Hybrid Niche Algorithm for Problem Inversion of Distributed Systems with Solution Multiplicity

1. Hybrid Niche Algorithm for Problem Inversion of Distributed Systems with Solution Multiplicity AIChE 2008 Annual Meeting Philadelphia, Pennsylvania, November 16 - 21 Paper 196e Applied Mathematics and Numerical Analysis (10D) #196 – Advances in Computational Methods and Numerical Analysis (10D02) Jeonghwa Moon, Libin Zhang, Andreas A. Linninger 11/18/2008 Laboratory for Product and Process Design, Department of Chemical Engineering, University of Illinois, Chicago, IL 60607, U.S.A. UIC

2. Motivation: Inversion problem Chemical Reactor: Data available: (at specific locations) Characteristics:- Partial information available- The system cannot be studied as a whole To determine: Reaction: • Kinetic constants: • Rate of the reaction (k1) • Other constants • Thermal constants: • Thermal conductivity (k) • Porosity (ε) • Diffusivity (D) mass: Energy balance What are the values of ke, and DA, ?

3. Mathematical formulation Physical properties FIRST PRINCIPLES DISTRIBUTED MODELS Transport and Kinetic Inversion (TKIP) Distributed system data s. t. Convection-Diffusion equation Data set obtained from experiments Model predicted data Covariance matrix of the measurements Direct analytical solution is challengeable!

4. Finite volume method (FVM) 2D transport problem with diffusion and reaction Transport Model Integrating over a control volume N n e w P E W s S Discretization Model

5. Multiplicity of solutions in TKIP Multiplicity in inversion solution is common due to (a) Model non-linearity(b) Error in experimental data(c) Multiple experimental datasets - Contour plot of the residual error in the space of the kinetic parameters (reaction constants) k1 and k11 for the H2/O2 combustion reactions- Shows physically possible multiple minima

6. Hybrid sequential niche genetic algorithm • Previous studies • Homotopy method (Sun and Seider, 1995), • Interval methods (Stadtherr et al., 1995), • Mixed Integer Nonlinear Programming (McDonald and Floudas, 1995) • Global terrain method (Lucia et al, 2008) • Two stage hybrid optimizer • To guess an initial point, a stochastic method is used. • To locate solution, a local deterministic method is used. • It finds all solutions including local minima (maxima) sequentially • Hybrid version of sequential niche technique (Beasley, 93) • Variable niche radius is used • No a priori- guess of number of solutions are needed

7. Two-stage hybrid genetic algorithms • Why hybrid? • Accuracy of solution • Produce fast convergence • Hybrid genetic algorithm- Two stage search START local deterministic optimizer Initial population Fitness evaluation Local optimizer Yes Convergent? exact Solution cost Hybrid GA Natural Selection END GA Mating Real solution No of iterations Mutation e.g: Li : GA+Gradient (2000) Sabatini: GA + Newton-Rhapson(2000) Sieary :GA + Simplex (2003)

8. Niches Niche method- multiple solutions • The key point of finding all solutions in stochastic method is how to maintain population diversity. • The meaning of niche • Different subspace that can support different types of life • In search space, each peak is niche • The number of individuals supported by niche • Proportional to Niche capacity • Niche capacity is determined by peak fitness

9. Given parameter set Solve for the state variables using the Finite volume method - Gradient based solvers (Newton’s method) Find least squares error (LSQ) Fitness=1/LSQ Suggested methodology for TKIP START Initial population of the model parameters Fitness evaluation Local optimizer Dense? Find exact Solution Yes No Duplicated? Natural Selection No Yes Mating Update Radius Add solution Set Radius Mutation Maximum iteration? Initialize population END

10. Case study I Multiplicity of Solutions in Catalytic Pellet Reactor

11. Cooling Outlet Multiscale Model B A Tubular Reactor Cooling inlet Packed Catalytical Pellet Bed Catalyst Pellet Micro Pores of Catalyst Catalytic pellet reactor Mathematical model s.t. Darcy’s law Mass and energy balance Pellet model Optimal Parameters: D= ? (m2/s) ; k =? (1/s)

12. 01 06 06 15 07 15 Multiplicity of inversion solutions-pellet reactor LPPD

13. Case study IISolutionMultiplicity in Plutonium Storage

14. Plutonium storage • DOE safety requirements regarding the safe storage of Plutonium dioxide powder for 50 years • Temperature increase could leads to an overall pressure increase: could lead to explosion • Determine values of: • The heat generation (q) • The wall heat transfer coefficient (U) • The overall thermal conductivity (k) Are we safe?

15. Problem formulation a) Nuclear heat generation, q, W/m3 b) Effective thermal conductivity, k, J/(K. s.m2) c) Wall heat transfer coefficients, U, W/m.K To determine: Math program: s.t. BC’s:

16. Results - Simulation model 10 multiple profiles Optimal Parameters h5 h4 h3 h2 h1 q: Nuclear heat generation U: Heat transfer coefficient, k1: Thermal conductivity k2: Radiative constant Experimental data and simulated radial temperature-profiles at different heights Normal Seq niche Residual error Hybrid Seq niche iteration

17. Conclusions • Non-linearity in the model and multiple erroneous datasets lead to multiplicity in TKIP • Solution multiplicity in distributed systems can be efficiently handled using hybrid niche methods • Future Work: • Consideration of dynamic systems • Larger number of parameters • Investigating solution-multiplicity in the TKIP for the human brain

18. UIC Thank you!