Probabilistic Meshless Methods for Bayesian Inverse Problems


Recent work establishes probabilistic foundations for models of the numerical error arising in application of the finite element method for solution of partial differential equations (PDEs). We construct similar probabilistic models of error for  a class of ‘meshless’ methods known as collocation. This allows us to establish rigorously the convergence properties of such solvers, as well as to propagate such numerical error into the posterior distribution in Bayesian inverse problems. We present applications of this method to linear PDE inverse problems, as well as an extension to inverse problems in a class of nonlinear PDEs.