Scientific and engineering models, which are generally partial differential equations (PDEs), often contain uncertain model parameters, initial conditions, and boundary conditions. These are often inferred by fitting to experimental or field observables. Bayesian inverse problems allow these unknowns to be modeled as random variables or fields and estimate a probability density for them. This is usually done via sampling, e.g., via a Markov chain Monte Carlo sampler. The probability density captures the uncertainty in the estimated quantities due to shortcomings of the model and sparsity of the data.
Computationally expensive PDE simulators do not allow their direct employment in samplers, and we often take recourse to statistical emulators. Training data for the emulators can be difficult to generate. We either reduce the dimensionality beforehand, or take recourse to sparse-grid sampling. Priors are generally known only as bounds, but arbitrary parameter combinations sampled from the resulting multidimensional uniform distributions may not be physically realistic and the PDE simulator may not even run.
We invite talks in dimensionality reduction, the construction of computationally inexpensive proxies of scientific/engineering simulators, strategies to fashion a physically realistic prior and other practical methods required to solve inverse problems of engineering/scientific interest. Examples where such methods have been used to solve inverse problems are also welcome.
16:30
Integrating Multifidelity Models into Sequential Tempered Markov Chain Monte Carlo
Thomas Catanach | Sandia National Laboratories | United States
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Thomas Catanach | Sandia National Laboratories | United States
Vo Huy | Colorado State University | United States
Bayesian updating for model calibration and system identification is held back by the computational cost of Markov Chain Monte Carlo (MCMC). MCMC often requires thousands to millions of sequential model evaluations. Sequential Tempered MCMC (Catanach 2018) and similar Sequential Monte Carlo methods have parallelized MCMC for Bayesian updating. In essence, these methods transform a sample population from the prior to the posterior using a series of annealing levels that gradually introduce the likelihood. While faster, these methods still require many sequential evaluations for each parallel Markov chain making them impractical for expensive models.
To speed up inference, we utilize lower fidelity but faster models. The Multilevel Sequential2 Monte Carlo (Latz 2018) algorithm demonstrates that at early annealing levels, lower accuracy models can be used. However, as the annealing level increases, we should use more expensive but higher accuracy models. The main challenge is knowing when to increase model fidelity. We develop an approach to select the appropriate fidelity using information theory and Bayesian model selection to estimate whether increasing the annealing factor with the current model will increase or decrease the information gained about the posterior. We demonstrate this approach using computational expensive problems in systems biology and seismic monitoring.
17:00
- CANCELED - Bayesian Inference and Optimal Experimental Design for System Identification of Material Physics Phenomena
Bobbie Wu | University of Texas at Austin | United States
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Bobbie Wu | University of Texas at Austin | United States
Zhenlin Wang | University of Michigan | United States
Wanggang Shen | University of Michigan | United States
Xun Huan | University of Michigan | United States
Krishna Garikipati | University of Michigan | United States
There is a collection of nonlinear, parabolic PDEs that, over suitable parameter intervals and regimes of physics, can resolve the evolution of patterns or microstructures in materials during processing and operation. The system identification problem thus entails determining which PDE best describes the data at hand. This question is particularly compelling because knowledge of the governing PDE immediately delivers insights to the physics underlying the systems. We present a Bayesian approach for system identification, and focus on distinguishing among diffusion-reaction, Cahn-Hilliard and Allen-Cahn dynamics, which respectively govern instability- and nucleation/growth-driven phase transitions and the resulting spatio-temporal microstructural patterns. Specifically, we target inference on coefficients for various differential operators from a dictionary of candidates, and employ Markov chain Monte Carlo to find their posterior distributions from a sparsity-inducing prior. The Bayesian framework allows for a flexible choice of data observables that best match the procedures and measurements from a laboratory setting. Furthermore, we demonstrate optimal experimental design to identify initial material concentrations and specimen acquisition times that offer a maximum expected information gain (uncertainty reduction) on the PDE coefficients.
17:30
- CANCELED - Bayesian filtering and parameter estimation without particles -- a tensor-based approach
Tiangang Cui | Monash University | Australia
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Tiangang Cui | Monash University | Australia
Sergey Dolgov | University of Bath | United Kingdom
Numerous real-world applications involve the filtering problem: one aims to sequentially estimate the states of a
(stochastic) dynamical system from incomplete, indirect, and noisy observations over time to forecast and control the underlying system.
Examples can be found in econometrics, meteorology, robotics, bioinformatics, and beyond. In addition to the filtering problem, it is often of interest to estimate some parameters that govern the evolution of the system. Both the filtering and the parameter estimation can be naturally formalized under the Bayesian framework.
However, the Bayesian solution poses some significant challenges. For example, the most widely used particle filters can suffer from particle degeneracy and the more robust ensemble Kalman filters rely on the rather restrictive Gaussian assumptions. Exploiting the interplay between the low-rank tensor structure and Markov property of the filtering problem, we present a new approach for tackling the Bayesian filtering and parameter estimation altogether. Our approach aims at exact Bayesian solutions and does not suffer from particle degeneracy.
18:00
VoroSpokes Sampling with Applications to Bayesian Inference
Nick Winovich | Purdue University | United States
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Nick Winovich | Purdue University | United States
Ahmad Rushdi | Sandia National Laboratories | United States
Eric Phipps | Sandia National Laboratories | United States
Jaideep Ray | Sandia National Laboratories | United States
Guang Lin | Purdue university | United States
Mohamed Ebeida | Sandia National Laboratories | United States
Bayesian inference provides a powerful framework for inductive reasoning which has proven applicable to an extensive range of disciplines and industrial applications. In this statistical theory, prior knowledge regarding quantities of interest are combined with observations to produce a full posterior distribution over the true values under consideration. MCMC methods have provided a practical framework for solving a very wide range of problems, yet there still remain a number of key difficulties and challenges for practitioners (e.g. issues with local trapping, inefficient mixing, correlated samples, and difficult convergence diagnostics).
Our novel VoroSpokes framework for Bayesian inference resolves these issues by removing the underlying reliance on Markov Chains from the posterior sampling procedure. The posterior is instead adaptively approximated using an implicit domain partitioning in the form of Voronoi tessellations to construct a Voronoi Piecewise Surrogate (VPS) model. The surrogate model and domain partitioning are then used to prescribe a hierarchical sampling procedure designed to efficiently draw independent samples from the approximate posterior distribution. In this talk we will present the VoroSpokes framework, its theoretical properties and demonstrate its superior performance over MCMC in practice.