Bayesian inverse problems of complex systems are usually intractable, mainly due to the expansive forward simulations of the system. In practice, both derivative-free methods (e.g., ensemble Kalman inversion) and fast adjoint methods serve as promising candidates of optimization scheme to approximately solve Bayesian inverse problems for complex systems. In this session, we include talks of solving inverse problems of complex systems by using either ensemble Kalman methods or fast adjoint method. With the forward simulations evaluated in the optimization scheme, it is possible to further build surrogate models for MCMC. In several talks of this session, physics-informed approaches are also discussed in the context of Bayesian inverse problems. On the other hand, high-fidelity data of complex systems are expensive to simulate or measure, making experimental design process critical in order to obtain the most information from the system under limited resources. This experimental design topic is also discussed in this session.
14:00
- CANCELED - Near surface site characterization using ensemble Kalman inversion
Elnaz Esmaeilzadeh Seylabi | University of Nevada | United States
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Authors:
Elnaz Esmaeilzadeh Seylabi | University of Nevada | United States
Andrew M. Stuart | California Institute of Technology | United States
Domniki Asimaki | California Institute of Technology | United States
We present an algorithm based on the ensemble Kalman inversion to estimate the near-surface shear wave velocity profile when heterogeneous data sets and a priori information in a form of equality and inequality constraints are available. Although non-invasive methods, such as surface wave testing, are efficient and cost-effective methods for inferring Vs profile, one should acknowledge that site characterization using inverse analyses can yield erroneous results associated with the inverse problem non-uniqueness. One viable solution to alleviate the inverse problem ill-posedness is to enrich the prior information and/or the data space with complementary data. In the case of non-invasive methods, the pertinent data is the dispersion curve of surface waves, typically resolved by means of active source methods at high frequencies and passive methods at low frequencies. Acceleration time-series recorded at downhole arrays, on the other hand, include body and surface waves, and encapsulate source, path and site effects. Due to the complementary characteristics of the body and surface waves- the former carrying information of travel time and the latter of near-surface dispersive characteristics- we show that formulating the inversion problem using both dispersion data and acceleration time-series can reduce the margins of uncertainty in the Vs profile estimation. We also show the proposed algorithm can systematically incorporate constraints and further enhance its well-posedness.
14:30
Fast methods for Bayesian inverse problems governed by random PDE forward models
Ilona Ambartsumyan | University of Texas at Austin | United States
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Ilona Ambartsumyan | University of Texas at Austin | United States
Omar Ghattas | University of Texas at Austin | United States
We consider Bayesian inverse problems governed by forward PDEs with random parameters. Specifically, given a prior probability of the inversion parameter m, a statistical model of observations d, and a forward model in the form of PDEs with a random parameter field k (representing model uncertainty), we wish to find the posterior probability of the inversion parameter. Besides the usual difficulties of Bayesian inversion, the random PDE forward model creates significant additional challenges. Finding the MAP point alone is a PDE-constrained optimization under uncertainty problem, and fully characterizing the posterior formally requires nested Monte Carlo sampling in random parameter and inversion parameter spaces. To address this challenge, we linearize the random-parameter-to-observable map, which leads to an explicit formula for the likelihood. New terms, involving the Frechet derivative J of d with respect to k, appear in the likelihood, in particular in the noise covariance operator, where they represent model uncertainty. The explicit computation of J is prohibitive, requiring as many PDE solves as the lesser of the data and random parameter dimensions. Instead, a low-rank approximation of J can be made efficiently via randomized SVD, typically requiring a small and k dimension-independent number of PDE solves. Finding the MAP point (with respect to m) then requires solving a deterministic many-PDE-constrained optimization problem.
15:00
Estimating model-form uncertainty for multi-scale systems
Jinlong Wu | California Institute of Technology | United States
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Jinlong Wu | California Institute of Technology | United States
Tapio Schneider | California Institute of Technology | United States
Andrew M. Stuart | California Institute of Technology | United States
Global climate models (GCMs) are widely used to simulate climate change. But their predictions have large uncertainties, arising primarily in subgrid-scale (SGS) parameterizations for globally unresolvable small-scale processes. Work to quantify parametric uncertainties in these parameterizations is underway, using methods from data analysis and machine learning. Quantifying structural or model-form uncertainties is more challenging. Here we propose a non-parametric approach to model structural errors that uses Gaussian processes and ensemble Kalman sampling and that respects physical constraints (e.g., energy conservation). We illustrate how this approach can be used to quantify model-form uncertainties in low-dimensional multi-scale systems and with an idealized GCM. The results demonstrate that this approach allows us to go beyond merely calibrating model parameters toward quantifying model uncertainty more broadly.
15:30
Bayesian optimal design to target simulation within an idealized climate model
Oliver Dunbar | California Institute of Technology | United States
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Authors:
Oliver Dunbar | California Institute of Technology | United States
Andrew M. Stuart | California Institute of Technology | United States
Tapio Schneider | California Institute of Technology | United States
Climate prediction relies upon closure models for subgrid-scale processes (e.g. turbulence, radiation, moist convection) that are unfeasible to resolve globally. These closures feature model parameters, and we are interested in quantifying the uncertainty of these parameters. In practice, we often observe only subsets of global data that are sensitive to the model parameters. We are interested in designing experiments (targeted high-resolution simulations) to more accurately and efficiently produce and incorporate this `local' informative data.
In this talk, we consider a closure for moist convection within an idealized aquaplanet general circulation model (GCM). We use a Calibrate -- Emulate -- Sample (CES) philosophy to feasibly perform uncertainty quantification on closure parameters, making use of Gaussian process emulation. We introduce Bayesian design to the CES framework to locate optimally informative data subsets. At these latitudes, we incorporate high-resolution simulation data to enrich posterior distributions of our closure parameters.