Inverse and big-data problems are widespread in computational sciences and engineering. Despite formidable advances in recent years on all frontiers, ranging from pure mathematics to computational sciences, significant challenges remain, especially when it comes to addressing data-driven problems. In inverse/learning problems, parameters are typically related to indirect measurements by a system of partial differential equations (PDEs) or a network, which could be highly nonlinear and nonconvex. Available indirect data are often noisy, and subject to natural variation, while the unknown parameters of interest are high dimensional, or possibly infinite-dimensional in principle. Bayesian inference provides a systematic framework that rigorously that allows us to quantify the uncertainty in the inverse/learning problems, and to assess model validity and adequacy. Since the amount of data we wish to process is only going to increase for the foreseeable future, there is a critical need for effective algorithms that integrate data with simulations and learning approaches that are computation- and data-scalable. This minisymposium aims to attract researchers at the forefront of inverse and learning problems, data science, and data-intensive problems to present their latest work on computation- and data-scalable algorithms in inverse problems and learning.
08:30
Uncertainty Quantification for Inverse Transport Problems
Andreas Mang | Department of Mathematics, University of Houston | United States
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Authors:
Jae Youn Kim | Department of Mathematics, University of Houston | United States
Andreas Mang | Department of Mathematics, University of Houston | United States
We discuss uncertainty quantification in the context of diffeomorphic image registration problems. Our formulation is a non-linear optimal control problem with initial value control; the control variable is the initial momentum (or the initial velocity) at t=0. The partial differential equation (PDE) constraints are the transport equations for the image intensities and the so-called Euler Poincare equations for diffeomorphisms (EPDiff) for the velocity. We exploit problem structure through derivative information of a local Gaussian approximation of the posterior probability density function (PDF). This allows us to efficiently manipulate the associated high-dimensional PDF that arises from the discretization of the corresponding Bayesian inverse problem. We study different numerical schemes to efficiently approximate the associated derivative information in terms of accuracy and computational requirements and make the proposed numerical strategy tractable. We report results for synthetic and real data.
09:00
Computational Bayesian Inversion for Nanocapacitor-Array Biosensors and Electrical-Impedance Tomography
Clemens Heitzinger | Vienna University of Technology | Austria
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Authors:
Clemens Heitzinger | Vienna University of Technology | Austria
Daniel Pasterk | Vienna University of Technology | Austria
Leila Taghizadeh | Vienna University of Technology | Austria
We discuss theory and numerical results for two data intensive applications of computational Bayesian PDE inversion, namely nanocapacitor-array biosensors and electrical-impedance tomography (EIT). In the first application, measurements from massively parallel nanosensor arrays are used to extract physical parameters based on 3D simulations of the sensors and an adaptive Markov-chain Monte-Carlo algorithm. In the case of EIT, we again employ actual measurements and a nonlinear Poisson equation that includes the effects of free ions in order to determine the relative sizes of tissues within body cross sections.
In order to solve the inverse problem, we first study the mathematical formulation of the Bayesian analysis in a measure-theoretic framework and in the infinite-dimensional setting to collect required assumptions. Then we state and prove the boundedness and Lipschitz
continuity of the solution of the physical model, i.e., the nonlinear Poisson equation, with respect to the parameters. This leads to the well-definedness and well-posedness of the Bayesian estimation method applied here for the basic model equations used in the applications.
Additionally, we show numerical results starting from validations where the true values are known. Very good agreement is achieved. Finally, a-posteriori distributions of the various physical and geometric parameters of interest calculated using real-world measurements are discussed for both problems.
09:30
Multilevel and Multiindex MCMC for High Performance Computing
Linus Seelinger | Institute for Scientific Computing, Heidelberg University | Germany
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Authors:
Linus Seelinger | Institute for Scientific Computing, Heidelberg University | Germany
Ole Klein | Institute for Scientific Computing, Heidelberg University | Germany
Robert Scheichl | Institute for Applied Mathematics, Heidelberg University | Germany
Assessing the uncertainty of model predictions is vital in many scientific and engineering applications. However, the well-established Markov Chain Monte Carlo (MCMC) method for solving Bayesian inverse problems arising in that context may require substantial amounts of model evaluations. In case of models that are themselves computationally challenging, for example finely resolved partial differential equations, the computational effort becomes
enormous.
One angle to mitigate this is by employing multilevel or multiindex MCMC methods (MLMCMC/MIMCMC) that exploit model hierarchies in order to estimate statistical properties more efficiently. In addition, both the model and the statistical algorithm need to be parallelized in order to exploit modern high performance computing architectures.
In this talk we give an algorithmic view of MLMCMC and MIMCMC and present a generic software framework based on the MIT Uncertainty Quantification library MUQ (http://muq.mit.edu/) providing fully parallelized versions of these algorithms, supporting arbitrary user-supplied models. We demonstrate the framework's usability and effectiveness on a number of applications including the high-performance PDE library DUNE (https://www.dune-project.org/).
10:00
Stochastic spectral embedding as a local surrogate model in MCMC based Bayesian model calibration
Paul-Remo Wagner | Chair of Risk, Safety and Uncertainty Quantification, ETH Zuerich | Switzerland
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Authors:
Paul-Remo Wagner | Chair of Risk, Safety and Uncertainty Quantification, ETH Zuerich | Switzerland
Stefano Marelli | Chair of Risk, Safety and Uncertainty Quantification, ETH Zuerich | Switzerland
Bruno Sudret | Chair of Risk, Safety and Uncertainty Quantification, ETH Zuerich | Switzerland
The calibration of the input parameters of a computational model is a classical problem in the applied sciences. One of the most powerful frameworks to conduct this so-called model calibration is Bayesian inference typically solved with Markov-chain Monte Carlo (MCMC) sampling methods. The main bottleneck in these algorithms is the high number of required model evaluations that frequently render them infeasible in practical applications.
To alleviate the computational burden associated with MCMC algorithms, one approach has been to replace the forward model with an initially constructed surrogate model. This surrogate model serves as a cheap to evaluate replacement for the original forward model and helps to reduce the overall computational cost of MCMC algorithms. Surrogate models that have been used in this context include Kriging, low-rank approximations and polynomial chaos expansions (PCEs).
Recently, a new surrogate modeling approach titled stochastic spectral embedding (SSE) has been proposed. It is based on PCE but replaces its global approach with an approximation based on a set of locally embedded PCEs. It can be used to produce surrogate models that are accurate in regions of interest while remaining coarse in others. In this talk, we will present this approach and showcase its potential in conjunction with MCMC algorithms to efficiently solve the Bayesian model calibration problem.