(parameters.keys()):
        print(f"{key}: {parameters[key]:.3e} ± {std[i]:.3e}")

# Now compare the uncertainty for the prior and the optimized design to see how much
# improvement we have achieved in terms of parameter uncertainty reduction.
cov_D = np.linalg.inv(FIM_unscaled_D)
std_D = np.sqrt(np.diag(cov_D))
std_prior = np.sqrt(np.diag(cov))
print("Parameter estimates with standard deviations for the prior:")
param_std(std_prior)

print(
    "\nParameter estimates with predicted standard deviations for the optimized design:"
)
param_std(std_D)

Parameter estimates with standard deviations for the prior:
Ua: 4.170e-02 ± 1.363e-05
Ub: 1.201e-02 ± 4.030e-04
inv_CpH: 1.675e-01 ± 3.212e-04
inv_CpS: 4.545e+00 ± 1.581e-01

Parameter estimates with predicted standard deviations for the optimized design:
Ua: 4.170e-02 ± 1.178e-05
Ub: 1.201e-02 ± 3.340e-05
inv_CpH: 1.675e-01 ± 2.949e-05
inv_CpS: 4.545e+00 ± 1.110e-06

Activity: Repeat the Design with A-Optimality¶

In this activity, we repeat the same workflow as above but change only the design objective. The repeated setup is shown again here so the section can be run independently.

# Create the design of experiments object using our experiment instance from above
TC_Lab_DoE_A = DesignOfExperiments(
    experiment=[
        doe_experiment_1,
        doe_experiment_2,
    ],  # We are optimizing two experiments simultaneously!
    step=1e-2,
    scale_constant_value=SCALE_CONSTANT_VALUE,
    scale_nominal_param_value=SCALE_BY_NOMINAL_PARAM,
    objective_option="trace",  # Now we specify the A-optimality objective, A-opt = "trace"
    prior_FIM=PRIOR_FIM_SCALED,  # We use the prior information from the existing experiment!
    tee=True,
    solver=solver,
)

TC_Lab_DoE_A.optimize_experiments()

Fetching long content....

WARNING:pyomo.contrib.doe.doe:No symmetry breaking variable specified. Automatically using the first experiment input 'U1[0.0]' for ordering constraints. To specify a different variable, add: m.sym_break_cons = pyo.Suffix(direction=pyo.Suffix.LOCAL); m.sym_break_cons[m.your_variable] = None

Ipopt 3.13.2: max_iter=3000
tol=1e-05
linear_solver=ma57
nlp_scaling_method=gradient-based


******************************************************************************
This program contains Ipopt, a library for large-scale nonlinear optimization.
 Ipopt is released as open source code under the Eclipse Public License (EPL).
         For more information visit http://projects.coin-or.org/Ipopt

This version of Ipopt was compiled from source code available at
    https://github.com/IDAES/Ipopt as part of the Institute for the Design of
    Advanced Energy Systems Process Systems Engineering Framework (IDAES PSE
    Framework) Copyright (c) 2018-2019. See https://github.com/IDAES/idaes-pse.

This version of Ipopt was compiled using HSL, a collection of Fortran codes
    for large-scale scientific computation.  All technical papers, sales and
    publicity material resulting from use of the HSL codes within IPOPT must
    contain the following acknowledgement:
        HSL, a collection of Fortran codes for large-scale scientific
        computation. See http://www.hsl.rl.ac.uk.
******************************************************************************

This is Ipopt version 3.13.2, running with linear solver ma57.

Number of nonzeros in equality constraint Jacobian...:    39418
Number of nonzeros in inequality constraint Jacobian.:        0
Number of nonzeros in Lagrangian Hessian.............:     3020

Total number of variables............................:    12984
                     variables with only lower bounds:        0
                variables with lower and upper bounds:     6914
                     variables with only upper bounds:        0
Total number of equality constraints.................:    10884
Total number of inequality constraints...............:        1
        inequality constraints with only lower bounds:        0
   inequality constraints with lower and upper bounds:        0
        inequality constraints with only upper bounds:        1

iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0  0.0000000e+00 6.24e+08 0.00e+00  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  0.0000000e+00 2.13e+02 1.02e-02  -1.0 1.25e+09    -  9.90e-01 1.00e+00h  1
   2  0.0000000e+00 2.24e+03 1.00e-06  -1.0 9.60e+02    -  1.00e+00 1.00e+00f  1
   3  0.0000000e+00 5.88e+04 5.77e-04  -1.0 2.60e+04    -  1.00e+00 8.03e-01f  1
   4  0.0000000e+00 8.00e+02 1.00e-06  -1.0 6.65e+04    -  1.00e+00 1.00e+00h  1
   5  0.0000000e+00 4.81e+01 1.00e-06  -1.0 1.61e+03    -  1.00e+00 1.00e+00h  1
   6  0.0000000e+00 6.39e-01 1.00e-06  -1.0 3.19e+02    -  1.00e+00 1.00e+00h  1
   7  0.0000000e+00 3.62e-07 2.00e-07  -1.7 4.21e-01    -  1.00e+00 1.00e+00h  1
   8  0.0000000e+00 5.96e-08 1.50e-09  -3.8 7.59e-03    -  1.00e+00 1.00e+00h  1
   9  0.0000000e+00 8.15e-10 1.84e-11  -5.7 1.23e-02    -  1.00e+00 1.00e+00h  1

Number of Iterations....: 9

                                   (scaled)                 (unscaled)
Objective...............:   0.0000000000000000e+00    0.0000000000000000e+00
Dual infeasibility......:   1.8449144625272179e-11    1.8449144625272179e-11
Constraint violation....:   8.1490725278854370e-10    8.1490725278854370e-10
Complementarity.........:   1.8450305182103931e-06    1.8450305182103931e-06
Overall NLP error.......:   1.8450305182103931e-06    1.8450305182103931e-06


Number of objective function evaluations             = 10
Number of objective gradient evaluations             = 10
Number of equality constraint evaluations            = 10
Number of inequality constraint evaluations          = 10
Number of equality constraint Jacobian evaluations   = 10
Number of inequality constraint Jacobian evaluations = 10
Number of Lagrangian Hessian evaluations             = 9
Total CPU secs in IPOPT (w/o function evaluations)   =      0.948
Total CPU secs in NLP function evaluations           =      0.023

EXIT: Optimal Solution Found.
Ipopt 3.13.2: max_iter=3000
tol=1e-05
linear_solver=ma57
nlp_scaling_method=gradient-based


******************************************************************************
This program contains Ipopt, a library for large-scale nonlinear optimization.
 Ipopt is released as open source code under the Eclipse Public License (EPL).
         For more information visit http://projects.coin-or.org/Ipopt

This version of Ipopt was compiled from source code available at
    https://github.com/IDAES/Ipopt as part of the Institute for the Design of
    Advanced Energy Systems Process Systems Engineering Framework (IDAES PSE
    Framework) Copyright (c) 2018-2019. See https://github.com/IDAES/idaes-pse.

This version of Ipopt was compiled using HSL, a collection of Fortran codes
    for large-scale scientific computation.  All technical papers, sales and
    publicity material resulting from use of the HSL codes within IPOPT must
    contain the following acknowledgement:
        HSL, a collection of Fortran codes for large-scale scientific
        computation. See http://www.hsl.rl.ac.uk.
******************************************************************************

This is Ipopt version 3.13.2, running with linear solver ma57.

Number of nonzeros in equality constraint Jacobian...:    39859
Number of nonzeros in inequality constraint Jacobian.:        2
Number of nonzeros in Lagrangian Hessian.............:     3080

Total number of variables............................:    13317
                     variables with only lower bounds:        5
                variables with lower and upper bounds:     7216
                     variables with only upper bounds:        0
Total number of equality constraints.................:    10915
Total number of inequality constraints...............:        1
        inequality constraints with only lower bounds:        0
   inequality constraints with lower and upper bounds:        0
        inequality constraints with only upper bounds:        1

iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0  1.6841287e+00 4.72e-03 6.67e-01  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  1.3939522e+00 2.31e-02 2.90e-01  -1.0 2.90e-01   0.0 9.90e-01 1.00e+00f  1
   2  1.2894587e+00 5.28e-02 6.27e-01  -1.0 2.37e-01   0.4 9.90e-01 1.00e+00f  1
   3  1.0062805e+00 3.21e-02 2.52e-01  -1.0 2.83e-01  -0.1 1.00e+00 1.00e+00f  1
   4  5.2248787e-01 9.75e-02 1.48e-01  -1.7 5.00e-01  -0.5 9.39e-01 1.00e+00f  1
   5  3.7208672e-01 9.22e-02 1.01e-01  -2.5 8.09e-01  -1.0 1.00e+00 1.00e+00h  1
   6  2.7016726e-01 9.25e-02 8.59e-02  -2.5 1.02e+00  -1.5 1.00e+00 1.00e+00h  1
   7  1.7878412e-01 1.49e-01 6.37e-02  -2.5 1.51e+00  -2.0 1.00e+00 1.00e+00h  1
   8  1.2404472e-01 7.81e-01 5.25e-02  -2.5 2.20e+00  -2.4 1.00e+00 1.00e+00h  1
   9  8.5578436e-02 3.60e+00 4.34e-02  -2.5 3.17e+00  -2.9 1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  10  5.9194480e-02 1.57e+01 3.61e-02  -2.5 4.56e+00  -3.4 1.00e+00 1.00e+00h  1
  11  4.1065400e-02 6.25e+01 3.01e-02  -2.5 6.54e+00  -3.9 1.00e+00 1.00e+00h  1
  12  2.8565185e-02 2.25e+02 2.53e-02  -2.5 9.37e+00  -4.3 1.00e+00 1.00e+00h  1
  13  1.9935228e-02 7.07e+02 2.14e-02  -2.5 1.34e+01  -4.8 1.00e+00 1.00e+00h  1
  14  1.3974359e-02 1.80e+03 1.83e-02  -2.5 1.90e+01  -5.3 1.00e+00 1.00e+00h  1
  15  9.8569496e-03 2.97e+03 1.57e-02  -2.5 2.67e+01  -5.8 1.00e+00 1.00e+00h  1
  16  7.0145499e-03 1.86e+03 1.37e-02  -2.5 3.71e+01  -6.3 1.00e+00 1.00e+00h  1
  17  5.0671402e-03 2.53e+02 1.17e-02  -2.5 5.04e+01  -6.7 1.00e+00 1.00e+00h  1
  18  3.7469374e-03 1.37e+01 9.86e-03  -2.5 6.59e+01  -7.2 1.00e+00 1.00e+00h  1
  19  2.8756900e-03 5.68e+00 7.70e-03  -2.5 8.04e+01  -7.7 1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  20  2.3371575e-03 4.88e+00 5.02e-03  -2.5 8.58e+01  -8.2 1.00e+00 1.00e+00h  1
  21  2.0521994e-03 2.61e+00 2.18e-03  -2.5 2.36e+02  -8.6 1.00e+00 1.00e+00h  1
  22  1.9459476e-03 2.76e+00 4.05e-04  -2.5 7.07e+02  -9.1 1.00e+00 1.00e+00h  1
  23  1.9257087e-03 2.59e+01 1.61e-05  -2.5 2.12e+03  -9.6 1.00e+00 1.00e+00h  1
  24  1.9237218e-03 2.67e+02 5.42e-07  -2.5 6.37e+03 -10.1 1.00e+00 1.00e+00h  1
  25  1.9228338e-03 8.51e+01 5.43e-07  -2.5 2.40e+03  -9.6 1.00e+00 1.00e+00h  1
  26  1.9165214e-03 2.35e+03 6.44e-06  -2.5 7.35e+03 -10.1 1.00e+00 1.00e+00h  1
  27  1.8979668e-03 3.25e+03 1.43e-02  -2.5 3.27e+03  -9.7 1.00e+00 5.00e-01h  2
  28  1.9400989e-03 2.46e+03 1.00e-01  -2.5 1.19e+03  -9.3 1.00e+00 2.50e-01h  3
  29  1.9499702e-03 2.36e+03 1.02e-01  -2.5 3.23e+02  -7.9 1.00e+00 6.25e-02h  5
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  30  1.9494784e-03 1.91e+03 1.47e-03  -2.5 1.08e+01  -5.7 1.00e+00 2.50e-01h  3
  31  1.9448676e-03 1.94e+03 2.44e-02  -2.5 4.25e+01  -6.2 1.00e+00 1.25e-01h  4
  32  1.9448863e-03 5.73e+03 2.03e-02  -2.5 2.48e+06    -  1.68e-01 1.53e-02h  5
  33  1.9002482e-03 4.32e+03 1.02e-01  -2.5 9.66e+01  -6.7 1.00e+00 2.50e-01h  3
  34  1.9005344e-03 9.26e+03 9.15e-02  -2.5 3.99e+06    -  7.49e-02 6.07e-03h  5
  35  1.4094709e-03 5.15e+03 4.08e-01  -2.5 5.30e+02  -7.1 1.00e+00 5.00e-01h  2
  36  1.0945097e-04 1.98e+03 1.24e-01  -2.5 1.41e+03  -7.6 1.00e+00 1.00e+00h  1
  37  6.3160986e-04 1.57e+03 1.52e+00  -2.5 4.60e+02  -4.5 1.00e+00 1.00e+00h  1
  38  6.0755954e-04 1.20e+03 1.16e+00  -2.5 5.34e+02  -5.0 1.00e+00 2.50e-01h  3
  39  4.7760239e-04 1.34e+03 1.74e-01  -2.5 7.23e+02  -5.4 1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  40  4.8061799e-04 1.34e+03 8.73e-02  -2.5 1.88e+02  -5.9 1.00e+00 5.00e-01h  2
  41  4.8982926e-04 3.76e+02 2.97e-02  -2.5 5.38e+01  -4.6 1.00e+00 1.00e+00h  1
  42  4.8857661e-04 3.74e+02 4.54e-02  -2.5 3.41e+01  -4.2 1.00e+00 2.50e-01h  3
  43  1.7742819e-03 2.68e+03 9.58e+00  -2.5 6.27e+03    -  3.78e-01 1.00e+00H  1
  44  8.8964522e-04 1.80e+03 7.96e+00  -2.5 1.95e+01  -4.6 1.00e+00 2.88e-01h  2
  45  2.5064247e-06 2.02e+02 2.41e+00  -2.5 1.23e+01  -5.1 1.00e+00 7.92e-01h  1
  46  2.2087717e-04 3.65e+01 9.65e-01  -2.5 8.48e+00  -5.6 1.00e+00 1.00e+00h  1
  47  2.2138987e-04 2.95e+00 1.99e-02  -2.5 2.45e-01  -5.2 1.00e+00 1.00e+00h  1
  48  2.2140819e-04 2.33e+01 2.96e-06  -2.5 4.22e-01  -5.6 1.00e+00 1.00e+00h  1
  49  2.2141169e-04 2.66e+00 1.52e-06  -2.5 1.66e-01  -5.2 1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  50  2.2142528e-04 2.23e+01 8.53e-07  -2.5 4.19e-01  -5.7 1.00e+00 1.00e+00h  1
  51  2.2146707e-04 6.17e+01 2.82e-06  -2.5 1.16e+00  -6.2 1.00e+00 1.00e+00H  1
  52  2.2159270e-04 2.96e+01 1.89e-06  -2.5 3.64e+00  -6.6 1.00e+00 1.00e+00H  1
  53  2.2159758e-04 4.88e-01 6.91e-07  -2.5 1.43e-01  -5.3 1.00e+00 1.00e+00h  1
  54  2.2161389e-04 1.10e+00 7.97e-07  -2.5 4.97e-01  -5.8 1.00e+00 1.00e+00h  1
  55  2.2161694e-04 2.72e+00 2.63e-03  -2.5 2.01e+00  -6.3 1.00e+00 6.25e-02h  5
  56  2.2163577e-04 6.37e+00 7.59e-03  -2.5 3.41e+00  -6.7 1.00e+00 1.25e-01h  4
  57  2.2208118e-04 3.05e+00 8.08e-07  -2.5 1.04e+01  -7.2 1.00e+00 1.00e+00h  1
  58  2.2343732e-04 1.32e+01 7.46e-06  -2.5 3.06e+01  -7.7 1.00e+00 1.00e+00h  1
  59  2.2346446e-04 1.44e+01 3.91e-01  -2.5 7.86e+02  -8.2 1.00e+00 3.91e-03h  9
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  60  2.2366985e-04 1.57e+01 4.42e-06  -2.5 4.38e+00  -6.9 1.00e+00 1.00e+00h  1
  61  2.2421594e-04 1.01e+01 1.65e-06  -2.5 1.21e+01  -7.3 1.00e+00 1.00e+00h  1
  62  2.2588231e-04 7.06e+00 1.10e-05  -2.5 3.70e+01  -7.8 1.00e+00 1.00e+00h  1
  63  2.3070847e-04 1.76e+01 9.05e-05  -2.5 1.05e+02  -8.3 1.00e+00 1.00e+00h  1
  64  2.3413101e-04 7.88e+01 5.41e-01  -2.5 2.83e+02  -8.8 1.00e+00 2.50e-01h  3
  65  2.4038125e-04 1.00e+02 2.15e+00  -2.5 1.01e+03  -9.2 1.00e+00 1.25e-01h  4
  66  2.4299742e-04 1.12e+02 7.33e-01  -2.5 3.96e+02  -8.8 1.00e+00 1.25e-01h  4
  67  2.4955244e-04 1.13e+02 2.08e+00  -2.5 9.75e+02  -9.3 1.00e+00 1.25e-01h  4
  68  2.7591434e-04 1.03e+02 6.89e+00  -2.5 3.72e+03  -9.8 1.00e+00 1.25e-01h  4
  69  3.6190160e-04 1.33e+02 2.46e-03  -2.5 1.17e+03  -9.3 1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  70  9.2125363e-04 1.49e+03 6.37e+00  -2.5 3.95e+03  -9.8 5.89e-01 1.00e+00h  1
  71  1.2022885e-03 5.57e+03 4.70e+00  -2.5 8.79e+03    -  1.00e+00 5.00e-01h  2
  72  2.9755666e-03 3.76e+03 4.16e+00  -2.5 1.96e+01  -4.9 3.14e-01 5.00e-01h  2
  73  4.2075541e-03 3.62e+03 4.69e+00  -2.5 1.03e+02  -5.4 1.00e+00 7.15e-02h  4
  74  4.5831056e-03 3.64e+03 3.73e+00  -2.5 7.10e+02  -5.8 3.21e-01 3.12e-02h  6
  75  4.9659478e-03 3.51e+03 4.07e+00  -2.5 1.26e+03  -6.3 1.00e+00 6.25e-02h  5
  76  5.3527133e-03 3.10e+03 3.19e+00  -2.5 3.12e+03  -6.8 1.00e+00 1.25e-01h  4
  77  5.3716332e-03 3.09e+03 3.33e+00  -2.5 3.05e+03  -6.4 9.51e-01 3.69e-03h  9
  78  5.3807001e-03 3.09e+03 2.99e+00  -2.5 3.02e+03  -5.9 8.41e-01 9.77e-04h 11
  79  5.3831460e-03 3.08e+03 3.51e+00  -2.5 3.04e+03  -6.4 1.00e+00 4.88e-04h 12
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  80  5.6956255e-03 2.70e+03 2.86e+00  -2.5 3.07e+03  -6.9 1.00e+00 1.25e-01h  4
  81  1.8093001e-02 5.75e+05 4.33e-01  -2.5 2.68e+03  -5.6 6.00e-01 8.97e-01w  1
  82  6.6500953e-02 1.52e+05 2.27e-01  -2.5 9.06e+01  -4.2 1.00e+00 1.00e+00w  1
  83  4.1380180e-02 4.00e+04 2.76e-01  -2.5 2.04e+02  -4.7 9.60e-01 1.00e+00w  1
  84  5.7440527e-03 2.69e+03 2.57e+00  -2.5 3.37e+02  -5.2 6.00e-01 3.51e-03h  8
  85  5.7604907e-03 2.69e+03 2.87e+00  -2.5 2.67e+03  -5.7 6.18e-01 1.28e-03h  9
  86  5.8199435e-03 2.68e+03 2.56e+00  -2.5 2.68e+03  -5.2 4.11e-01 3.39e-03h  9
  87  5.9115681e-03 2.66e+03 3.02e+00  -2.5 2.67e+03  -5.7 1.00e+00 7.81e-03h  8
  88  7.3051075e-03 1.99e+03 1.92e+00  -2.5 2.99e+03  -6.2 1.00e+00 2.50e-01h  3
  89  7.3055113e-03 1.99e+03 1.95e+00  -2.5 4.11e+03  -6.7 9.49e-02 9.04e-05h 11
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  90  7.6304372e-03 1.74e+03 1.98e+00  -2.5 2.21e+03  -6.2 1.00e+00 1.25e-01h  4
  91  7.7406764e-03 1.37e+03 9.86e-01  -2.5 1.98e+03  -6.7 1.00e+00 5.00e-01h  2
  92  7.5477594e-03 1.42e+03 7.71e-01  -2.5 9.99e+02  -6.3 1.00e+00 2.50e-01h  3
  93  8.0842601e-03 6.33e+02 1.27e-01  -2.5 7.50e+02  -5.0 1.00e+00 1.00e+00h  1
  94  4.5180579e-03 1.56e+02 2.42e-02  -2.5 1.73e+02    -  1.00e+00 1.00e+00h  1
  95  5.4731948e-03 3.00e+01 1.77e-05  -2.5 3.81e+00  -5.4 1.00e+00 1.00e+00h  1
  96  5.4591696e-03 3.76e+01 1.67e-04  -2.5 2.15e+00  -5.0 1.00e+00 5.00e-01h  2
  97  5.3325833e-03 2.85e+01 7.63e-05  -2.5 4.58e+00  -5.5 1.00e+00 1.00e+00h  1
  98  5.2912221e-03 3.87e+01 2.90e-02  -2.5 1.20e+01  -6.0 1.00e+00 1.25e-01h  4
  99  5.1485987e-03 7.56e+01 9.39e-05  -2.5 7.71e+00  -5.5 1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
 100  4.8221539e-03 2.66e+01 4.45e-04  -2.5 1.25e+01  -6.0 1.00e+00 1.00e+00h  1
 101  4.8274747e-03 4.03e+00 1.41e-05  -2.5 6.19e-01  -4.7 1.00e+00 1.00e+00h  1
 102  4.7924336e-03 2.08e+00 1.03e-05  -2.5 1.51e+00  -5.2 1.00e+00 1.00e+00h  1
 103  4.6962875e-03 1.61e+00 4.45e-05  -2.5 4.22e+00  -5.6 1.00e+00 1.00e+00h  1
 104  4.4561234e-03 7.88e+00 2.90e-04  -2.5 1.10e+01  -6.1 1.00e+00 1.00e+00h  1
 105  3.9903659e-03 1.37e+01 1.22e-03  -2.5 2.41e+01  -6.6 1.00e+00 1.00e+00h  1
 106  3.3642797e-03 4.84e+00 2.82e-03  -2.5 4.19e+01  -7.1 1.00e+00 1.00e+00h  1
 107  2.7580323e-03 2.93e+00 3.91e-03  -2.5 6.02e+01  -7.6 1.00e+00 1.00e+00h  1
 108  2.2939462e-03 9.39e+00 3.62e-03  -2.5 7.24e+01  -8.0 1.00e+00 1.00e+00h  1
 109  2.0016315e-03 3.39e+01 2.18e-03  -2.5 7.75e+01  -8.5 1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
 110  1.8711176e-03 9.14e+00 6.06e-04  -2.5 4.26e+01  -9.0 1.00e+00 1.00e+00h  1
 111  1.8314458e-03 1.42e+00 6.22e-05  -2.5 1.84e+01    -  1.00e+00 1.00e+00h  1
 112  1.8309752e-03 3.20e-01 2.83e-08  -2.5 5.59e+01    -  1.00e+00 1.00e+00H  1
 113  1.8309753e-03 9.61e-03 2.83e-08  -2.5 2.92e+00    -  1.00e+00 1.00e+00h  1
 114  4.2381088e-04 8.22e+01 6.89e-02  -3.8 4.25e+02    -  7.50e-01 1.00e+00h  1
 115  3.1180115e-04 2.21e+02 5.04e-03  -3.8 1.05e+03    -  1.00e+00 1.00e+00h  1
 116  4.8712274e-04 5.42e+00 5.07e-03  -3.8 2.33e+00  -5.8 1.00e+00 1.00e+00h  1
 117  3.3448460e-04 6.25e+01 5.19e-03  -3.8 7.17e+02    -  1.00e+00 1.00e+00h  1
 118  3.7590403e-04 3.47e-01 6.91e-05  -3.8 5.89e-01  -6.3 1.00e+00 1.00e+00h  1
 119  2.7609681e-04 5.23e+01 1.93e-03  -3.8 7.58e+02    -  1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
 120  2.5209517e-04 2.07e+01 3.60e-04  -3.8 5.95e+02    -  1.00e+00 1.00e+00h  1
 121  2.5197789e-04 1.67e-01 4.59e-07  -3.8 7.31e+01    -  1.00e+00 1.00e+00h  1
 122  2.5197919e-04 6.59e-07 1.50e-09  -3.8 2.71e-01    -  1.00e+00 1.00e+00h  1
 123  1.1170942e-04 3.56e+02 5.63e-02  -5.7 2.43e+03    -  7.81e-01 1.00e+00h  1
 124  7.1610780e-05 8.29e+02 1.79e-03  -5.7 5.16e+03    -  1.00e+00 1.00e+00h  1
 125  9.5309511e-05 1.37e+01 2.67e-02  -5.7 3.70e+00  -6.8 8.56e-01 1.00e+00h  1
 126  6.2410071e-05 4.92e+02 1.76e-03  -5.7 4.83e+03    -  1.00e+00 1.00e+00h  1
 127  7.1952766e-05 3.54e+00 3.22e-05  -5.7 1.88e+00  -7.3 1.00e+00 1.00e+00h  1
 128  4.7287696e-05 7.12e+02 1.34e-03  -5.7 6.88e+03    -  1.00e+00 1.00e+00h  1
Reallocating memory for MA57: lfact (410779)
 129  5.4350435e-05 5.24e+00 3.32e-05  -5.7 2.29e+00  -7.8 1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
 130  4.2273183e-05 1.29e+02 8.62e-04  -5.7 8.21e+03    -  1.00e+00 1.00e+00H  1
 131  4.2156018e-05 1.26e-01 1.65e-06  -5.7 3.54e-01  -8.2 1.00e+00 1.00e+00h  1
 132  3.5659446e-05 3.83e+02 2.59e-04  -5.7 7.12e+03    -  1.00e+00 1.00e+00h  1
 133  3.4720745e-05 6.01e+01 1.47e-05  -5.7 3.51e+03    -  1.00e+00 1.00e+00h  1
 134  3.4731686e-05 5.86e-02 2.80e-09  -5.7 1.61e+02    -  1.00e+00 1.00e+00h  1
 135  3.4731694e-05 2.38e-07 1.84e-11  -5.7 5.75e-02    -  1.00e+00 1.00e+00h  1

Number of Iterations....: 135

                                   (scaled)                 (unscaled)
Objective...............:   3.4731694252493991e-05    3.4731694252493991e-05
Dual infeasibility......:   1.8449144625279314e-11    1.8449144625279314e-11
Constraint violation....:   2.3841857910156250e-07    2.3841857910156250e-07
Complementarity.........:   1.8449144625279519e-06    1.8449144625279519e-06
Overall NLP error.......:   1.8449144625279519e-06    1.8449144625279519e-06


Number of objective function evaluations             = 242
Number of objective gradient evaluations             = 136
Number of equality constraint evaluations            = 348
Number of inequality constraint evaluations          = 348
Number of equality constraint Jacobian evaluations   = 136
Number of inequality constraint Jacobian evaluations = 136
Number of Lagrangian Hessian evaluations             = 135
Total CPU secs in IPOPT (w/o function evaluations)   =     21.977
Total CPU secs in NLP function evaluations           =      0.407

EXIT: Optimal Solution Found.

Analyze the Results¶

multiexp_results = extract_plot_results(None, TC_Lab_DoE_A)

Depending on whether you used parameter scaling in the DesignOfExperiments

FIM_A = np.asarray(TC_Lab_DoE_A.results["solution"]["param_scenarios"][0]["total_fim"])

if SCALE_BY_NOMINAL_PARAM:
    D = np.diag(theta_ref)
    FIM_unscaled_A = D @ FIM_A @ D / SCALE_CONSTANT_VALUE**2
else:
    FIM_unscaled_A = FIM_A / SCALE_CONSTANT_VALUE**2
results_summary(np.asarray(FIM_unscaled_A))

======Results Summary======
Five design criteria log10() value:
Pseudo A-optimality: 10.23559046813628
A-optimality: -7.344252849475594
D-optimality: 36.137496106376815
E-optimality: 7.357207085369849
Modified E-optimality: 2.6003313400615897

FIM:
 [[ 7.17766371e+09  1.00811755e+08 -2.18902125e+08 -1.78696757e+06]
 [ 1.00811755e+08  8.99331580e+08  1.51888668e+08  2.53869117e+06]
 [-2.18902125e+08  1.51888668e+08  5.69021857e+07  1.94676190e+06]
 [-1.78696757e+06  2.53869117e+06  1.94676190e+06  9.06855902e+09]]

eigenvalues:
 [7.18585487e+09 2.27618253e+07 9.25277825e+08 9.06856198e+09]

Eigenvector matrix:
          eigvec_1  eigvec_2  eigvec_3  eigvec_4
Ua         0.9994    0.0326    0.0098   -0.0010
Ub         0.0153   -0.1743   -0.9846    0.0003
inv_CpH   -0.0304    0.9842   -0.1747    0.0002
inv_CpS    0.0010   -0.0002    0.0004    1.0000

# Now compare the uncertainty for the prior and the optimized design to see how much
# improvement we have achieved in terms of parameter uncertainty reduction.
std_D = np.sqrt(np.diag(cov_D))
std_A = np.sqrt(np.diag(np.linalg.inv(FIM_unscaled_A)))
print("Parameter estimates with standard deviations for D-optimal design:")
param_std(std_D)

print("\nParameter estimates with standard deviations for A-optimal design:")
param_std(std_A)

Parameter estimates with standard deviations for D-optimal design:
Ua: 4.170e-02 ± 1.363e-05
Ub: 1.201e-02 ± 4.928e-05
inv_CpH: 1.675e-01 ± 2.065e-04
inv_CpS: 4.545e+00 ± 2.855e-03

Parameter estimates with standard deviations for A-optimal design:
Ua: 4.170e-02 ± 1.363e-05
Ub: 1.201e-02 ± 4.881e-05
inv_CpH: 1.675e-01 ± 2.064e-04
inv_CpS: 4.545e+00 ± 1.050e-05

Discussion: How does changing the objective change the result? Which one is better for reducing the uncertainty in our experiments?