08:30
A generalized Bayesian multi-fidelity framework for high-dimensional inverse problems and uncertainty quantification
Jonas Nitzler | Technical University of Munich | Germany
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Authors:
Jonas Nitzler | Technical University of Munich | Germany
Jonas Biehler | Technical University of Munich | Germany
Wolfgang Wall | Technical University of Munich | Germany
Phaedon-Stelios Koutsourelakis | Technical University of Munich | Germany
State-of-the-art methods for uncertainty propagation and Bayesian model calibration face serious difficulties when dealing with expensive numerical models due to the small number of model runs that can be practicably performed.
Such problems are amplified in cases of very high stochastic dimension which are for instance introduced by random boundary conditions or correlated variations in material properties. To overcome these challenges we propose a novel, generalized Bayesian multi-fidelity framework that can exploit automatically-generated, lower-fidelity model variants of the original problem by relaxing of the numerical coupling constraints as well as coarsening of the spatial and temporal discretization.
Rather than explicitly accounting for the high-dimensional input space - as most current multi-fidelity schemes do - we motivate a joint space of latent features and model responses that allows us to find a simpler manifold encoding in the small data regime by leveraging the Nash-Kuiper embedding theorem. We show that a projection of the manifold back on the subspace of model outputs is equivalent to finding the complex conditional density of model responses.
Low-dimensional, informative features of the input are discovered by employing supervised dimensionality reduction techniques. We demonstrate our approach in challenging numerical examples such as direct Navier-Stokes flow simulations or monolithic fluid-structure interaction problems.
08:50
Information field theory: recovering fields & their uncertainty from data and knowledge
Torsten Enßlin | Max Planck Institute for Astrophysics | Germany
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Torsten Enßlin | Max Planck Institute for Astrophysics | Germany
A physical field varies continuously over space and time. Recovering fields from finite data is an ill-posed problem addressed by information field theory (IFT). IFT, the information theory for fields, uses Bayesian probabilities in conjunction with knowledge about statistical and physical field properties to recover fields from data and to quantify the remaining uncertainties. The mathematical formalism of IFT borrows from quantum field theory, while the numerical machinery to address real world problems has parallels to deep learning frameworks. A number of successful applications of IFT to astrophysics and other areas testify its fidelity and versatility.
09:30
Total variation Bayesian learning via synthesis
Victor Churchill | Dartmouth College | United States
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Victor Churchill | Dartmouth College | United States
Anne Gelb | Dartmouth College | United States
We present a sparse Bayesian learning algorithm for inverse problems in signal and image processing with a (high order) total variation sparsity prior that can provide both accurate estimation as well as uncertainty quantification. Sparse Bayesian learning often produces more accurate estimates than the typical maximum a posteriori Bayesian estimates for sparse signal recovery. In addition, it also provides a full posterior distribution which aids downstream processing and uncertainty quantification. However, sparse Bayesian learning is only available to problems with a direct sparsity prior or those formed via synthesis. We build upon a recent paper to demonstrate how both 1D and 2D problems with a (high order) total variation sparsity prior can be formulated via synthesis, and develop a method that combines this to form a synthesis-based total variation Bayesian learning algorithm. Numerical examples are provided to demonstrate how our new technique is effectively employed.
10:10
Bayesian inverse problems in scalar conservation laws
Duc-Lam Duong | University of Sussex | United Kingdom
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Duc-Lam Duong | University of Sussex | United Kingdom
Masoumeh Dashti | University of Sussex | United Kingdom
We consider some inverse problems for scalar conservation laws. Specifically, we look at recovery of the initial field given observation of the (function) solution. We use a Bayesian approach and show the well-posedness of the solution and convergence of appropriate approximations. The key point for the well-posedness and the consistency of the posterior is based on the properties of continuity or Lipschitz continuity of the forward operator. Under only measurability of the forward map, we prove that the posterior is well-defined and is stable under changes in data. For the consistency, we propose some different ways to approximate the posterior without assuming the strong continuity of the forward model. One of those is to study the Lagrangian representation of the solution for scalar conservation laws in which the data is given by tracking a specific particle moving along the flow.