A New Scheme for Multifidelity Optimization Incorporating Pattern Search and Space Mapping

1. A New Scheme for Multifidelity Optimization Incorporating Pattern Search and Space Mapping Joseph P. Castro Jr. *, Genetha Gray g, Anthony Giunta b, Patricia Hough g, and Paul Demmie a Sandia National Laboratories: * Computational Sciences, a Computational Physics/Simulation Frameworks, b Validation & Uncertainty Quantification Processes, g Computational Sciences & Math 2005 SIAM Conference on Computational Science and Engineering February 13, 2005 Orlando, FL *Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000.

2. Finite Element Models of the Same Component High Fidelity 800,000 DOF Low Fidelity 30,000 DOF Multifidelity Surrogate Models • The low-fidelity surrogate model retains many of the important features of the high-fidelity “truth” model, but is simplified in some way. • decreased physical resolution • decreased FE mesh resolution • simplified physics • An MFO approach optimizes an inexpensive, low fidelity model while making periodic corrections using the expensive, high fidelity model. • Works well when low-fidelity trends match high-fidelity trends.

3. pruned node parent node dislocated node child node APPS Allows Us To Integrate a Low-Fidelity Response For Multifidelity Optimization • Pattern search is a non-gradient optimization search with pre-determined patterns. • Asynchronous Parallel Pattern Search (APPS)*, takes advantage of non-dependent responses with very different compute times • Ideal fit for use with multifidelity optimization • APPSPACK is open source software that implements the APPS algorithm • Does not assume homogeneous processors (MPI implementation) *Developed by Patricia Hough, Tamara Kolda, Virginia Torczon

4. xH xH high-fi model mapped low-fi model xL=P(xH) RL(P(xH))~RH(xH) such that RL(P(xH)) RH(xH) We’re using the mapping Space Mapping* Provides a Conduit Between The Design Spaces of the Low and High Fidelity Models xL x – design variables R - response P - mapping ? P(xH) low-fi model RL(xL) • Space mapping* is a technique that maps the design space of a low fidelity model to the design space of high fidelity model such that both models result in approximately the same response. • The parameters within xH need not match the parameters within xL *Developed by John Bandler, et. al.

5. Oracle The APPS/Space Mapping Scheme Outer Loop Inner Loop multiple xH,f(xH) Space Mapping Via Nonlinear Least Squares Calculation High Fidelity Mode Optimization via APPSPACK a,b,g Low Fidelity Model Optimization a (xH) b + g xHtrial

6. High Fidelity Model: Low Fidelity Model: A Simple Polynomial was Used to Study Space Mapping Sensitivities • Ideally a0=a0* , g0=g0*, etc... (fH = fmapped) • Studied the space mapping sensitivities to various inputs • # high fidelity responses used for the mapping • scaling of the mapping parameters (size of offset between the low and high fidelity models) • starting point • Compare the optimum found and the number of high fidelity runs required to reach the optimum Mapped Space (a*,b*,g* calculated via Least Squares):

7. The APPS/Space Mapping Scheme Improved Optimization Performance and Value g ~O(1);a, b=1 Starting Point = (-2.0,-2.0)

8. Plot of Best Points Found With APPS/Space Mapping Scheme Polynomial Model with g~O(1);a,b=1, starting point = (-2,-2) • In all cases the inner loop call finds a best point with the first call • All inner loop calls beyond this do not find a best point (APPS dominates at this point) 13 17 27 27 43

9. Comparison of Design Space of High and Low Fidelity Polynomial Models with a, g~O(1);b=1 View of Unmapped Low Fidelity Design Space View of High Fidelity Design Space

10. Plot of Best Points Found With APPS/Space Mapping Scheme Polynomial Model with a, g~O(1);b=1, starting point = (-2,-2) • Though there is an improvement with the inner loop, the performance is not as great as with the previous case • The APPS only case had the best optimal value as well 47 51 34 53

11. Best Case: # response points = 8 2 calls to inner loop Approximate Inner Loop Call Locations within Hi-Fi Model (-0.8,-1.2) 1 2 (-0.76,2.0) 1 • The numbered white boxes show approximately where the inner loop was called • The point in red brackets is where APPSPACK is before the inner loop call • The point in green was found by the inner loop (-0.56,1.6) 2 (-0.61,1.25)

12. Penetrator Case: 3-D Model of Steel Earth Penetrator Striking a Concrete Target • Steel Penetrator composed of multiple materials entering a concrete target • High Fidelity Model = elastic EP body • ~40 minute calculation time • Low Fidelity Model = rigid EP body • ~10 second calculation time V rc rp rN • A computational cost ratio of 1:240 • The low-fidelity model gives the same general response trends as the high-fidelity model • These factors makes these models prime candidates for multifidelity optimization b -y Target

13. Acceleration Comparison with Varying Mesh • Rigid body response follows the trend of elastic body response

14. Minimize Acceleration with Varying DensityHi-Fidelity Model = Elastic Model (Fine Mesh)Low-Fidelity Model = Rigid Model (Fine Mesh) • A series of calculations were done minimizing acceleration and maximizing displacement • displacement 2-3x speed up • acceleration 1-2x speed up • For this case, the APPS/Space Mapping scheme took longer to converge but a better optimum was found • provides a type of global search capability to get past the local “noise”

15. Ongoing and Future Work • Study spaces defined using different constraints. • Implement a generic oracle in APPSPACK. • Include a space mapping that does not require domain spaces to be defined by the same numbers of parameters. • Apply our multifidelity optimization schemes to some real world problems: • Earth penetrator analysis • Groundwater problems including well field design & hydraulic capture • Circuit system design

16. References and Contact Information APPSPACK: Software Website APPSPACK 4.0  http://software.sandia.gov/appspack/version4.0 This website includes the software itself (open-source) and instructions for downloading, installing, and using it.  It also has a complete list of references to papers on the software development and convergence analysis.  DAKOTA: Software Website http://endo.sandia.gov/DAKOTA ORACLE:  Overview Paper Kolda, Lewis, and Torczon, "Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods," SIAM Review, 45(3):385-482, 2003. SPACE MAPPING:  Bakr, Bandler, Madsen, and Sondergard, "An Introduction to Space Mapping Technique, " Optimization & Engineering, 2:369-384, 2001. Contact Info: Joseph Castro: jpcastr@sandia.gov