1. Bounded Nonlinear Optimization to Fit a Model of Acoustic Foams
Andrew Willemsen
Yinan Zuo
4/27/2011

2. Outline Problem Background Cost Function and Constraints
Frequency Range Considerations
Description of Optimization Algorithms
Active Set
SQP
Interior Point
Trust Region Reflective
Results of Model Parameter Optimization
Initial Conditions and Global vs. Local Minimas
Comparison of Algorithms
Conclusions
References
Nonlinear Optimization of Foam Model
4/27/2011

3. Problem Background
The amount of dissipation of acoustic energy provided by a porous material is commonly modeled based upon 5 physical properties of the foam.
Model is complex valued and frequency dependent
5 model parameters are real valued and independent of frequency
These five model parameters are difficult to measure, while the resulting acoustic properties are relatively easy to measure.
Thus, optimization strategies are used to estimate the model parameters by minimizing the difference between predicted and measured acoustic properties.
Nonlinear Optimization of Foam Model
4/27/2011

4. Cost Function
Sum of squares of model-measurement differences for many frequencies:
Nonlinear Optimization of Foam Model
4/27/2011

5. Bound Constraints Each of the 5 model parameters is bounded:
Nonlinear Optimization of Foam Model
4/27/2011

6. Frequency Range Considerations
Although the model parameters are constant at all frequencies, the dependence of the model on these parameters changes as a function of frequency.
Can result in scaling issues and poor parameter estimates.
Choosing the correct frequency range over which to optimize the function is crucial to accurate parameter estimation.
Ideally, all parameters would equally effect the value of the model.
In reality, the parameters can be grouped into primary effects, secondary effect, and non-effects depending on frequency
Nonlinear Optimization of Foam Model
4/27/2011

7. Frequency Range Considerations
Relevant frequency range can be broken into 3 zones:
Nonlinear Optimization of Foam Model
4/27/2011

8. Primary Influences
At all frequencies, x1 has the most influence on the value of the model.
In Zone 2, x2 has a comparable influence on the value of the model.
Nonlinear Optimization of Foam Model
4/27/2011

9. Secondary Influences
At all frequencies, x3 has a secondary influence on the value of the model.
In Zone 1, x2 and x5 also have a secondary influence.
In Zone 2 and 3, x2 and x4 also have a secondary influence.
Nonlinear Optimization of Foam Model
4/27/2011

10. Selection of Best Frequency Range
It was determined that using Zones 1 and 2 together would be best for the model optimization.
All parameters have at least a secondary influence on the model value over some part of this frequency range.
The measured values peak in this range, so less influence from measurement noise.
Nonlinear Optimization of Foam Model
4/27/2011

11. Active Set Definition: is active if and is inactive if
Lagrange function:
Algorithm:
While the solution is not good enough
1. Solve the equality problem defined by the active set
2.Compute the Lagrange multipliers of the active set
3. Remove the constraints with negative Lagrange multipliers
4. Search for infeasible constraints
End
Nonlinear Optimization of Foam Model
4/27/2011

12. Sequential Quadratic Programming
An iterative method. Requirement for the problem: Twice continuous differentiability of the function and continuous differentiability of the constraints.
The problem is solved by solving a sequence of subproblems, the search direction at each iterate is defined by
subject to
where
Nonlinear Optimization of Foam Model
4/27/2011

13. Interior Point Method
Implemented by Matlab’s fmincon function for constrained minimization.
Barrier function (μ > 0) used to redefine problem as:
The inequality constraints are made equality constraints by the use of of vector of slack variables s ≥ 0.
Nonlinear Optimization of Foam Model
4/27/2011

14. Interior Point Method
The algorithm uses two types of subproblem solvers to compute the step at each iteration.
Direct Step – KKT system solver (H positive definite)
Conjugate Gradient Step – Trust-region (H indefinite)
subject to linearized constraints and the trust region size
Nonlinear Optimization of Foam Model
4/27/2011

15. Nonlinear Least Squares Methods
The cost function is instead a vector-valued function and the problem is of the form:
The Newton direction for this problem becomes:
J is the Jacobian of the vector-valued function f.
Implemented in Matlab by lsqnonlin.
Nonlinear Optimization of Foam Model
4/27/2011

16. Trust Region Reflective Method
Implemented by the lsqnonlin function in Matlab for bound constrained problems
Modification of trust-region approach to unconstrained problem.
Scaled modified Newton step:
Reflections modify steps crossing any bounds:
Nonlinear Optimization of Foam Model
4/27/2011

17. Global and Local Minima and Initial Point
Nonlinear Optimization of Foam Model
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18. Global and Local Minima and Initial Point – Active Set
Nonlinear Optimization of Foam Model
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19. Global and Local Minima and Initial Point – SQP
Nonlinear Optimization of Foam Model
4/27/2011

20. Global and Local Minima and Initial Point – Interior Point
Nonlinear Optimization of Foam Model
4/27/2011

21. Global and Local Minima and Initial Point – Least Squares
Nonlinear Optimization of Foam Model
4/27/2011

22. Comparison of Algorithms
Each of the three fmincon algorithms and the one lsqnonlin algorithm were used to optimize the cost function using the same initial point.
The primary stopping conditions for all algorithms were the first order optimality conditions, although by default not all algorithms used this stopping condition.
Nonlinear Optimization of Foam Model
4/27/2011

23. Comparison of Algorithms
Function Value
Parameter Values
Iterations
First Order Optimality
Function Evaluations
Active Set
[46088, , 4, , 1] 31
7.8883E-03
191
SQP 36
2.8610E-06
231
Interior Point
[46087, , , , ] 40
2.0000E-06
253
Trust-Region Reflective
[46262, , 4, , 1] 73
5.3090E-01
444
Nonlinear Optimization of Foam Model
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24. Active Set Method Convergence
Nonlinear Optimization of Foam Model
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25. SQP Method Convergence
Nonlinear Optimization of Foam Model
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26. Interior Point Method Convergence
Nonlinear Optimization of Foam Model
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27. Trust-Region Reflective Method Convergence
Nonlinear Optimization of Foam Model
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28. Conclusions
An effective means to estimate the properties of an acoustic foam is to optimize the acoustic material model.
Zone 1 and 2 is the best frequency range to optimize over since all parameters contribute significantly to the model value in this range.
Final solution depends on initial conditions
If x1 is within a certain range, global minimization guaranteed
For this size and type of problem:
Active set method was the fastest/most efficient
Least squares (trust region reflective method) was the least efficient and least accurate
SQP and Interior Point methods resulted in the most accurate minimums and were nearly as efficient as active set.
Nonlinear Optimization of Foam Model
4/27/2011

29. References
Y. Atalla and R. Panneton, “Inverse acoustical characterization of open cell porous media using impedance tube measurements”, Canadian Acoustics, 33(1), (2005).
J. Nocedal and S. J. Wright, Numerical Optimization, Springer, New York, (2006).
T. F. Coleman and Y. Li, “An interior trust region approach for nonlinear minimization subject to bounds”, SIAM J. Optimization, 6(2), , (1996).
Matlab Documentation, “Constrained Nonlinear Optimization Algorithms”, (2011).
Matlab Documentation, “Least Squares (Model Fitting) Algorithms”, (2011).
Nonlinear Optimization of Foam Model
4/27/2011