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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Global and Local Minima and Initial Point Nonlinear Optimization of Foam Model 4/27/2011

Global and Local Minima and Initial Point – Active Set Nonlinear Optimization of Foam Model 4/27/2011

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

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

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

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

Comparison of Algorithms Function Value Parameter Values Iterations First Order Optimality Function Evaluations Active Set 33.4230 [46088, 0.8436, 4, 5.8401, 1] 31 7.8883E-03 191 SQP 36 2.8610E-06 231 Interior Point 33.4234 [46087, 0.8436, 3.9999, 5.8406, 0.9995] 40 2.0000E-06 253 Trust-Region Reflective 33.4254 [46262, 0.8432, 4, 5.8162, 1] 73 5.3090E-01 444 Nonlinear Optimization of Foam Model 4/27/2011

Active Set Method Convergence Nonlinear Optimization of Foam Model 4/27/2011

SQP Method Convergence Nonlinear Optimization of Foam Model 4/27/2011

Interior Point Method Convergence Nonlinear Optimization of Foam Model 4/27/2011

Trust-Region Reflective Method Convergence Nonlinear Optimization of Foam Model 4/27/2011

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

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), 418-445, (1996). Matlab Documentation, “Constrained Nonlinear Optimization Algorithms”, http://www.mathworks.com/help/toolbox/optim/ug/brnoxzl.html, (2011). Matlab Documentation, “Least Squares (Model Fitting) Algorithms”, http://www.mathworks.com/help/toolbox/optim/ug/brnoybu.html, (2011). Nonlinear Optimization of Foam Model 4/27/2011