Comparative Study: Wavelet-Adaptive Multiple Shooting vs. Single Shooting in Matlab-EMSO Environment

1. A Comparative Study between Wavelet-Adaptive Multiple Shooting and Single Shooting Implemented in a Matlab-EMSO Environment Lizandro S. Santos, Argimiro R. Secchi, Evaristo C. Biscaia Jr. Chemical Engineering Program – PEQ/COPPE Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil.

2. Outline Outline • Introduction • Dynamic Optimization with Wavelets • Algorithm Structure • Results • Conclusions Outline

3. Goals • Implementation of Wavelet-Adaptive Algorithm[1] to solve optimal control problems using both single shooting and multiple shooting strategies; • Using software EMSO as modeling environment for dynamic optimization formulation; • Comparing CPU performance of wavelet-adaptive single and multiple shooting strategies. [1] Santos et al., 2014, Computer Aided and Chemical Engineering, v. 33, p. 247 - 252. Goals

4. Mathematic Formulation – Dynamic Optimization General Definition equality inequality terminal Mathematic Formulation

5. Dynamic Optimization Methods Dynamic Optimization Methods Bellman (HJB) Indirect Methods Pontryagin (EL) Dynamic Optimization (Sequential) Single Shooting Direct Methods Multiple Shooting Simultaneous Dynamic Optimization Methods

6. Sequential Method (Single Shooting) Control vector parameterization (e.g., piecewise constant approximation) Stage 2 Stage 3 Stage 1 Time domain discretization Number of stages u tF t0 time Sequential Method

7. Single Shooting Sensitivity Equations Non-Linear Programming ex.: Interior Point, SQP Sequential Integration u tF t0 time Sequential Method

8. Multiple Shooting Sensitivity and Constraints Non-Linear Programming ex.: Interior Point, SQP parallel Parallel Integration tF t0 time Multiple Shooting

9. Characteristics Characteristics

10. Wavelet Analysis for Image and Signal Processing Wavelet Decomposition Original Image Processed Image Resolution levels Noisy signal Wavelet Adaptation

11. How Wavelets Work Wavelet-Adaptive Dynamic Optimization NLP solver NLP solver Iteration 2 Wavelets Wavelets Iteration 1 Control profile Wavelet Adaptation

12. Wavelets Thresholding Analysis Considering a function , it can be transformed into wavelet domain as (discrete wavelet transform): detail coefficients control variable Inner product Vector of wavelets details Resolution Position where is the maximum level resolution. Wavelet Adaptation

13. Wavelets Thresholding Analysis Haar[1] wavelet has been considered: Orthogonal basis [1] Daubechies, I., 1992, Ten Lectures on Wavelets, Philadelphia, Society for Industrial and Applied Mathematics. Wavelet Adaptation

14. Wavelets Thresholding Analysis details Thresholding: some details are eliminated. New thresholded control profile Wavelet Adaptation

15. Thresholding Strategies Wavelets Thresholding Analysis soft threshold:[2] detail coefficients Choice of threshold: Visushrink[3] Sureshrink[4] Cross Validation[5] Fixed Threshold[6] [2] Wieland B., 2009, PhD Thesis, Universität Ulm, Deutschland. [3] Donoho, D.L. and Johnstone, I.M., 1994, Biometrika, v. 81, pp. 425-455. [4] Donoho, D.L., 1995, IEEE Transactions on Information Theory, v. 41, n. 3, pp. 613-627. [5] Jansen, M., 2000, PhD Thesis. [6] Schlegel, M, et al., 2005, Computer and Chemical Engineering, v. 29, pp. 1731-1751. Wavelet Adaptation

16. Control Vector Parameterization Shrink Threshold (CVPS) Wavelets Daubechies Wavelet Adaptation

17. Algorithm Structure (MATLAB) Based on wavelets analysis Santos et al. (2011, 2014) Enviroment for including DAE model Algorithm Structure

18. Case Studies Case Studies

19. Thresholdings with Single Shooting Results: overview CPU normalized by the uniform discretization with 128 stages

20. Results (Single Shooting: 1 CPU, Multiple Shooting: 2 CPUs) Illustrative example: Semi-batch Isothermal Reactor [2] Piecewise-constant interpolation [2] Srinavasan, B. et al., 2003, Computer and Chemical Engineering, v. 27, pp. 1-26. Illustration

21. Illustration Illustration of Wavelet Refinement Algorithm Example: Semi-batch Isothermal Reactor [2] [2] Srinavasan, B. et al., 2003, Computer and Chemical Engineering, v. 27, pp. 1-26. Illustration

22. Illustration Illustration of Wavelet Refinement Algorithm Example: Semi-batch Isothermal Reactor [2] Analytical [2] Srinavasan, B. et al., 2003, Computer and Chemical Engineering, v. 27, pp. 1-26. Illustration

23. Example 1 [3] [3] Diehl et al., (2006), Fast Direct Multiple Shooting Algorithms for Optimal Robot Control Example 1

24. Example 1 [1] State variable at the first iteration of MS method State variable at the first iteration of SS method time time Uncontrollable growth of x for any [1] Diehl et al., (2006), Fast Direct Multiple Shooting Algorithms for Optimal Robot Control Example 1

25. Example 1 [1] Optimal control profile Optimal state profile u time time [1] Diehl et al., (2006), Fast Direct Multiple Shooting Algorithms for Optimal Robot Control Example 1

26. Example 1 [1] [1] Diehl et al., (2006), Fast Direct Multiple Shooting Algorithms for Optimal Robot Control Example 1

27. Example 2 Example 2

28. Example 2 – Evolution of Control Profiles Iteration 2 Iteration 3 Example 2

29. Example 2 – Evolution of Control Profiles Iteration 4 Iteration 5 Example 2

30. Example 2 Example 2

31. Example 3 [3] [2] Srinavasan, B. et al., 2003, Computer and Chemical Engineering, v. 27, pp. 1-26. Example 3

32. Example 3 Example 3

33. Example 3 Example 3

34. Example 3 Example 3

35. Example 4 [4] [4] Banga, J.R., et al. 2005, Journal of Biotechnology, v. 117, pp. 407-419. Example 4

36. Example 4 [4] [4] Banga, J.R., et al. 2005, Journal of Biotechnology, v. 117, pp. 407-419. Example 4

37. Example 4 [4] [4] Banga, J.R., et al. 2005, Journal of Biotechnology, v. 117, pp. 407-419. Example 4

38. Example 4 [4] [4] Banga, J.R., et al. 2005, Journal of Biotechnology, v. 117, pp. 407-419. Example 4

39. Conclusions • The wavelet-adaptive algorithm generated a non-uniform control parameterization for both multiple shooting and single shooting strategies allowing a faster convergence to the optimal solution; • The combination of wavelet-adaptive strategy with parallel integration makes multiple shooting attractive and a good alternative for single shooting method; • It has been observed that single shooting is faster than multiple shooting for small and smooth NLP problems; however, for large-dimension or non-smooth problems multiple shooting might be faster than single shooting approach. Conclusions

40. A Comparative Study between Wavelet-Adaptive Multiple Shooting and Single Shooting Implemented in a Matlab-EMSO Environment Optimal Wavelet-Threshold Selection to Solve Dynamic Optimization Problems Thank you for your attention! arge@peq.coppe.ufrj.br

41. Visushrink Threshold (VS) Minimizes the probability that any noise sample will exceed the threshold. Given a sequence of variables, the expected value of the maximum increase with the length of the signal: Standard deviation estimate for normal distribution of small details Inverse of cumulative normal distribution

42. Sureshrink Threshold (SS) Sureshrink is an adaptive thresholding strategy. It is based on Stein’s Unbiased Risk Estimator[7] (SURE), a method for estimating the optimal threshold at each wavelet level: where [7] Stein, C M., 1981, The Annals of Statistics, v. 9, pp. 1135-1151

43. Cross Validation Threshold (CV) In short, the concept of CV is given by the construction of several estimates, never using the whole dataset. Using these estimates one predicts what the expelled data could have been and compares the prediction with the actual values of the expelled data.[8] where [8] Wieland B., 2009, Speech signal noise reduction with wavelets, PhD Thesis, Universität Ulm, Deutschland

44. Fixed Threshold Use a user specified fixed threshold rule.[9] The threshold does not change with the wavelet level. The threshold value must, at least, be set between max (d) and min (d). [9] Schlegel, M, et al., 2005, Computer and Chemical Engineering, v. 29, pp. 1731-1751.