Graph-based Prior and Forward Models for Inverse Problems on Manifolds with Boundaries

by   John Harlim, et al.

This paper develops manifold learning techniques for the numerical solution of PDE-constrained Bayesian inverse problems on manifolds with boundaries. We introduce graphical Matérn-type Gaussian field priors that enable flexible modeling near the boundaries, representing boundary values by superposition of harmonic functions with appropriate Dirichlet boundary conditions. We also investigate the graph-based approximation of forward models from PDE parameters to observed quantities. In the construction of graph-based prior and forward models, we leverage the ghost point diffusion map algorithm to approximate second-order elliptic operators with classical boundary conditions. Numerical results validate our graph-based approach and demonstrate the need to design prior covariance models that account for boundary conditions.


page 9

page 18

page 21

page 22


Ghost Point Diffusion Maps for solving elliptic PDE's on Manifolds with Classical Boundary Conditions

In this paper, we extend the class of kernel methods, the so-called diff...

Unsupervised Deep Learning Algorithm for PDE-based Forward and Inverse Problems

We propose a neural network-based algorithm for solving forward and inve...

Enforcing Boundary Conditions on Physical Fields in Bayesian Inversion

Inverse problems in computational mechanics consist of inferring physica...

Kernel Methods for Bayesian Elliptic Inverse Problems on Manifolds

This paper investigates the formulation and implementation of Bayesian i...

On parameter identification problems for elliptic boundary value problems in divergence form, Part I: An abstract framework

Parameter identification problems for partial differential equations are...

MCMC for a hyperbolic Bayesian inverse problem in traffic flow modelling

As work on hyperbolic Bayesian inverse problems remains rare in the lite...

Efficient Graph-based Tensile Strength Simulations of Random Fiber Structures

In this paper, we propose a model-simulation framework for virtual tensi...