Probabilistic Particle Swarm Optimization Pro PSO for Using

Probabilistic Particle Swarm Optimization (Pro. PSO) for Using Prior Information and Hierarchical Parameters Jaehoon Lee, Tapan Mukerji Stanford Center for Reservoir Forecasting

What Is Particle Swarm Optimization (PSO)? Pros Individual Learning ü ü Simple and easy to be hybridized Global optimizer with multiple solutions Favorable for parallel computing Faster than other evolutionary methods Cons Solution ü Computational cost ü No easy way to use probabilistic prior information Neighborhood’s Learning Stanford Center for Reservoir Forecasting

Objective of Probabilistic PSO (Pro-PSO) How to incorporate a probabilistic form of prior information into PSO Probabilistic Prior More Probable Particles Solution Stanford Center for Reservoir Forecasting

Incorporating Prior Probabilities Prior Probability Conditional probability given prior data dprior: P(x|dprior) Prior probability PSO algorithm PSO Algorithm Newtonian physics : Perturbation of candidate solutions Movement of particles What is the probabilistic form of PSO? Stanford Center for Reservoir Forecasting

General PSO Algorithm xi(t+1) : position of particle i at iteration t+1: vi(t+1) : velocity of particle i at iteration t+1: where Personal best position of particle i : total population of the swarm : inertia constant : personal and global acceleration constants Global best position : diagonal matrices with random numbers from uniform distribution U[0, 1] Stanford Center for Reservoir Forecasting

Derivation of Pro-PSO Total number of parameters = N Particle i Personal Current k attractor, pi (t) position, xik(t) k axis wvik(t) Global attractor, gk(t) Acceleration Sampling of perturbation 6 Stanford Center for Reservoir Forecasting

PSO Derivation of Pro-PSO Convolution Pro-PSO Stanford Center for Reservoir Forecasting

Derivation of Pro-PSO Total number of parameters = N Particle i Personal Current k attractor, pi (t) position, xik(t) k axis wvik(t) Global attractor, gk(t) Pro-PSO 8 Stanford Center for Reservoir Forecasting

Incorporating Prior Probabilities Combined probability PSO probability Prior Probability Conditional probability given prior data dprior: P(x|dprior) PSO Probability Conditional probability given the vectors at iteration t: P(xi(t+1)|xi(t), vi(t), pi(t), g(t)) Combined Probability xi(t) = 0, pi(t) = 5, g(t) =7 P(xi(t+1)|xi(t), vi(t), pi(t), g(t), dprior) Tau Model (Journel, 2002) Stanford Center for Reservoir Forecasting

Comparison of Pro-PSO and PSO 2 D Synthetic Reservoir Seismic Inversion Acoustic Impedance Flow History Matching Production Rates Stanford Center for Reservoir Forecasting

Comparison of Pro-PSO and PSO 2 D Synthetic Reservoir Principal Component Analysis(PCA) 90% variation : 300 PCs 2501 realizations Fernández-Martínez et al. (2010), Suman (2013), and Sarma (2006) Stanford Center for Reservoir Forecasting

Prior Probabilities from Realizations Pro-PSO can utilize the distributions In PSO, particles move within the boundaries Stanford Center for Reservoir Forecasting

Pro-PSO and PSO: Search Space Projected 300 -dimensional Search Space : each vertical line represents one axis In PSO, particles move within the boundaries Stanford Center for Reservoir Forecasting

Pro-PSO and PSO: Search Space Projected 300 -dimensional Search Space The score distributions are incorporated into Pro-PSO Stanford Center for Reservoir Forecasting

Pro-PSO and PSO: Performance (Seismic) Misfit = 0. 7312 Misfit = 0. 2181 Pro-PSO accomplished the minimum misfit of PSO after 12 iterations Stanford Center for Reservoir Forecasting

Pro-PSO and PSO: Best Models Reference with 300 PCs Pro-PSO Stanford Center for Reservoir Forecasting

Pro-PSO and PSO: Variogram Major Direction (45°) Minor Direction (135°) Stanford Center for Reservoir Forecasting

Pro-PSO and PSO: Performance (Flow) Pro-PSO accomplished the minimum misfit of PSO after 24 iterations Misfit = 0. 00220 Misfit = 0. 00068 Stanford Center for Reservoir Forecasting

Pro-PSO and PSO: Production Rates Pro-PSO Stanford Center for Reservoir Forecasting

Pro-PSO with Hierarchical Use of Parameters Pro-PSO runs on only part of parameters first and then includes the rest of the parameters hierarchically. Hierarchy 1 - PSO with part of parameters, k = 1, …, h 1 - Global and personal best positions : pki(h 1), gk(h 1) for i = 1, …, S and k = 1, …, h 1 Hierarchy 2 - Add the next part of parameters, k = 1, …, h 2 - Prior probability : P(xki(h 2)|pki(h 1), gk(h 1)) or P(xki(h 2)|gk(h 1)) for i = 1, …, S and k = 1, …, h 1 Stanford Center for Reservoir Forecasting …

Pro-PSO and PSO: Performance (Seismic) Pro-PSO uses the first 22 PCs (50% variance) and the 278 PCs are added when the best 50 models fall within the 80% of the search space. Misfit = 0. 7312 Misfit = 0. 5301 Stanford Center for Reservoir Forecasting

Pro-PSO and PSO: Best Models Reference with 300 PCs Pro-PSO w/ Hierarchies PSO Stanford Center for Reservoir Forecasting

Pro-PSO and PSO: Variogram Major Direction (45°) Minor Direction (135°) Stanford Center for Reservoir Forecasting

Conclusions and Future Work • Reformulated PSO from a probabilistic point of view. • Allows incorporating probabilistic prior information. • Investigated the possibility of the hierarchical use of parameters. • Analyze the performance and stability of PSO. • Can be a pseudo-sampler. Stanford Center for Reservoir Forecasting