Posterior Sampling using Particle Swarm Optimizers and Model Reduction Techniques

Inverse problems are ill-posed and posterior sampling is a way of providing an estimate of the uncertainty based on a finite set of the family of models that fit the observed data within the same tolerance. Monte Carlo methods are used for this purpose but are highly inefficient. Global optimization methods address the inverse problem as a sampling problem, particularly Particle Swarm, which is a very interesting algorithm that is typically used in an exploitative form. Although PSO has not been designed originally to perform importance sampling, the authors show practical applications in the domain of environmental geophysics, where it provides a proxy for the posterior distribution when it is used in its explorative form. Finally, this paper presents a hydrogeological example how to perform a similar task for inverse problems in high dimensional spaces through the combined use with model reduction techniques.

[1]  I. Jolliffe.  Principal Component Analysis , 2002 .

[2]  Esperanza García Gonzalo,et al.  The PSO family: deduction, stochastic analysis and comparison , 2009, Swarm Intelligence.

[3]  Benoit Noetinger,et al.  The FFT Moving Average (FFT-MA) Generator: An Efficient Numerical Method for Generating and Conditioning Gaussian Simulations , 2000 .

[4]  Juan Luis Fernández-Martínez,et al.  Application of Global Optimization Algorithms to a Salt Water Intrusion Problem , 2009 .

[5]  Albert Tarantola,et al.  Monte Carlo sampling of solutions to inverse problems , 1995 .

[6]  Ioan Cristian Trelea,et al.  The particle swarm optimization algorithm: convergence analysis and parameter selection , 2003, Inf. Process. Lett..

[7]  M. Ghiassi,et al.  A Web-Enabled, Mobile Intelligent Information Technology Architecture for On-Demand and Mass Customized Markets , 2011 .

[8]  Goldberg,et al.  Genetic algorithms , 1993, Robust Control Systems with Genetic Algorithms.

[9]  Juan Luis Fernández-Martínez,et al.  Theoretical analysis of particle swarm trajectories through a mechanical analogy , 2008 .

[10]  J. F. Martínez,et al.  The generalized PSO: a new door to PSO evolution , 2008 .

[11]  Albert Tarantola,et al.  Inverse problem theory - and methods for model parameter estimation , 2004 .

[12]  Juan Luis Fernández-Martínez,et al.  Particle Swarm Optimization (PSO): a Simple And Powerful Algorithm Family For Geophysical Inversion , 2008 .

[13]  Rainer Storn,et al.  Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces , 1997, J. Glob. Optim..

[14]  Zhihua Cui,et al.  Particle swarm optimization with FUSS and RWS for high dimensional functions , 2008, Appl. Math. Comput..

[15]  G. Mariéthoz,et al.  Bayesian inverse problem and optimization with iterative spatial resampling , 2010 .

[16]  J. L. F. Martínez,et al.  Anisotropic Mean Traveltime Curves: A Method to Estimate Anisotropic Parameters from 2D Transmission Tomographic Data , 2009 .

[17]  M. Sambridge.  Geophysical inversion with a neighbourhood algorithm—I. Searching a parameter space , 1999 .

[18]  Alexandre Boucher,et al.  Applied Geostatistics with SGeMS: A User's Guide , 2009 .

[19]  Juan Luis Fernández-Martínez,et al.  What Makes Particle Swarm Optimization a Very Interesting and Powerful Algorithm , 2011 .

[20]  Nicoletta Sala,et al.  Reflexing Interfaces: The Complex Coevolution of Information Technology Ecosystems , 2008 .

[21]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[22]  Luis Tenorio,et al.  Prior information and uncertainty in inverse problems , 2001 .

[23]  Seifedine Kadry.  Diagnostics and Prognostics of Engineering Systems: Methods and Techniques , 2012 .

[24]  Clifford H. Thurber,et al.  Parameter estimation and inverse problems , 2005 .

[25]  R. Parker.  Geophysical Inverse Theory , 1994 .

[26]  G. Mariéthoz,et al.  Reconstruction of Incomplete Data Sets or Images Using Direct Sampling , 2010 .

[27]  Maurice Clerc,et al.  The particle swarm - explosion, stability, and convergence in a multidimensional complex space , 2002, IEEE Trans. Evol. Comput..

[28]  Sebastien Strebelle,et al.  Conditional Simulation of Complex Geological Structures Using Multiple-Point Statistics , 2002 .

[29]  T. Mukerji,et al.  How to design a powerful family of particle swarm optimizers for inverse modelling , 2012 .

[30]  Franco Scalzone,et al.  Thinking Animals and Thinking Machines: What Relations? (with Particular Reference to the Psychoanalytical Point of View) , 2008 .

[31]  Aran Lunzer,et al.  Interdisciplinary Advances in Adaptive and Intelligent Assistant Systems: Concepts, Techniques, Applications, and Use , 2010 .

[32]  James Kennedy,et al.  Particle swarm optimization , 2002, Proceedings of ICNN'95 - International Conference on Neural Networks.